AN

ELEMENTARY,- PRACTICAL AND
THEORETICAL

TEEATISE ON NAVIGATION:

NEW AND EASY PLAN

fob rnrDive

DIFF. LAT., DEP., COURSE, AND DISTANCE BY PROJECTION.

BY MTOYMATJRY,

IiISVT., V. 8. HATY.

SECOND EDITION, REVISED AND CORRECTED.

-o-o©e-€H

PHILADELPHIA:
EDWARD C. BIDDLE, NO. 6 SOUTH PIPTH STREET.

1843.

NOTICES

OF FIRST EDITION OF

MAURY'S NAVIGATION

*"VN/N*\*V^^N/N/V%*S/S^**O^^W\/\*W>/>^\/>/«

« U. S. N. 8., New York, January 19, 1836.

"Dear Sir, — I have had much pleasure in the perusal of your "New
Theoretical and Practical Treatise on Navigation;" the plan and arrange-
ments of which are original; it contains little or nothing superfluous, and
every part of it appears to he as clear and intelligible as the nature of the
subject will admit. Such a work has long been wanted in our Naval Schools,
and on board our vessels of war. I intend to make use of it in the Naval
School on this station; and I recommend it to be used by all the professors
of Mathematics, and Nautical Science in the Navy of the United States.
Yours respectfully, EDW. C. WARD.

" Passed Midshipman M. F. Maury. Prof. Math. U. S. Navy."

U. S. Navy."

" V. 8. Navy Yard, Gosport, March 7, 1836.

" I have examined a Treatise on Navigation written by M. F. Maury of
the U. S. Navy; and have no hesitation in recommending it to the students
of that science. The explanations are clear, the rules are illustrated by
many examples, and the new arrangement of some of the tables exemplify
the calculations of the navigator. Mr. Maury is deserving of great credit
for the work, and I wish him every success*

P. J. RODRIGUEZ.

"Navy Department, April 9, 1830.

** Sir, — I have to request that you will add the " New Theoretical and
Practical Treatise on Navigation," by M. F. Maury, Passed Midshipman,
to the list of books furnished vessels of the navy going to sea.
I am respectfully yours.

(Signed,) M. DICKERSON."
" Com. John Rodgers,

President of the Board of Navy Commissioners."

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Entered according to Act of Congress, in the year 1896, by

KEY & BIDDLE,

In the Clerk's Office of the District Court of the Eastern District of Pennsylvania.

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PREFACE.

The object of the present volume, is to place in the hands of stu-
dents, and especially of the Midshipmen of the United States Navy,
a Text Book, in which the theory as well as the practice of Navi-
gation, is explained and taught.

It is not pretended that new theories are set forth, or that new
principles are established in this work ; but it is believed that those
which have already been established, are here embodied in such
a form, that the means of becoming a theoretical as well as a prac-
tical navigator, are placed within the reach of every student.

For this purpose the works of Bonnycastte, Colburn, Hutton,
Legendre, Davies, Bowditch, Lardner, Hassler, Kelly, Keith, and
La Place, have been consulted. m

Care has been taken to introduce only those theorems upon

which the problems in Navigation immediately depend, and which

y ' it is necessary to understand, in order, satisfactorily to comprehend

„ the principles of Mathematics and Astronomy, involved in the

: solution of these problems.

A The fear of introducing more than is essentially requisite for this
. purpose, may have led to an error on the other extreme, by causing
V) something to be omitted, which should have been inserted ; but if
or> such a fault be detected in the work, while it will be readily ad-
\ mitted on one hand to be a fault, it can scarcely be unjust on the
\ * other, to say that the error is on the safe side ; especially when
1 they who judge, are reminded, that there is not throughout the
-vj. whole work, a single principle laid down which does not serve at
once as a rule, or as the basis of a rule, or as reference in some suc-
ceeding demonstration, position, or explanation either to prove,
establish, or elucidate ; and moreover, that there are many, (perhaps
the greater number,) of those for whose benefit the work is chiefly
designed, who, during the whole period of their service at sea, never
have the advantage of instruction from a teacher of Navigation;
consequently they have to depend upon their own exertions,* and
the books before them, for their proficiency as Navigators. How
necessary is it then, that the work on Navigation for them, should
be an elementary one, adapted to the capacity of all, and that it
should not embrace the widest range ; more particularly so, as there
is not yet any regular system of education provided for the Navy.
The idea of such* work as the present, grew out of the author's
own experience, and was suggested to him by his own wants while

384603

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vi PREFACE.

a student of Navigation ; if it be not sufficient for the supply of
similar wants on the part of others, it is hoped that it will, at least,
serve to provoke some more capable pen to undertake and complete
what is here attempted.

A more elementary work than any hitherto published on Naviga-
tion is much required as a school book in the United States. The
attention of teachers of Navigation throughout the country is respect-
fully invited to it.

These pages were written chiefly on board of a man-of-war, in
the midst of the various calls of duty, and the thousand interruptions
incident to such a place ; the author trusts that this circumstance
will ensure him on the part of his brother officers, and those into
whose hands his work may fall, the indulgence usually claimed for
inexperienced authors.

-s

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CONTENTS.

ALGEBRA.

Definitions .... 3

Addition .... 4

Subtraction. .... 5

Multiplication ... 6

Division - ... 7

Equations .... 8

Proportions - - - - 10

GEOMETRY. Part I.

Definitions (of rectilineals) - 15

Axioms ----- 18

Propositions - - - 19

GEOMETRY. Pabt II.

Definitions (of the circle and its

parts) - ... 33

Propositions 35

Proportions and ratios 38

Axioms and propositions 39

LOGARITHMS.

Nature and use of - - - 49

Multiplication and division - 57
Involution and evolution - 67 & 58

Log. sines, tangs., etc. 68

PLANE TRIGONOMETRY. .

Cases and problems ... 65

Examples for practice 74

SPHERICS.

Definitions

81

Right Aroled Trigonometry 85
The six cases and Napier's rules 85

The five circular parts
Solutions of the cases
Equivalents for sine, cos., etc

Obliq.ue Trigonometry
The six cases - - -
Solutions of the cases

PAOE

87
88
95

99

99

100

NAUTICAL ASTRONOMY.

121

Figure and Motions of the earth

Day, astronomical, sea, civil, and
sidereal -

Equation of time

Ecliptic and signs of the zodiac

Equinoctial and solsticial points

Year, solar and sidereal -

Primary planets and nodes

Equator and the poles

Tropics and zones -

Latitude, meridians, and longitude 127

Declination, right ascension, and
horary angles

Colures, cardinal points and hori-
zon

Zenith, azimuth circles, and prime
vertical - - - .

Altitude, refraction, and parallax

122
122
123
123
124
124
125
126

- 128

129

130
131

Variation of the Compass - 132

Stereographic projection - - 135

Azimuths and amplitudes - 137

Of the 8ur's Risiko ard Set-
tiro .... 140
Time and degrees ... 140

Of the Plarets, Moor, etc. 142

Jupiter and its satellites - - 143

The moon's motions and phases 143

The motion of light - - 143

Radius vector - * - - - 144

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Vlll

CONTENTS.

PAGE

Or firdirg the Tiirt or Day 145
Of Long, bt Chronometers 147
The rate and error of chronometer 146

Latitude bt Meridian Alti-
tudes - - - 151

Latitude bt Double Altitueb 153
of the sun - - 154
of two stars - - 158
of the son and a star 161

Luvabs - - - - 165

Latitude, time, ete, by lnnan - 170'

Latitude bt the North 8tab 178
Tables for finding tame - • 180

PAGE

Tide* 182

Table for time »f high water - 185

NAVIGATION.

Of course, distance, etc. - - 108

Explanation of Plate I - - 191

Loxodromic sailing - - - 105

Op turning Def. irto Diff.

Lorg. - - - - 100

Mercator's Sailing - - 210

Surystieo - - - - 218

Baseline .... 218

Triangulating ... 214

Reducing sounding! to low water 215

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ALGEBRA.

B

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ALGEBRA.

§ I. Algebra is a method of computation, in which magnitudes, or
quantities, are represented by means of the letters of the alphabet.
These letters have no positive or fixed value ; they only stand for
the quantities to be computed.

§ II. In algebra, known quantities are expressed either according
to their numerical value, or by the first letters a, 6, c, etc., of the
alphabet.

§ III. And quantities of unknown value are usually represented
by the last letters, x, y, z, etc.

§ IV. In algebraical computations, certain characters, called
signs, have been introduced, and are used in the place of written
words: thus,

+ {plus) is the sign of addition.

— (minus) is the sign for subtraction.

X or . is the sign for multiplication.

as is the sign of equality.

-f- is the sign for division.

a + b — c x d = x -*- y, or -, is read a plus b minus c mul-
tiplied by d equals x divided by v.

§ V. a > b signifies that the former quantity (a) is greater than
the latter (b).

a <b signifies that the latter quantity (b) is greater than the
former (a).

§ VI. (:) is to (: :) as, represents equality of ratio, and denotes
proportion : thus a: b::c: J, (read a is to b as c is to d) 9 signifies that
the ratio of a to b is equal to the ratio of c to J, and that these four
quantities are proportional.

$ VII. */> represents the difference between any two unknown
quantities between which it is placed ; thus, x s> y denotes the dif-
ference between x and y.

$ VIII. oo signifies that the quantity standing before it, thus,
a; co, is infinite in value.

$ IX. The numbers 2, 3, 4, etc., placed after a letter, thus, a s ,
a 3 , a 4 , denotes the 2d, 3d, 4th, etc., power of the quantity which
that letter represents. The second power is the square, the 3d the
cube, the 4th the biquadrate: a x a = a % ,

a 8 x a as a 3 ,
a 3 x a =s a*.

§ X. The numbers 2, 3, 4, etc., placed as above, are called the
indices of the quantities to which they are affixed.

$ XI. The quantity which is a constant multiplier in the involu

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4 INTRODUCTION.

tion of a power, is called the root of that power : thus, ($ IX.,) a is
the cube root of a', and the square root of a 9 .

$ XII. </ i* called the radical sign ; and </x, or */#, also a?i,
denotes that the square root of x is the quantity alluded to; so <f y,
or y\ y denotes the cube root of y : and so on with the other num-
bers, or with letters thus placed.

The number or letter placed over the radical sign, thus, (/,![/,
or placed thus, x$ 9 etc., is called the exponent

$ XIII. A surd is a quantity to which the radical sign (vO i*
prefixed, and whose root cannot be expressed by numbers ; thus,
%/3, ^5, are surds.

$ XIV. A term is any quantity that is separated from another by
a sign; thus, a, b and x 9 are terms in the compound quantity
a + b — x.

$ XV. The number, (5,) or the letter, (c), that is prefixed to any
quantity, (5 a, or c a,) is called the coefficient of that quantity.

$ XVI. Like quantities consist of the same letters as a b + 4
a b — 2 a b.

$ XVII. Unlike quantities consist of different letters, as x + 2
y xa + b.

$ XVIII. When a quantity has no sign prefixed to it, -f , or
plus, is under stood.

$ XIX. y/ x* + y, or (x* + y) I, denotes that these two quan- .
tities are as one, and that the square root of their sum is alluded to
by the radical sign, or the exponent $. Thus, the square root of
64 + 36, or v/ 64 + 36, is 8 + 36 =- 42, but the square root of
64 + 86, or (64 + 36) J, under a vinculum, is 10, for 64 + 36
« 100, and the square root of 100 is 10.

$ XX. Addition is performed by collecting several quantities in
a more simple form :3a + 10 a + a = 14 a, (read fourteen
times a). 1 is understood to be the coefficient of every quantity that
has no coefficient prefixed to it.

$ XXI. When the quantities to be cast up have unlike signs, the
problem is solved, partly by addition and partly by subtraction. If
the negative be greater than the positive quantities, the negative
sign must be retained in the result. The sum of 6 — 10 + 9, is 6 ;
for 6 and are here positive quantities, and make 16; and 15 —
10=- 5: and the sum of 6 — 10— 9, is — 13; for— 10 — 0» — 10;
and — 10 -f 6, or 6 — 19 «■ — 13. Every quantity, which has
not a sign prefixed to it, is understood to be positive.

To add together, 4* — 9 x + x + 3 x — 6 a?. Collecting all
the positive quantities into one sum, and all the negative quantities
into another, then taking the less from the greater, the remainder,
(prefixing the sign of the greater,) is the answer. Thus :

4*— 9 x
ar— 5 x

3«

8 x — 14 x ss» — 6ar

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ALGEBRA. 5

_6 ay 2 10 b a — a c

4 ay 9 2b a lac

9 ay 2 —19 ba ^9oc

Ans. 7 ay 2 — 7 6a — 3 a c

$ itXIl. If all the terms be not like quantities, the positive and
the negative of those that are like must each be added up separately ;
then these two sums must be subtracted, the one from die other,
and the remainder thus obtained, connected by its proper sign to the
unlike quantities, shows the answer. To add together :

4x + 6a— c 2 + 9 y

— 10 a? — 5 a + c 2 — 5 y

x — a + 4 c 2 + 3 y

2 x + 2a

~3 a? + 2a + 4c 2 +7y

The positive x'a are, 4 x + x -f* 2 x = 7 x; the negative are,
— 10 x. The difference is, (7 a: — 10a:=), — 3 x. The positive a's
are, 6 a + 2 a = 8 a; the negative are, — 5a — a = — 6 a. The
difference is, (8 a — 6 a =), 2 a. The positive c*'s are, c* + 4 c*
= 5 c 8 ; the negative is, — c* . The difference is, (5 c 2 — c 2 =), 4c 2 .
The positive y's are, 9 y + 3 y = 12 y; and the negative are, — 5 y.
The difference is, (12 y — 5 y =), 7 y.

To add together:

v/«6 — a 2 +2ay — 6a?+10a 2 — 9xy — a* b + y — a

— v' 6a6 + 3a» + ^/5a6 — ar+a:y + a 2 6 — 4a?y — 2x

a — c 2 + 4y — y/2ab — 2 a 2 — 3xy + 4x+14xy + a 2 b

</ ab — a* + 2xy + 6x —a 2 b + a — c 2 + 4 y
— v/ 6a6 + 10a 2 — 9a?y — a? — 2 a* b + a + y

</5ab+, 3a*+xy — 2 x + a % b
— y/ 24nb — 2 a 2 — 4xy + 4x

— 3 x y

14 x y

— ^/4a6 + 20a 2 +2a?y + 2g— a 2 6 + 2a— 2c 2 +"T0 y

To add together : To add together :

y + ab + dx % — z* a a + b+ c — x

y + c b + b x* — z* x 2a — 6 — 2 c — 2 x

2y + (a+c)b + (d + b)x*—z*(a + x) -a + a b — c — 4 x
= 2a + ab—2c — 7x

$ XXIII. Subtraction is the reverse of addition. The method of
performing it consists in changing, or reversing, all the signs of the
subtrahend, (i. e., making the positive negative, and the reverse,)
and then proceeding as in addition ; viz., by adding together like
quantities that have like signs ; and subtracting from each other like
quantities that have unlike signs. The quantity or the quantities
that result from this operation is the remainder.

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INTRODUCTION.

To subtract a — 6 from a + 6.

a + b
Changing the signs of the subtrahend a — b

The remainder =s 2 6

This will be readily understood by ascribing a numerical value to
a and b. Let a = 6 and b = 4 ; now subtract 6 — 4 from 6 + 4 ;
6 — 4 ■ 2, and 6 + 4 = 10, and 10 — 2 = 8 (or twice 4 = 2 6.)

To subtract a + b from a — 6.

a — b
a + b

— 26

To subtract : To subtract :

2xy + 6 2 c — 3 a 10 a: — y — ac

xy — b* c + a — Sx — 2y — 4a c

xy + 2b % c — 4 a 13 x + y + 3 ac

§ XXiV. In the multiplication of letters, the product of any two
factors is obtained and expressed by prefixing, as coefficients, the
several letters contained in the multiplier, to those of the multipli-
cand. Thus :

x + v
by a + b

ax + ay + bx + by.

$ XXV. Multiplication is denoted thus, a x 6/ or thus, a • b;
or thus, a b.

f 1. By recollecting that a times 6, and b times a, are expressions
for the same product, just as 4 times 9, and 9 times 4, express the
same number, it will at once be understood how there is no differ-
ence in the value of the quantity, whether it be expressed a 6, or
b a. But when several letters are contained in the same term, as
a x c x b 9 it is generally expressed by the letters arranged in al-
phabetical order ; thus, a b c.

$ XXVI. To multiply x + y by a — 6. Arts, ax + ay — 6a:
— by.

When terms of unlike signs are multiplied together, their product
is negative, and (— ) must be prefixed to it.

$ XXVII. The product of x + y by x + y is x* + 2 xy + y»,
orxx + xy + xy + yy. The former, being the snorter method
of writing it, is the better. The small figures that are affixed show-
how many times the letter with which they are connected enters as
a factor in the term.

J XXVIII. The product of x+ yby — x—y 9 is— x % — 2xy

— y % .

% XXIX. The product of x + y by x — y, is x % — y* ; for x y
occurring twice in the product, and with unlike signs, (+ and — ),
cancels itself.

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ALGEBRA.

To multiply x* — y 8 + 3 xy
by x + y

a? 3 — xy* + 3x* y

x* + 2a?y 8 + 4a? 8 y — y 3

§ XXX. Unless the quantity be under a vinculum, the index ap-
plies only to that letter with which it is in juxta-position. The
term x y % denotes the product of x and y * ; and x* y denotes the
product of the two factors x* and y. The term (x y) 2 , or x y 2 , de-
notes the square of the product of the two factors x and y. Let
x = 4 and y = 3 : x* = 16 and y* = 9. The value then of the
first term (x y 8 ) is 4 x 9 ( =36 ;) of the second, (x 8 y), it is 16 x 3
(= 48 ;) and of the third, (a? y 8 ), it is the squareof the product, (12),
of 3 and 4, (=144).

To multiply a x + a b — a*
by a + x + b

a* x + a* b — a 3

ax* + abx — a 2 x

ab x + ab % — a* b

ax 2 — a 3 + 2abx + ab*

To multiply — 5 a — 4 x
by 2 — a — x

— 10 a — 8 x

5 a 8 + 4 a a?

5 a x + 4 a: 2

— 10 a + 5 a* — 8a? + 9aa: + 4a? 8

$ XXXI. When the same letter enters into both factors, the pro-
duct is obtained by adding the indices or exponents of the letter, as
a» x a* = a 5 ; for 3 + 2 is 5. And x x x is x* ; I is the index
and coefficient of every letter which has no other index or coef-
ficient expressed.

$ XXXII. Division is the converse of multiplication.

4 times b divided by 4 gives b. So a times b divided by a gives b.

4ax -i-2x = 2a
4ax -±-2a = 2x

$ XXXIII. Division being the converse of multiplication, the
product of the divisor by the quotient gives the dividend. Thus,
2xx2a = 4(ix.

$ XXXIV. When the dividend and divisor are powers, or are
roots, of the same quantity, the index, or exponent, of the divisor,
minus that of the dividend, is the quotient. Thus, a: 3 + x* ■+■ x*
■b x + x s .

$ XXXV. When all the terms of the dividend have letters that
are common to the divisor, the operation of dividing is performed

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8 INTRODUCTION.

by striking out from the dividend the letters of the divisor, and
dividing the coefficients. Thus,

Sxy -*- 2y a= 4a?
\2axz-7-3ax = 4z
Qxy*z + 2xyss:3yz.
$ XXXVI. Division is sometimes expressed without being per-
formed, as x + y -T- a, or x "*"^«

a
§ XXXVII. This last quantity is a fraction. Fractions in algebra
are multiplied, divided, etc., after the same manner by which such
operations are performed' in common arithmetic.

§ XXXVIII. Two or more quantities, with the sign (=) of
equality between them, constitute an equation; as a: + 10 a 14
+ 6.

$ XXXIX. All the quantities or terms that are on either side of
the sign (=) constitute a member of the equation ; x+ 10 is Hie first,
and 14 + 6 is the second, member of the equation x + 10 = 14 +
6. The two members, necessarily, are always equal to each
other.

$ XL. In the process of solving, or reducing, an equation, the
known quantities, or terms, of the equation, are all collected and
arranged with their proper signs, in one member of the equation,
and the unknown quantities in the other member, as £ = 14+6— 10
The value of x in this equation, is therefore equal to 10.

$ XLI. In transposing a term from one member to another, the
equality of the two members of the equation is preserved,by changing
the sign of the term transposed. By transposition, positive terms
become negative, and the reverse ; also the sign (x) of multiplica-
tion becomes (-*-) die sign of division, and vice versH. Thus, in
the equation above x + 10 = 20, (or 6 + 14:) by transposition,
x s- 20 — 10. Also, in the equation 12x4 = 8x6, by trans-
posing we have V = | ; and the equation a •+■ b sb x -*- y , by trans-
position, becomes a x y == x x b.

To find the value of £ in the equation a + x— c m* b + 14 ; by
transposing a and — c, and changing their signs, the value of x is
obtained: it stands thus, x ■■ b + 14 — a + c.

Ex. a— x % + y «= 00 + a — 40. To find v ; a being in each
member, and having the same sign, cancels itself, and may therefore
be stricken from the equation ; for if transposed, the expression
would be a — a, which two balance each other. The value of
y is obtained then, by transposing — x* 9 when the equation stands
thus, y a 90 — 40 + x* ; in its most simple form, thus, y a 50
+ X s .

Ex. 8 x — 10 as 80 + 2 x. To find x ; transposing and placing
the known and unknown quantities on opposite sides, the equation
stands, S x — 2 # = 80 -f 10. Subtracting and adding, it becomes
6 x ■* 90 ; and Xal5, (by division).

Ex. x -*- 2 + 10 = 14 + 8 . To find x; ix or | =* 14 + 8

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ALGEBRA. 9

— 10. The value of the second member of the equation is 12 ;

X

therefore ^=\2; and by multiplication, x = 24.

From the two last examples, this general conclusion may be
drawn, viz. :

$ XL1I. When a multiplier of either member is transposed, it
becomes divisor to the other member ; and vice versa.

Ex. 8x* + ax — 10a: =14* + 4s* + 16*. To find a:.
x is common in every term of each member of the equation. Then,
dividing by x, the equation becomes Bx +a — 10 = 14 + 4 x +
10; transposing and placing all the a: terms alone in one mem-
ber, 8 x — 4 x = 14 + 16 + 10 — a; subtracting and adding,

4 x s= 40 — a; dividing, x =s 10 — -..

Ex. 3 b x — a b =s 3 y. To find x. Transposing, 3 b x =

3 y + 9 a b; dividing, x » y j g * = | + 3a. Therefore,

x = I + 3 a.

x . ae

Ex. ^ + - ss 10. To find x. Clearing the equation of frac*

3 x
tions; lst,ar + — ss 30; 2d, 2 a? + 3 a? = 60; adding, 5ia=

60; dividing, x = 12.

$ XLIII. Thus an equation is cleared of fractions by multiplying
every term (except the fraction itself) by the denominator.

Ex. 1. The commander of a man-of-war is desirous of having
his ship calked, that he may proceed on his voyage. His own
calkers can finish the job of calking in 10 days. But he employs
a gang from the shore, that could finish the whole work in 6 days,
to assist his. How many days' job are there for both gangs to-
gether?

Let x denote the job of work. Then j^ is one day's work for the

ship's calkers; g- is one day's work for the shore gang; and jg + ^

« 1 day's work for both gangs together. Clearing this equation
of fractions, 6 x + 10 x «* 60; and 16 x = 60, or x *= 3||,
the number of days.

Ex. 2. A vessel, after an engagement, could muster only 238
able bodied men ; on examining her list of sick and wounded', she
found her loss in killed to be -J of her whole crew, and in wounded
$ of the whole crew. What crew had she when she went into
action ?

Let x denote her crew when the action commenced. Thus,

5 + 3 +238 = x. Clearing the equation of fractions, 3 x + 5

x + 3570 >s 15 a: ; transposing, 3570 = 7 a:, or 510 =■ x 9 the
whole crew when the action commenced.

C

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to INTRODUCTION.

$ XLIVr When the value of more than one unknown quantity is
required, the problem, if definite, comprises conditions for as many
equations as there are quantities required. In such cases the un-
known quantities (#, y, z, etc.) have the same value in all the
equations ; i. e. x in one equation is equal to a? in another of the
same set When the value of x y y , or z, is found, its value is sub-
stituted in its stead.

Ex# ^JL 3 ^ JJ^Tofindarandy; x - 44 — y, and x -

36 rf 3 y; thus 44 — y = 36 + 3 y ; transposing, 4 y = 8, or
y = 2 ; and substituting, x = 44 — y , or x = 42.

Ex. x + y + * = 60 ~)

* + 4 y + 3 z = 144 vTo find a?, y, and * ; x =* ©0 —
2 a? + y + 8*»132j

y — *; a? «= 144 — 4 y — 3 z; and a? s* 66 — | — 4 *. Elimi-
nating x; 60 — y — * = 144 — 4 y — 3 * ; and 144 — 4 y — 3
z a 66 — • | — • 4 z. Transposing, to find the value of y ; 3 y =
144 — 60 — 2 z, or y = 28 — |s again, 7 y » 288 — 132 —

o i a 156— 2 *' ,. . ,. 0£ > 2* 156 — %x

8 * + 6 s, or y «= ? ; ehmmating y,28 — — =- ? — *

clearing the equation, 14 z — - 588 a 468 — 6 * ; transposing, 20
* a 120 or z =s 6.

£ j

For *, in the equation y = 28 — y, substituting its value (6),

y a 28 — y , orjj* 24. And for z and y, in the equation a; »
60 — y-—z, substituting their values (24 and 6), a? =» 60 — 24 —
6, or a; s 30.

$ XLV. There is another manner of expressing certain equa-
tions which do not involve more than 4 terms, and when thus ex-
pressed the terms are said to be proportional. Thus the propor-
tion 3 : 4 : : 6 : 8, is but another method of expressing the equation
3 -4- 4 = 6 -$- 8, or £ = |. The dots (:) being an abbreviation of
the sign (-«-) of division; and the dots (: :) being another form for
expressing the sign (=) of equality, to show that the ratio be-
tween the quantities on each side of it is the same.

$ XL VI. Whence it may be inferred as a general rule, that, if
the quotient of two quantities be equal to the quotient of two
other*, these four quantities are proportional. And,

f 1. That the ratio between either divisor and its dividend, is
equal to the ratio between the other divisor and its dividend.

$ XLVII. By this rule we have 4 : 3 : : 8 : 6, for # = f. Thi*
form of expression for the proportion (3 : 4 : : 6 : 8) first quoted,
is called " invertendo" (from inverting the divisor and dividends,)
thus 3 + 4 a 8 + 8, and inversely 4 -*- 3 = 8 -*- 6.

$ XL VIII. By the same rule we also have 3 : 6 : : 4 : 8. This
form of expression for the proportion (3 : 4 : : 6 : 8) is called

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ALGEBRA. 11

" alternando" for by taking the terms alternately we have 3-4-6
= 4-1-8.

$ XLIX. Whence also another general rule in proportions : that,
if four quantities be proportional, they are also proportional when
taken inversely, or when taken alternately.

§ L. 3 -s- 4 =s 6 -*- 8, by transposition 3x8 = 6x4; where-
fore, also, if the product of two factors be equal to the product of
two other factors, those four factors are proportional, as 3 : 6 : ;
4:8.

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GEOMETRY.

PART I.

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GEOMETRY.

DEfiNrnoNs.

$ 1. A point is an atom of space.*

$ 2. .4? Itne ( ) is length without breadth.

§ a. The track of a moving point would be a line.
$ b. The extremities of every line are points.
§ e. A line may be of any length.
§ d. Lines are either straight or curved.

§ 3. A straight or right line ( ■ — ) is the shortest line thai

can be drawn from one point to another.

$ a. If two lines meet each other they form an angle. 4

§ 4. A plane is an extent, (as the surface of this page), on which
the straight line will lie that joins any two points which are in
that extent.

$ a. A plane is without thickness, and it may be of any length
and breadth.

§ b. The limits of every plane are lines.

§ c. If two planes cross each other, the line of their intersection
is a straight line.

§ 5. Two lines diverging from, or meeting in, the same pointy

form an angle*

$ a. The point in which the lines join or cross each other, i*
called the angular point.
§ b. Angles are either right, acute, or obtuse.

$8. When one straight line stands
Upon another, without inclining to either
side, the angle, or angles, which these
two lines form, are called right angles.
§ a. All right angles are equal, and
every one contains '90° (degrees). (Vide
MM<*0

• According to the atomic theory of matter, all bodies are composed of mdi-
tinble and impenetrable particles, called atoms; an atom then is to matter what
a point is to space ; hence the idea of an atom of space*

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16 DEFINITIONS.

§ b. Straight lines that form right angles
with each other, are said to be perpendicular
the one to the other; and those that are per-
pendicular are at right angles with each
other.

§ 7. Every angle which is less than a right
angle, is an acute angle.

§ a. Angles are generally particularized by
means of letters ; as the angle B. But when
there are more angles than one at the same
angular point, the angle to be particularized
is made known by placing the letter at the
angular point between the letters which
stand for the lines that form the angle. As
the angle ABC,orCBD.
$ 8. Every angle which is greater than a right angle
is an obtuse angle; as the angle 0.
§ a. An oblique angle may be either acute or obtuse.
§ b. Two angles are equal, when they contain the
same number of degrees (°), minutes ('), and seconds
(")> or when the lines which form an angle have the
same divergence from each other, which the lines have
that form the other angle.
$ c. The difference between an oblique angle and a right angle
is die complement of the oblique angle.

§ 9. Parallel lines are lines that have
always the same distance between them.
They lie in the* same direction, and if
lengthened, ad infinitum, would neither

approach, or recede from, each other,

§ a. The distance of two parallel lines from each other is mea-
sured by any straight line (p) that may be drawn between them,
perpendicularly from one to the other*

§ 10. A figure is any extent bounded by one or more lines, or
surfaces.
§ a. The space included by a figure is called its area.
§ 11. A superficies is the surface of a figure.
§ a. A superficies and a plane coincide, when a straight line,
that joins any two points in the superficies, lies on that surface.
The superficies of a figure is limited to the extent of the surface of
that figure ; but its plane is infinite.

$ b. When the superficies and the plane of a figure coincide, the
former is called a plane superficies.

§ 12. A plane triangle is a figure that is
formed by the intersection of three straight lines,
any two of which intersect the other in different
points.

§ a. The intercepted parts of these lines are
called the sides of the triangle.

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GEOMETRY.

17

$6. Every triangle has six parts; viz.,
three sides, and three angles.

§ c. Triangles, with regard to their sides,
are either equilateral, isosceles, or scaline;
with regard to their angles, they are either
acute, right, or obtuse angled.

§ 13. An equilateral triangle has its sides
all equal to each other ; viz., b equal to a or e.
§ 14. An isosceles triangle has
two equal sides ; as, c and a.

§ a. An isosceles triangle may be
either acute, right, or obtuse an-
gled. Its third side (6) is its
base.

§ 15. A scaline triangle has none
of its sides equal.

§ a. A scaline triangle may also be right or
oblique angled.

§ 16. An acute angled triangle has each of its
angles less than a right angle.

§ a. The vertex of a triangle is the angle that is
opposite to the base of the triangle.

§ b. Any angle may be called the vertical angle,
and consequently any side may be made the base.
§ 1 7. A right angled triangle has an an-
gle (B), that is, a right angle.

§ a. The side (b) which subtends the
right angle, is called the hypothenuse;
the two other sides (c and a) are called
legs.

§ 18. An obtuse angled triangle has an
angle (G) that is obtuse.

§ a. The altitude of a trian-
gle is the perpendicular dis-
tance (p) of the vertical angle
(C) (§ 16, 5 b.) froth the base

ic). The base must be pro-
uced to meet the perpendicu-
lar, if the perpendicular fall without the triangle.

§ b. As any side (§ 16, $ b.) may be made the base of a triangle,
the perpendicular distance of ai y angle from its opposite side may
be called the altitude, or height cf the triangle.

§ 19. A parallelogram is a
right lined (§ 3.) quadrilateral
figure, the opposite sides of which
are equal and parallel.

§ a. The altitude of a parallel-
ogram is the distance (p) ($ 9. $ a.) between either pair of its oppo-
site sides. To show the altitude of a parallelogram, either of two

D

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18

AXIOMS.

opposite sides may be produced until it meets at right angles, a
perpendicular from the other side.

§ b. The measure or area of a parallelogram, is
the product of its length and breadth, or of its base
and altitude.

§ c. Any side of a parallelogram may be made
its base.

§ 20. A square is a parallelogram of which all
the sides and angles are respectively equal.

§ a. Every angle of a square is a right
angle.
\e£ / § 21. A rhombus is a parallelogram

that has all of its sides equal to each
other; but its angles are not right an-
gles.

§ 22. A trapezoid is also a four-sided
figure, but only two sides of it are pa-
rallel, though they are not equal.

§ a. A diagonal is a straight line
(d) that joins two opposite angles in

L

a four-sided figure.

§ b. The space included by a parallelogram is called (§ 10. § a.)
its area, or measure, and is expressed (§ 19. § b.) by the product of
its base and altitude.

$ 23. Two figures are equal when every part in one is equal to
the part in the other, which corresponds to it; and two equal figures
are always of the same magnitude.

$ a. And two figures are of the same magnitude when their areas
are equal : a triangle and a parallelogram may be of the same mag-
nitude, but they cannot be equal. A triangle only can be equal to
a triangle ; a parallelogram to a parallelogram, etc.

§ b. Two figures are similar when every angle in one is equal
to the angle which corresponds to it in another figure. Figures
only of the same class are similar, viz. : triangles are similar to tri-
angles, parallelograms to parallellograms, etc.

§ c. Homologous
sides, or angles, are
the sides, or angles,
which, in two equal or
similar triangles, cor-
respond by their rela-
tive positions to each
other; thus c and h
are homologous sides, also b and e, and a and d.

§ d. Homologous angles are equal to each other. B and G are
homologous angles ; so also are A and D, and C and H.

AXIOMS.

§ 24. Axioms are self-evident truths, such as :

31

/v

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GEOMETRY. 19

§ a. Things that are equal to the same, or to equal things, are
themselves equal.

$ b. If equals he added to, or subtracted from, or substituted for,
multiplied or divided by, the same or equal quantities, the sums or
remainders, quotients or products, will be equal.

§ c. A part is less than the whole.

§ d. All of the parts are equal to the whole, and the whole to all
of its parts.

PROPOSITIONS.

PROPOSITION I.

$ 25. Two straight lines which cross each other, make the two
angles that are on the same side of either line, either two right an-
gles, or equal to two right angles.

§ a. Let A B and F 6 be two straight
lines that cross each other in C ; the two
angles (F C A, F C B) on the same side
of either line (A B) are either two right
_3 angles, or are equal to two right angles.
§ b. If A B and F 6 be perpendicular
to each other, they cut each other at right
angles (§ 6. § b.) ; and consequently each
of the angles F C A and F C B is a right
angle. But if A B and F G be not perpendicular to each other, from
C, the point of their intersection, draw C D, which shall be perpen-
dicular to (A B) one of them, then will D C A and D C B, the
whole angular space from C, on one side of A B, be two right
angles (§ 6. § b.) ; F C A and FOB, together, also comprehend
the same angular space ; therefore they are equal to D C A + D C
B (§ 24. § c), which are two right angles. In a similar manner, it
may be proven, that BCF + BCG, orGCB + GC A,orA
C G + A C F, are equal to two right angles.

PROPOSITION II.

§ 26. All the angles, that any number of straight lines, which
cross each other in the same point, make with each other, are
together equal to four right angles.

§ a. Since the two angles (§ 25.)
F C A, F C B, are equal to two right
angles, and also G C B, G C A,
equal to two right angles ; the four
angles F C A, F C B, G C A, G C
B, which two straight lines make by
crossing each other, are equal to four
right angles. If these four angles
be divided into any number of other
angles, by straight lines a 6, a 6, crossing in the point C, the

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20 PROPOSITIONS.

sum of the divisions thus made will be equal (§ 24. § d.) to four
right angles.

PROPOSITION III.

§ 27. When two straight lines
A B, D E, cross each other, they
make the angles D C B and ACE,
or A C D and B C £, that are
vertically opposite, equal to each
other.

§ a. The angles A C D and
D C B, (§ 25.) are together equal
to two right angles ; by the same

proposition A C D + A C E, are also equal to two right angles ;

wherefore (§ 24. $ c.) iA C D + D C B = A C D + A C E;

and A C D is a term in each member of the equation, and being

taken away or cancelled, we have (§24. § b.) D C B as A C E.
§ b. It may be demonstrated in a similar manner, that the vertical

and opposite angles A C D and B C E, are equal to each other.

PROPOSITION IV.

I K § 28. When a straight line (A C)

/ crosses two others (B E and G D)

A »/ x that are parallel, it makes the angles

._..............--* (ABE and A C D), which are on the

m ..-J— — ■ b same side of the two lines, equal to

* I each other.

/ § a. If the straight line A C cross

/ the two others perpendicularly, the

l*f proposition becomes evident, for the

angles formed would be (§ 6. § b,) right angles, and consequently

equal.

§ 6. But if they cross obliquely, it is obvious that if two lines be
parallel to each other, they must have the same divergence from
any straight line which crosses them; wherefore (§ 8. $ 6.) A B E
«b A C D; for B E and C D have the same divergence from A C.

§ c. In the same manner it may be shown that E B / is equal to
DC/.

$ d. The angles (A B E, h B/, A C D, g C /, etc.) taken alter-
nately on each side of A/, are called alternate angles.

$ e. The angles (A B A, AB E,/ C g, and /C DO on the out-
ride of the two parallel lines are called the external or exterior
angles.

§/. And the others are called internal or interior angles.

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GEOMETRY.

21

PROPOSITION V.

$ 29. A right line that crosses two others which are parallel,
makes the interior angles (§ 28. $ /.) (A H G and D F E) that are
alternate (§ 28. § d.) equal to each other.

§ a. A H 6 and E H B are oppo-
site and vertical angles, wherefore (§
27.) they are equal to each other,
* and E H B (§ 28.) is equal to D F
E; therefore (§ 24. § a.) A H G is
equal to D F E.

§ b. In the same manner it may be
proven that the alternate and interior
angles B H F and C F H are equal to
each other.

PROPOSITION VI.

§ 30. If a straight line cross two
others that are parallel, it makes the
sum of the two internal angles (B H
F and HFC) that are on the same
side of it, equal to two right angles ;
and the alternate (§ 28. $ d.) angles
(E H B, A H F, H F C, and D F G)
equal to each other.

§ a. The angles B H E and B H
F (§ 25.) are together equal to two right angles, and (§ 28.) B H E
is equal to H F C, therefore (§ 24. § 6.) the internal angles (B H F
and HFC) that are on the same side of E G, are together equal to
two right angles.

§ b. The alternate angle EHB = AHF,andHFC = DFG,
because (§ 27.) they are vertically opposite; and (§ 29.) AHF =
HFC; wherefore (§ 24. § a.) the alternate angles EHB.AHF,
HFC, and D F G, are equal to each other.

§ c. After the same manner of demonstration, it may be proven,
that the alternate angles A H E, B H F, H F D, and C F G, are
equal to each other.

§ d. Cor. If a straight line (E G) cross two others (A B and D C)
so as to make the sum of the internal angles (§ a.) (B H F and H
F C) on the same side of it, equal to two right angles ; or the alter-
nate angles (§ b.) (E H B, A H F, H F C, and D F G,) equal to
each other; or an exterior angle (§ 28.) (E H B) equal to its alter-
nate and internal angle (H F C), these two straight lines are pa-
rallel.

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22 PROPOSITIONS.

PROPOSITION Til.

§ 31. The exterior angle (D B G) which is formed by producing
any side (A B) of a triangle, is equal to the two interior and remote
angles (A and D) of the triangle.

§ a. From B let B E be
drawn parallel to A D;
also let A B be produced to
*C.

§ b. Because A D and B
E are parallel (§ a.) and
A C crosses them, the an-
gles A and E B C (§ 28.)
•■ are equal to each other;

moreover D B, by crossing the same two parallels (D A and B E),
makes (§ 29.) the alternate and interior angles (D and D B E) equal
to each other ; the exterior angle D B C is made up of D B E and
E B C, then substituting the whole for its parts (§ 24. § d.) we have
(§ 24. § 6.) the exterior angle D B C = A + D, the two interior and
remote angles in the triangle A D B.

§ c. If either of the two other sides be produced, the proposition
is proven in the same way.

PROPOSITION Till.

§ 32. The sum of the angles of a triangle is 180°, or two right
angles.

§ a. Produce a side (c)
of any triangle to D. Then
C B A and the exterior an-
gle C B D (§ 25.) are equal
to two right angles: and
the exterior angle C B D,
($31.) is equal to the two
remote angles C and A:
therefore (§ 24. § 6.) C, A,
and C B A are equal to two right angles : and C, A, and G B A are
the three angles of the triangle (A G B) proposed.

§ b. Cor. If two angles of one triangle be known, the third is also
known. It is found by subtracting the sum of the two known an-
gles from 180°.

$ c. Cor. And the two acute angles of a right angled triangle
(§ 17.) are together equal to one right angle.

§ d. Wherefore the acute angles of any right angled triangle are
the complements ($ 8. § c.) of each other.

§ e. It is also evident from the above, that if two angles in one
triangle be equal to two angles of another, the remaining angles are
also equal to each other.

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GEOMETRY.

23

PROPOSITION IX.

§ 33. A diagonal (A C) of a parallelogram divides the parallelo-
gram into two equal (§ 23.) triangles (A C D and A C B).

§ a. The opposite sides of a
parallelogram (§ 19.) are parallel
and equal to each other ; where-
fore D C = A B, A D «. C
B, and A C is common to the
two triangles (A C D and A C B);
therefore their sides are equal,
those of the one to those of the other. The angles D C A and C A
B are equal to each other (§ 30.), for they are alternate angles (§ 28.
$ d.) made by A C crossing the two parallels D C and A B. For
the same reason the alternate angles D AC and A C B made by A
C with the parallels A D and C B, are equal to each other. Where-
fore, the two angles (D C A and D A C) in one triangle being equal
to two angles (A C B and C A B) in another, the remaining angles
(D and B) (§ 32. § e.) are also equal to each other. Therefore, the
triangle A C D having its sides and angles respectively equal to the
sides and angles of the triangle A C B, is equal (§ 23.) to the latter,
and the diagonal (A C) divides the parallelogram DABC into these
two equal triangles.

§ b. Scholium. The area (§ 10. § a.) of a parallelogram (§ 19.
$ b.) is the product of its base and altitude. The triangle ABC and
the parallelogram DABC have the same base (A B) and altitude
(/>), and the triangle is proven to be half of the parallelogram ;
Therefore,

§ c. Cor. The area or magnitude of a triangle is the product of its
base by half its altitude.

§ d. Cor. The triangles into which a diagonal divides a parallelo-
gram are both equal and (§ 23.) of the same magnitude.

$ e. Scholium. D AC + B A C (§ a.) = D C A + B C A,and
D = B. Therefore,

§/. Cor. The opposite angles of a parallellogram are equal.
§ g. Cor. If the opposite angles of a quadrilateral figure be equal,
figure is a parallelogram.

proposition x.

§ 34. If a parallelogram (D A C B) and a triangle (A B C) stand
upon the same base (A C), and between the same parallels (D B and
A C), they have the same altitude (/>), and the triangle is equal to
half the parallelogram.

§ a. If one side
B (A B) of the triangle
be diagonal to the
parallelogram, the
triangle falls within
the parallelogram,
and (§ 33.) the pro-
position is proven.

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24 PROPOSITIONS.

§ b. But if the vertex (H) of the triangle proposed, fall without
the parallelogram, as A H C, continue one of the parallels (D B) on,
through the vertex H, to E ; and upon the base A C construct the
parallelogram A H £ C, to which the side H C of this triangle is
diagonal (§ 22. § a.), therefore A H C (§ 33.) is half of the parallelo-
gram A H E C. The parallelograms A H E C and D A C B f § 19.
§ a.) have the same base (A C) and altitude (/>), and the area of each
(§ 19. § b.) is the product of A C by p ; wherefore (§ 24. § b.) these
two parallelograms are of the same magnitude ; therefore (§ 24. § a.)
the triangle A H C is also equal in magnitude to half of either pa-
rallelogram, say D B C A ; and'/* (§ 18. § a.) is also the altitude of
the triangle A H C.

§ c. Cor. If a triangle and a parallelogram have their bases and
altitudes equal, the triangle is equal in magnitude to half the paral-
lelogram.

§ d. Cor. Triangles which have the same or equal bases and alti-
tudes, are of the same magnitude.

PROPOSITION XI.

§ 35. When two sides (A C and C B) of any triangle (A C B)
are equal to two sides (E A and A C) of another (A E C), if the an-
gles (A C B and C A E), which these sides contain, be equal, the
two triangles are equal.
K ■ytf § fl - ket one of the equal sides

\ ^/\ (A C) be made common to the

\ ^^ \ two triangles proposed, by con-

\ ^/^ \ structing them on opposite sides

\ S^ \ of it; the figure A E C B, thus

A.v£~ i* formed, will be a quadrilateral.

% b. By the conditions of the proposition, the angle E A C = A
C B, and they are alternate angles, made by A C with the equal lines
E A and C B, therefore (§ 30.) E A and C B are parallel ; and
the opposite straight lines (E C and A B) which join their ex-
tremities are also parallel, wherefore (§ 30.) the alternate angles E
C A and C A B are equal to each other.

§ c. E and B are the remaining angles of the two triangles, and
(§ 32. § e.) they are equal to each other, for (§ b.) E A C + E C A
bsACB + CAB, therefore the angles of the two triangles pro-
posed are equal to each other.

§ d. Now, since EAC + BAC = ECA + BCAthe whole
(§ 24. $ d.) E A B is equal to the whole E C B ; and (§ c.) E — B;
these four are the opposite angles of the quadrilateral A E C B,
which therefore ($ 33. § g.) is a parallelogram, and (§ 19.) E C «=
A B, and (§ 33.) A C divides the parallelogram into the two equal
triangles A C B and A E C.

§ e. Cor. If the opposite sides of a quadrilateral figure be either
parallel or equal, the figure is a parallelogram.

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GEOMETRY. 35

PROPOSITION XII.

§ 36. Either side (6) of any triangle is less than the sum of the
two other sides (a & c).

.* § a. The straight line

6, which joins the two
points A and C, is less
than the two straight

aZ ^ c < lines c and a which join

the same two points,
because the shortest distance between any two' points (A and C)
(§ 3.) is the straight line (6) which joins them ; wherefore b is less
than the sum of the two other sides of the triangle, or of any two
lines that can join A and O.

PROPOSITION XIII.

§ 37. Any two triangles (CAB and D C A) are equal, if a side
(A C), and the two angles (B A C and B C A) adjacent to it, in the
one, be respectively equal to a side (C A) and the two angles (A C
D and CAD) adjacent to this side in the other.

A » § a. Let the equal

"7 side (A C) be made
common, by construct-
ing the two proposed
triangles upon it, so
that one of them may
be on each side of the
common line A C ; the
/ IX figure (B A D C) thus

B £ w J / < formed is a quadrilate-

° ral, and the side A C,

which is common to the two proposed triangles, is a diagonal of it.
§ b. By the conditions of the proposition, the angles B A C and
A C D are equal, and (§ 28. § d.) they are alternate angles, where-
fore (§ 30. § d.) the two opposite sides (B A and C D) of the qua-
drilateral, are parallel. By supposition also, B C A and C A D are
equal, and they are likewise alternate angles, wherefore (§ 30. § d.)
the other two opposite sides (B C and A D) of the figure, are pa-
rallel.

£ c. If the opposite sides of a quadrilateral figure (§ 35. § e.) be
parallel, the figure is a parallelogram ; therefore B A D C is a paral-
lelogram, and it is divided by the diagonal A C into the two pro-
posed triangles CAB and D C A, which (§ 33.) are therefore equal
to each other.

PROPOSITION XIV.

§ 38. Any triangles (B C A and D B C) are equal, if two angles
( A & A B C) and an opposite side (C B), in one of the triangles

E

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S6 PROPGMITONS.

be respectively equal to two angles (D & B C D) and the corre-
sponding side (B C), of the other triangle.

c » § a* Let the equal

^ "side (O B) be made
,^^"' common to the two tri-

angles, by constructing
them so that one will
be on each side of C B.
Then C B becomes diagonal to the quadrilateral figure A C D B,
which is formed by thus constructing the two proposed triangles.

§ 6. By supposition, A = D, and ABGaBCD; wherefore
($ 32. § e.) the remaining angles A C B and C B D are equal to each
other, and they are alternate angles, therefore (§ 30. $ dJ) the two
opposite sides, C A and D B, of the four sided figure, are parallel ;
the other two opposite sides, C D and A B, are also parallel, the
alternate angles ABC and BCD being equal, by the conditions
of the proposition. Therefore, the quadrilateral A C D B (§ 35. § e.)
is a parallelogram, and is divided by the diagonal B C, (§ 33.) into
the two equal triangles B C A and DBG.

§ c. Cor. Hence it is inferred, that if two angles and a side of one
triangle be equal to two angles and a side of another, the two triangles
are equal.

PROPOSITION XV.

§ 89. Any two triangles (BAG and B D C) are equal, if every
side of the former be equal to its corresponding side in the latter :
e. g. AB« DC; AC = BD; andBC = CB.

* __ _ ^n § a. Let one of the

common to the two
triangles proposed, by
£'" constructing one of

them on each side of
B C ; the figure (A B D C) thus formed is a quadrilateral, and thft
common side B C is a diagonal of it

§ b. By the conditions of the preposition A B is equal to D C, and
A C to B D, and they are opposite sides of the quadrilateral figure
A B D C ; and if the opposite sides of a quadrilateral figure be equal
(§ 35. § e.), the figure is a parallelogram ; therefore A B D C is a
parallelogram, and its diagonal B C divides it into the two proposed
triangles B A C and B D C, which (§ 38.) are therefore equal.

§ c. Scholium. Since the corresponding parts in equal figures
(§ 23.) are equal, it follows :—

§ d. Cor. That, if the sides of one triangle be equal to the sidea
of another triangle, the angles opposite the equal sides are equal.

PROPOSITION XVI.

$ 40. Every equilateral triangle (A B C) is also equiangular.

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GEOMETRY.

27

§ a. Let an equilateral tri-
angle (E D H) be drawn,
having its sides equal to those
of the proposed triangle. The
two triangles (§ 39.) are then
equal, and the angles A and D
which correspond, (§ 39. § d.)
are equal to each other.
§ b. By the conditions of the proposition each side of A B C is
equal to the same thing, and by construction, equal to either side of
£ D H ; then A B = E H ; and the two triangles (§ a.) being
equal the angles D and O, which are opposite to those equal sides
(§ 39. § <?.), are equal. Wherefore A and C are each equal to D,
and, therefore, (§ 24. § a.) equal to each other. In the same manner
B and A may be proven to be equal to each other. Wherefore
and B are each equal to A, and consequently the equilateral triangle
A B C is equiangular.

§ c. Cor. Every equiangular triangle is also equilateral.
§ d. Scholium. Since the three angles of any triangle (§ 32.)
are together equal to 180°, and the three angles of an equilateral or
equiangular triangle, are equal to each other:

§ e. Cor. Every angle in any equilateral or equiangular triangle;
contains 60°.

PROPOSITION XVII.

§ 41. The angles (A & G) at the base of an isosceles triangle
A B C (§ 14.), are equal to each other.

§ o. Let the base (A C) of the pro-
posed triangle, be divided into two
equal parts ( A D & t> C) bf a straight
line (B D) drawn fnrni the vertex (B)
of the triangle.

§ b. This line (B D) divides the
isosceles triangle inttf the two tri-
angles A B D and D B C, in Which
every side of the one, is equal to the side which corresponds to it
in the Other ; viz : A B =* C B (§ 14.) for they are the legs of the
isosceles triangle A B C ; A D == D C (§ a.), by construction ; and
B D is common to both of the triangles ; therefore (§ 39.) these two
triangles (A B D shid D B C) aire equal, and the corresponding angles
A and C, being opposite to the common side B D, are therefore
(§ 80. § d.) equal to each other.

§ c* Scholium. A B D = C B D, because (§ 39. § tf.) they are
apposite to the two equal sides A D and DC; for a similar reason
B D A and B D C are also equal to each other, and (§ 25.) these
are together equal to two right angles, therefore each of them is a
right angle, and the right line B D (§ 6. § 6.) is perpendicular to
the base (A C) of the isosceles triangle ABC; wherefore :—
§ d. Cor. A straight line drawn from the vertex, so as to bisect

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28

PROPOSITIONS.

the base of an isosceles triangle, is perpendicular to the base, and
it divides the vertical angle (A B C) into two equal parts, and the
isosceles into two equal triangles. Also ;

§ e. Cor. A straight line (BD) that is drawn perpendicularly from
the vertex, to the base of an isosceles triangle, bisects the vertical
angle and the base.

PROPOSITION XVIII.

§ 42. If only two angles (A & B) of a triangle (A C B) be equal,
the triangle is isosceles.

r § a. Call the other angle (C) (§ 16.

§ b.) the vertex of the proposed triangle ;
and from it, let the straight line C D be
drawn, so as to bisect the vertical angle
(C) and divide the proposed triangle
into the two C A D and C B D, in

which, by construction, the side C D is
common, the angles A C D and B C D equal, and (§ 42.) A = B ;
wherefore in these two triangles (C A D & CBD), two angles
and a side of the one are equal to the homologous angles and side
of the other, and therefore (§ 38. § c.) these two triangles are equal
to each other ; consequently (§ 23.) the corresponding sides, C A &
C B, are equal, and hence, (§ 14.) the triangle A C B is isosceles.

PROPOSITION XIX.

$ 43. In every right angled triangle (A C B), the square (D A
B E) of the hypothenuse (§ 17. § a.) is equal in magnitude to the
squares (ACFH&BCQP) of the two legs.

$ a. Let D A B E represent the square (§ 20.) of the hypothe-
nuse (A B), and A C F H & B C Q P, the squares of the two legs
A C and B C ; then (§ 43.) DABE = ACPH + BCQP.

§ b. From the right angle (A C B) let C R be drawn parallel to
the parallels (§ 20. & § 19.) A D & B E, join C D and H B.

_ § c. The angles of a square

(§ 20. § a.) are right angles, and all
right angles (§ 6. § a.) are equal,
therefore HAC = DAB; to
each of these equals add the angle
B A C, and the sums H A B and
D A C ($ 24. § b.) will be equal ;
also HA = AC, because (§ 20.)
they, are sides of the same square
A F; and D A= A B, because they
also are sides of a square A E.

5 d. Wherefore, in the two tri-
angles A H B and A D C, the two
sides H A and A B of the one,

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GEOMETRY. 20

are respectively equal to the two sides G A and A D, of the other,
and the angles H A B and D A C, contained by these sides, are also
equal (§ c), therefore (§ 35.) the two triangles are equal.

§ e. The triangle A D C is equal in magnitude (§ 34.) to half of
the parallelogram A R, for they stand upon the same base D A, and
between the same parallels (§ b.) C R and A D. And the triangle
A H B is equal in magnitude to half of the parallelogram A F, be-
cause they stand upon the same base A H, and between the same
parallels H A and F C B.

§ /. The parallelograms A F and A R, being each double of either
of the equal triangles (§ d.) A H B and A D C, are therefore (§ 24.
§ b.) equal to each other in magnitude ; but the parallelogram A F
(§ a.) is the square of the leg A C ; wherefore the square of the leg
A C, and the parallelogram A R, are of the same magnitude.

§ 8- fy joining A P and C E, it may be demonstrated in the
same manner, that the parallelogram B R, and the square (B P Q C)
of the other leg B C, are of the same magnitude.

§ h. The parallelograms A R and B R make up the square (D A
B E) of the hypothenuse (§ a.) A B, therefore (§ 24. § a.) the square
of the hypothenuse (A B) is equal in magnitude to the sum of the
squares of the two legs (A O & C B).

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GEOMETRY.

PART II.

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GEOMETRY.

DEFINITIONS.

$ 49. A circle is *
figure that is bounded
by a line of uniform
curvature: all parts
of this line are equi-
distant from a point
(a) within it, which
is called the centre.

$ a. If a straight
line of any determin-
ate length were to re-
volve in a plane, and
on the point at one of
its own extremities*
its other extremity
would describe the
circumference (A D
C H) of a circle;
the point (a) on
which the line would
revolve would be the
centre of the circle ;
"* the length of the re-

volving line would be the radius (a B) of the circle ; and the plane
in which the line would revolve, would be the plane of the circle.

$ b. The circumference (A D C H) of every circle is divided into
300 equal parts, called degrees (°) ; every degree is divided into
00 equal parts, called minutes (' ) ; every minute is divided into 00
equal parts, called seconds ( " ) ; and these, when calculations are
performed in which great nicety is required, are divided into halves,
tenths, or hundredths.

$ c. Whether the circumference of a circle be infinitely great, or
infinitely small, a degree is still the same ; and the circumference
of the smallest is equal to the circumference of the largest circle,
when the two circumferences are compared in degrees, minutes,
and seconds, to each other ; for a degree is the 300th part of the
circumference of every circle.

$ d. Degrees (°), minutes ^ ' ), and seconds ( " ), are, the terms .
in which the value of angles is expressed. *

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34 DEFINITIONS.

§ e. When the straight lines which contain an angle, are produced
to the circumference of any circle that may be described from the
angular point as a centre, that part of the circumference which the
two lines intercept, contains the degrees, etc., which express the
value of said angle. An angle is said to stand upon the part of
the circumference that is thus intercepted. Neither the length of
the lines which contain the angle, or the distance of it from the cir-
cumference by which it is measured, affects its angular value.

§ 50. The radius of a circle is a straight line (e) that extends
from the centre to the circumference of the circle.

$ a. All the radii of the same, or equal circles, are themselves
equal.

$ 51. The diameter of a circle is a straight line (A C) that passes
through the centre of the circle, and is terminated at each end by the
circumference of the circle.

$ «. Either of the two parts (A B C & A H C) into which the
diameter divides the circle, is a semicircle.

$ ft. A segment of a circle is any part (g) cut from the circle by
a line (f)$ or a plane, which crosses the circle.

$ 53. Jin arc of a circle is any part of its circumference. That
part (A H) of 'the circumference which bounds a segment (g) is an
arc ; and the straight line (/) which joins the extremities of an
arc, is a chord.

$ a. The complement of an arc, or angle, is the difference be-
tween either and 90° ; and the supplement is what either wants of
being 180°.

$ ft. The radius and centre of a segment, or of an arc, are the
radius and centre of the circle, of which the segment, or the arc, is
a part.

$ c. Every arc or angle has its sine and co-sine, tangent and co-
tangent, secant and co-secant, besides its versed sine and semi-tan-
gent.

$<f. The " co " is an abbreviation for complement: the co-sine,
co-tangent, etc., of an angle or an arc, are the sine, tangent, etc., of
the complement of that arc or angle.

$ e. The sine, tangent, secant, etc., of an arc, are also the sine,
tangent, secant, etc., of the supplement of that arc.

$ 53. A chord is a straight line (t) (§ 52.) that joins the extre-
mities of an arc (B C).

$ a. Every arc has its chord.

5 ft. When a chord passes through the centre of a circle, it be-
comes a diameter, and the arc it subtends is a semicircle.

$ 54. A sine is a straight line Co) that extends from one extre-
mity (B) of an arc (B C) perpendicularly to the radius (a C) that
joins the other extremity.

$ 55. A versed sine is that part (• C) of the radius which is in-
tercepted between the sine and the extremity of the arc.

§ 56. A tangent is a line (ft) that touches one extremity of an
arc (B G), is perpendicular to the radius (a C) at that extremity,

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GEOMETRY. 95

and extends to another radius (e) produced through (B) the other
extremity of the arc.

§ 57. A secant (a B c) is the produced radius (c) that intersects
the tangent.

§ a. The co-sine, co-tangent, and co-secant, of the arc B C, are
p, D t, and a t \ and » (§ 19.) is equal to a*.

$ b. The sine, co*sine, tangent, co-tangent, etc., of an angle, is
the sine, co-sine; tangent, co-tangent, etc., of the arc which subtends
that angle.

$ c. These are sometimes called trigonometric functions.

§ 58. A sector is a figure (h) contained by two radii (a B, a C),
and the arc (B 0) between them.

proposition 1.

$ 59. An angle (A B C) at the circumference, is equal to half of
an angle (A H C) at the centre, if they stand upon the same arc
(C A), of a circle.

§ a. Draw the chord A C, and from
B let a diameter (BHD) be drawn.
The radii HA, H C, and HB (§50.
§ a.) are equal, and the triangles A H B
and C H B (§ 14.) are isosceles. The
exterior angle C H D (§ 31 .) is equal to
the two interior angles (II B C and H
C B) of the triangle CHB; for the
same reason the exterior angle (A H D)
of the triangle A H B, is equal to its two
interior angles H A B and H B A, which
are equal to each other, because (§41.)
they are at the base of the isosceles
triangle A H B ; therefore A H D is double of either of them, say
of H B A. For the same reason H B C and H C B are equal to
each other, and C II D is double of either, say of H B C.

The difference between CUD and A H D, is C H A, and the
difference between their equals, viz., twice HBC and twice H B A,
is twice ABC. Wherefore C H A (§ 24. § b.) is equal to twice
ABC; or, which is the same thing, the angle (A B C) at the cir-
cumference, is equal to half of the angle A H C at the centre of a
circle, and both of them stand upon the same base.

§ 6. An angle at the centre of a circle, is measured (§ 49. § e.) by
the arc it stands upon ; Wherefore —

§ c. Cor. An angle at the circumference of a circle is measured
by half the arc which subtends it.

§ d. Cor. All angles are equal that are at the centre, or all angles
are equal that are at the circumference, if they stand upon the same,
or equal arcs.

§ e. Cor. An angle that is at the circumference, and that stands
upon a semicircle (§ c), is a right angle.

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36

PROPOSITIONS.

proposition n*.

$ 60. If the radius (A C) of a circle cut a chord (B H) at right
angles, it bisects that chord and its arc (B C H)..

§ a. l>t the radii, AB and AH, be
drawn to join to extremities of the
chord (B H). The triangle (B A H)
thus formed (§ 50. § a. & $ 14.) is
isosceles. And the straight line A F
C bisects the vertical angle H A B
(§ 41. $ e.) ; for by the conditions of
the proposition, A C is perpendicular
to the chord B H, which is the base
of the isosceles triangle BAH.
Wherefore (§ 41. § d.) the two trian-
gles B F A and H F A are equal, B H
is bisected, and the angles BAG and
H A C are equal to each other, and being equal (§ 59. § d.) they
stand upon equal arcs (B C & C H). Therefore the chord (B H),
and its arc (B C H), are bisected by the radius A C.

$ b. Cor. If a radius bisect an arc, it also bisects the chord of
that arc, and cuts the ehord at right angles.

$ c. Cor. If a radius bisect a chord, it also bisects the arc of that
chord, and cuts the chord perpendicularly.

§ d. Cor. If a radius bisect a chord, or its arc, it also bisects the
angle at the centre, which stands upon that arc.

proposition ni.

$ 61. B C, half the chord (B D) of an arc (B E D), is the sine of
CAB, half the angle (D A B) which that arc subtends.

$ a. Let the chord be bisected at C,
and through C let the radius ACE
be drawn, then this radius (§ 60. $ c.)
bisects the arc B E D, cuts the chord
B D at right angles, and also (§ 60.
§ d.) bisects the angle (D A B) which
this arc subtends. Wherefore, B A C
is half of the angle DA B, BC is half
the chord of B E D, and it is perpen-
dicular to the radius A C E ; therefore
(§ 54.) B C, half the chord of B E D,
is the sine of C A B, which is half of
the angle (DAB) proposed.

proposition rv.
$62. In any citcle, radius (AD), the sine (AH) of 90°, the

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GEOMETRY.

37

chord (B D) of 60°, and the tangent (D C) of 45°, are all equal to
each other.

§ a. Let A be the centre of the cir-
cle, the arc H B E D a* 90°, the
ajc B E D = 60°, and the arc E D =
^5°, and join B A and C A. The
angles at the centre (A) ($ 40. § e.)
are measured by the arcs they stand
upon ; therefore the angle DAC =
45°, D A B = 60°, and D A H = 90° ;
A H then (§ 6. § b.) is perpendicular
to A D, and A H (§ 54.) is the sine
of the arc H B E D = 90°, it is a
radius, and therefore ($ 50. § a.) equal
to AD.

§ b. The radii A B and A D being equal ( §'50. § a.), makes the
triangle BAD (§14.) isosceles; consequently ($41.) the angles
ABD and AD Bare equal. The angle DAB = 60° (§49. §e.)
because it stands upon the' arc BED; wherefore (§ 32.) ABD-f
A D B = 120°, and being equal, the value of each is 60° ; there-
fore the triangle B A D is equiangular, and (§ 40. § c.) also equi-
lateral ; consequently (§ 13.) the chord (B D) of 60° is equal to
radius (A D).

$ c. C D is the tangent of E D = 45°, and consequently (§ 56.)
is perpendicular to A D ; wherefore (§ 6. § b. & § a.) the angle C
D A «= 90°, and (§ 17.) the triangle A D C is right angled. The
angle CAD = 45° (§ 49. § tX because it stands upon the arc E D,
therefore (§ 32. § d.) the angle AC D is also equal to 45°, and is
equal to CAD, and (§42.) the triangle ADC is isosceles, and
(§ 14.) the tangent (D C.) of 45° is equal to radius (A D).

§ d. Therefore the tangent of 45° (§ c), the chord of 60° (§ &.),
and the sine of 90° (§ a) being each equal to radius, are (§ 24. § a.)
equal to each other.

proposition v.

§ 63. In every triangle, the angle (A) that is opposite to the
greatest side (a) is the largest, and (B), that which is opposite to
the smallest side (6), is the smallest angle of the triangle.

§ a. Describe a circle (A C H B)
about the proposed triangles B A C, the
circumference of which touches the
three angular points (B, A, & C) of
the triangle. Each side of the triangle
then becomes the chord to the arc,
which subtends the angle that is oppo-
site to it

§ b. The angle A stands upon the
arc B H C, which is greater than either
of the arcs (B A, A C) upon which the

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38 DEFINITIONS.

two other angles (G & B) of the triangle stand. Each of these
three angles (§ 59. § c.) is measured by half the arc that subtends
it. Therefore A, which stands upon the greatest arc, and subtends
the greatest side (a), is the largest angle of the proposed triangle.

§ c. Half the smallest arc (A. C) (§ 50. $ c.) measures the angle
(B) that stands upon it ; therefore B (the angle that is opposite to
the smallest side) (6) is the smallest angle of the triangle.

DEFINITIONS.

$ 64. The multiple of a magnitude is the product of this magni-
tude and any other factor.

$ «. Equimultiples of magnitudes are the product of each of these
magnitudes by the same or equal multipliers.

§ b. Thus, 15 and 30 are equimultiples of 3 and 6 ; for 3 x 5= 15,
and 6x5=30. Consequently (§ 24. § 6.)—

$ 65. The same or equal multiples of equal magnitudes are equal
to each other.

$66. Ratio is the relation which the value of any magnitude
bears to that of another, and it is shown by dividing one magnitude
by the other.

$ a. Thus, the ratio of a to b is a -*■ b, and is expressed thus,-—
a : b.

$ b. And if the ratio between two quantities (a & 6) be equal
to the ratio between two other quantities (c & d), this equality of
ratio is expressed (§ XLV. Algebra) by writing the sign ( : : )
between the two former and the two latter quantities* thus,—
a : b : : c : d.

$ 67. Proportion consists in the equality of the ratios between
magnitudes.

$ a. Thus, the ratio (§ 66.) of 3 to 4 is j ; and the ratio of 6 to 8
is f ess | ; and these quantities are proportional ; t. 0. 3 : 4 : ; 6 : 8,

$6. Four quantities are in direct proportion* when the 4th is
equal to the quotient, which arises from dividing by the 1st quantity,
the product of the 2d and 3d. Thus (3 : 4 : : 6 : 8), 4 x 6 = 24,
and 24 -j- 3 = 8. So, also, a : b : : c : d , which in Algebra
(§ XLV.) is but another form for expressing that a«t-0saC-*-(2; and
by transposition a x d « c x b ; also, (c x b) -*- a «= d.

§ c. The first and third (a Si c) of four magnitudes that are in
direct proportion, are called antecedents ; and the second and fourth
(6 & d) are called consequents,

$ d. Also the first and fourth (a & d) are called extremes; and
the second and third (6 & c) are called means.

$ e. It is a rule in proportion, that the quantities for calculation
be 90 arranged that the product of the two extremes be equal to the
product of the two means. Thus, axd=bxc ; and as axdssdxa.
(§XXV. fl. Algebra), dxa*=cxb. Wherefore the means may
be made extremes, and the extremes means ; as-, b : a : : d : c
($ XLVIIJ. Alg.) ; the antecedents may be made consequents and
the coaaequenta antecedents ; as, d : c : : b : a, without inter-
rupting the harmony of the proportion between the quantities.

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I

GEOMETRY. 89

§/. Hence if the value of three magnitudes be known, the un-
known value of the fourth, which is in the same ratio of proportion,
is determinable. It is the quotient that arises (§ ft.) from dividing
the product of the two means by the known extreme. Thus,
3 : 4 : : 6 — ; a 4th quantity, 4 x 6 = 24, and 24 -h 3 = 8, the 4th
quantity* Generally speaking, the magnitude whose value is re-
quired is expressed last in the order of proportion.

AXIOMS.

§ 60. Among quantities that are proportional, equals may be sub-
stituted for equals.

$ a. Magnitudes that are proportional to the same two, or to other
magnitudes equal to these, are proportional to each other.

§ b. Equal magnitudes have the same ratio to the same magni-
tudes.

PROPOSITION VI.

( 70. The same, or equimultiples (C and D), of any two magni-
tudes (A and B), are to each other as the magnitudes themselves.
§ a. Let C be the same multiple, say the 5th, of
A, that D is of B ; then (§ 64. $ a.) A x 5 = C, and
Bx5=D ; by transposition C-s-A=5, andD-f-B=
5 ; wherefore (§24. § a.) C -$- A = D -*- B ; then
a XLVI.) C : A : : D : B, and alternately (§ XLVIII.)
C : D : : A : B. Therefore, the equal multiples are
as the magnitudes.

§ b. Scholium. The magnitude of a triangle (§ 33.
§ c.) is the product of its base and half of its altitude.
§ c. Wherefore the magnitudes of triangles of the
same altitude, are equimultiples of their bases. And
so of parallelograms of equal altitudes ; and therefore,
$ d. Cor. Triangles, or parallelograms, that have
the same altitude, are, in magnitude to each other as their bases ; or
those that have equal bases are to each other as their altitudes.

PROPOSITION VII.

$71. When the product of any two quantities (a&b) equals
the product of two others (c & d), the ratio of the greater multipli-
cand to the greater multiplier, is equal the ratio of the less multi-
plicand to the less multiplier | and the four quantities are propor-
tional.

$ a. Let ax&=*cX<f be the four quantities proposed, and let a be
the greater multiplicand, then d will be the greater multiplier, b the
less, and t the less multiplicand ; and (§ 71.) a : d : : c : b.

$ b. By the conditions of the proposition a x b ** e x d , and by
transposition (§" XLII.) a -*- d == c -$- b ; therefore these quantities
($ XLV.) are proportional; i. e., a : d : : c : b, and consequently
(§ 67. § a.) their ratios are the same.

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40 PROPOSITIONS.

PROPOSITION VIII.

$ 72. If a straight line (D E) be drawn parallel to the base (C fl)
(§ 16. $6.), and so as to cut the two sides (A C and A B) of any
triangle (A B C), the sections (AD, D C, A E, & E B) of these
two sides will be proportional ; i. e., A D : D C : : A E : E B.

$ a. Join D B and C E ; and
let E F be drawn from the ver-
tex (E) of the two triangles A E D
and C E D, perpendicularly upon
A C, then (§ 18. § a.) E F is the
altitude of each of these two
triangles. In the same manner,
by drawing D H perpendicular
to A B, it is shown Jhat the two
triangles A D E and B D E have
c /r**~~ • ■ ^T'A B the same altitude (D H).

§ b. The triangles A E D and C E D, having the same altitude (E F),
are to each other (§ 70. § d.) as their bases ;u.,AED:CED
: : A D : D C. And the triangles A D E and B D E are to each
other also as their bases; t. e., ADE : BDE : : AE : EB.

$ c. The triangle A D E is a magnitude in each set of these pro-
portions ; and the triangles C E D and B D E are of the same mag-
nitude (5 34. §<*.), for they stand upon the same base (D E), and
between the same parallels (§ 72.) D E and C B.

$ d. Wherefore A D and D C, A E and E B, are proportional to
the same or equal magnitudes ; they are therefore (§69. §a.) pro-
portional to each other ; t. «., A D : D C : : A E : E B.

§ e. By drawing a straight line from the vertex B, perpendicu-
larly upon A C, as a base 5 and another from C as a vertex, per-
pendicularly upon A B, it may be proven in the same way, that
AC: DC :: AB:EB. And also that A C : AD : : AB : AE.
Wherefore, —

§/. Cor. If the two sides of a triangle be crossed by a line that
is parallel to the base, these two sides will be proportional to their
sections.

PROPOSITION IX.

$73. The homologous (§ 23. $c.) sides of similar triangles (AB
C & D E F) are proportional to each other, viz., A B : c : : A C :

b : : C B : a. .

C a. By the conditions of the proposition, the triangles ABC
and D E F are similar, wherefore ($ 23. $ 6.) their corresponding
angles are equal, viz., B — E, A * D, and C — F. Upon the two
sides A B and A C of the greater triangle, (A B C) let A 6 be set
off, equal to c, and A H = b ; let G H be joined, then (§ 35.) the
two triangles AGHandDEFare equal to each other ; therefore
($23.) GH a o, the angle AG H = E, etc.

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GEOMETRY.

41

§ b. The angle A
G H (§ a.) =* E, and
B=E, therefore (§42,
§ a.) A G H = B,
and (§30. §ef.) GH
is parallel to B C.
Consequently (§ 72.
§/.) A B : A G : : A
C : A H ; by con-
struction A G = c,
and A H :±± 6 ; then substituting for A G and A H their equals, we
have (§69.) the proportion A B : c : : A C : b.

§ c. Then by drawing frorh G, G P parallel to A C, the paral-
lelogram G P C H (§ 35. § e.) is formed ; and (§ 72. §/) we have
A B : A G : : C B : C P. But (§ 19.) C P = G H, and (§ a.) G H
= a\ also A G = c. Therefore (§69.) A B : c : : C B : a.

§ d. Cor. Whence it may be inferred, that if a straight line be
within a triangle and parallel to its base, the ratio of the base to
this line will be equal to the ratio of the sides to their sections <

PROPOSITION X.

§ 74. In every triangle (A B C) the sides are proportional to the
sines of their opposite angles.

§ a. From C and A, as centres,
and with A C as the radius, let
the arcs A D and C E be described,
and draw A F perpendicular upon
B C, and G H perpendicular upon
A B, then (§ 54.) A F is the sine
of the angle A C B, and C H is
, the sine of the angle CAB.

§ b. Of the two triangles B H C
and B A F, (he angle at B is com-
mon, the right angles A F B and
C H B are equal, wherefore (§ 32. § e.) the remaining angles H C B
and F A B are equal, and (§ 23. § b.) these two triangles are similar.
§ c. The homologous sides of similar triangles (§ 73.) are pro-
portional ; wherefore AB:CB::AF:CH; and A F & CH
(§ a.) are the sines of the angles A C B and CAB; therefore
A B : C B : : sine A C B : sine CAB. In the same manner it

may be proven that A C

: sine
: C B : : sine ABC: sine CAB.

§ d. If the proposed triangle be right
angled, and the hypothenuse be made ra-
dius, the hypothenuse becomes (§ 62. § a.)
the sine of the right angle, and the legs (c
and a) the sines of their opposite angles (C
& A), whence we have a self-evident truth,
viz., A B : C B : : c : a ; but c and a are the
sines of C and A, therefore A B : C B : :
sine C : sine A. Also AC : A B : : 6 : c ;
G

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42

PROPOSITIONS.

bat b is the sine of B, and c of C, therefore A C : A B : : sine B :
sine C ; and (§ 62.) the sine of B is equal to radius. By like rea-
soning, it is shown that AC : C B : : sine B : sine A. Wherefore,
§ e. Cor. If an hypothenuse be made radius, that hypothenuse
is to either leg as radius is to the sine of the angle opposite that leg.

PROPOSITION XI.

§ 75. In every right angled triangle, the sides are proportional to the
several trigonometric functions (§ 57. § c.) of the two acute angles.

$a. Let the hypothenuse (6)
of the proposed triangle* be made
radius (A C) to an arc C D = A.
Then (§64. & (57. $«0 C B
is the sine, and A B the co-sine
of A ; likewise A B is the sine,
and C B the co-sine of C.

§6. Now(§XXV,tl.)»xa«
CBxAC; then($71.)6?AC
: : C B : a; by construction (§ a.)
A C is radius, and C B is sine A
or co-sine G ; therefore b : radius : : sine A or co-sine G ; a. Again,
6 X c = A B x A C, and A B (§ a.) is the co-sine of A, and sine of
G ; therefore, b : radius : : co-sine A or sine G : c. And these
quantities (§ XLVII.) are also proportional, when taken inversely
or alternately.

$ c. Let either of the legs (c) be made radius (A B) to another
arc E B = A. Then G B becomes tangent ($ 56.) to A, and (§ 57.
$ b.) co-tangent to G ; and A C (§ 57.) the sec. of A, and co-sec. of
C. And (§ XXV. n0»

1st. ABxo=CBxc, and (§71.) AB : c:: C B : a.
2d. ABx6 = AC xc, „ AB:c::AG:6.

3d. A C x a — C B x 6, ,, A C ; b : : C B : a.

Now ($ c.) A B is radius, C B is tangent A, or co-tangent C, and
A C is sec. A, or co-sec. C. Therefore in these three sets of pro-
portions, the first is, radius : c : : tangent A, or co-tangent G : a.
The second is, radius : c : : sec. A, or co-sec. G : 6. And the
third is, sec. A, or co-sec. C : b : : tangent A, or co-tangent G : a.
And ($ XL YIII.) these quantities are also proportional, when taken
inversely, or alternately.

§ d. If B C be made radius, the same method of demonstration
will show that radius, the tangent and secant of G, and of its com-
plement, are proportional to the sides (a, b & c) of the proposed
triangle.

proposition xn.

$ 76. The chords (a & 6), sines (d p & e *), tangents (e a
& B C), etc., of equal ares (d a & « B), which have different
radii (A B, A «), are to each 04her, as the radius of one of the arcs
is to the radius of the other.

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GEOMETRY.

48

§ d. The proposed arcs (cf * & e B)
are concentric ; c * is the tangent of
d *, and C B of e B ; by the definition
of a tangent (§ 56.), they are perpendi-
cular to A B ; they are therefore pa-
rallel; and (§73. § d.) ae : B C : :
radius A * : radius A B.

§ b. The sines (dp & e *) of these
two arcs, are also parallel, and by the
same proposition ($73. §d.) dp : es
:: As: AB.

$ «. The chords a and b, are also parallel, and, for the reason
above, are to each other as, A * : A B.

$ d. By similar methods of demonstration the secants, co-secants,
co-sines, co-tangents, etc., of the two arcs, may be proven to have
the same proportion to each other, which the two radii A * and A B
have.

PROPOSITION XIII.

5 77. The sum (B C) of two sides (C A & A H) of any triangle
(A G H), is to the tangent of half the sum of their adjacent angles
(H & C), as the difference (C p) between the same two sides, is
to the tangent of half the difference between the same two angles ;
orBC : tangent i (ACH+AHC) :: Cp : tangentl (ACH*>
AHC).

$ a. With the greatest side
(A H) of the proposed triangle, as
radius, describe the circle BHjo,
about the centre A. The side A C
is extended both ways, until it
forms the diameter (B p) of the
circle. Join 'B H, Up. DC is
drawn parallel to H P.

§ b. Because A H = A B (§ 50.
§a.) ; C A + A B, or C B is the
sum t)f the two sides C A & A H.
And because A H = A p ; A p — -
A C, or C p is the difference be-
tween the same two sides.
$ e. The angle B H p (§ 59. $ e.) is a right angle ; then C D is
perpendicular to B H, for it is drawn parallel to H p ; therefore
(§ 30.) C D B is a right angle ; and B D is tang, of B C D.
t $ d. The exterior angle H A B (§ 31.) is equal to the sum of the
adjacent angles (A C H & A H C) ; H p B ($ 59.) is equal to half
of H A B, for they stand upon the same arc (B H) ; then (§ 24. §a.)

^

HpB

ACH + AHC
2

Because D C and H p(§a.)are pa-

rallel, BCD (§28.) is equal to H p B ; therefore B C D is equal to

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44 PROPOSITIONS.

half the sum of the two angles A C H and AHC. And in the right
angled triangle B D C, C B is the sum (§ b.) of the two sides A H,
A C of the proposed triangle A C H, and B D (§ c.) is the tangent
of B C D, which is half the sum of A C H and A H C.

§e. By§31,theangleACH = C/)H+CHj0. Because AH«
A p (§ 50. § a.), the triangle H A p (§ 14.) is isosceles ; and (§ 41.)
C/>H=AH/>. Then (§24. § a.) AC H=A H/>+C Hjo; from
this sum , subtracting A H C, the remainder is twice OH/), which
is the difference between the two angles A C H and A H C ; then
C H p is half this difference. And (§29.) CHp = HCD;
therefore H C D is half the difference between the angles A C H
and A H 0. Now making C D radius, D H (§ 56.) becomes tan-
gent of H C D.

§/. Now C p being the difference (§6.) between the two sides
A H and A G ; and D C being parallel (§ a.) to the side H p of the
triangle HBp;we have (§72,) B C : B D : : C p : D H. Or,

AHC + ACH
C A + A H : tangent ^ : : C A» A H : tangent

A H C m A C H.

2

§g. Scholium. The half sum (§(f.) BCD added to the half dif-
ference H C D (§ 6.) makes the greater angle A C H ; and the half
difference C H p (§ c.) subtracted from the half sum A H p (§ c),
gives the less angle AHC. Wherefore,

§ A. Cor. Half the difference and half the sum of any two mag-
nitudes being added to each other, give the greater, and being sub-
tracted from each other, give the less of the two magnitudes.

PROPOSITION XIV.

^—*.^^ § 79. The base (c) of any trian-

■Jr^ ^% % £ le ( A B C) is to the difference be-

/ /\ \ tween the two sides (6 & c), as their

/ / \ \ sum is to twice the distance from

/ / \ \ the middle of the base to a perpen-

/ .• \ i dicular (A a) falling upon the base

j / 5fc • from its opposite angle (A) ; that is,

\ / \r \ \* / e : b « c : : b + e is to twice the

\ / yS j \ / distance of a from the middle of

ytis I X- ■yic the base.

\ * m \ J' §a. With a radius equal to the

% v, % ^^ longer side (b) of the proposed tri-

*"* **'* angle, the circle E C R H is de-

scribed about the angle (A) that is opposite to the base. Then b
(§ 50. § a.) is equal both to A E and to A R ; and the difference
between the two sides b and c is (A R — c =) B R ; and their sum
is(c + AE=)BE;

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GEOMETRY.

45

$b. Because the perpendicular
A a is drawn from the centre of
the circle upon the chord C H
($60.), it bisects it. Then the
distance of the perpendicular at a
from the middle of the base («),
plus, half the base, makes up
(C a) half the chord G H. Dou-
ble this, and we have the whole
chord C H, which is made up of
twice the distance from a to the
middle of the base, and twice the
half of the base e. Twice the
half, is equal to the whole base «. Therefore subtracting the base e
from the chord C H, the remainder (B U) is twice the distance of
the perpendicular at a from the middle of the base.

§ c. The two sides C B, B E of the triangle E B 0, are the base
(«), and the sum of the two sides (b & c) of the proposed triangle*
And the two sides BR,BH of the triangle H B R, are the dif-
ference between those sides (6 & c,) and twice the distance (§ b.) from
the perpendicular to the middle of the base («). These two triangles
(E B C & H B R) (§ 23. $6.) are similar; for the vertical angles
EBC&HBR (§27.)are equal; and HCE=»H RE ($59. $<*.),
they stand upon die same arc (H E) ; and the remaining angles
C E R, C H R, ($ 32. $ e.) are equal. Therefore (§ 73.) the homo-
logous sides of these two similar triangles are proportional ; that is,
C B : B R : : B E : B H ; or, e : ba> c :: b + a twice the dis-
tance of the perpendicular at a from the middle of the base.

$ d. Scholium. Then if the distance of the perpendicular
from the middle of the base be added to half the base, the sum
will be the greater segment (C a) ; or, if it be subtracted, the dif-
ference will be the smaller segment (B a) ($ 77. $ A.).

$ e. The distance of each angle from the perpendicular are the
segments of the base.

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LOGARITHMS.

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to

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LOGARITHMS.

$80. The purpose of logarithms is to facilitate arithmetical cal-
culations. The term is derived from two Greek words {logos and
arithmos), and may rightly be called the language of numbers ; for
by means of logarithms, not only multiplication and division, but
also the tedious operations of involving powers, extracting roots,
etc., are simplified. They are performed by the simple process of
addition and subtraction, multiplication and division.

§ a. Through the intervention of logarithms, angular and linear
magnitudes are compared with each other, as quantities of the same
denomination are in common arithmetic ; and the unknown value of
a line, arc, or an angle, may be deduced from the known ratio be-
tween other lines and angles.

$©. Before the invention of Lord Napier had introduced into
mathematical calculations the use of logarithms, the solution of tri-
gonometrical problems was referred to synthesis, and obtained by
construction. The process of finding the value of unknown quan-
tities in trigonometry was tedious ; and the result, owing to the
mechanical manner in which the operation was conducted, was sub-
ject to partial inaccuracies.

§ c. If the logarithms that correspond to a series of numbers in
geometrical progression, be taken out and noted down, they will be
found to constitute another series of numbers in arithmetical pro-
gression. And if two series of numbers be arranged, one in the
order of geometrical progression (the first term of which shall be
unity or 1, and the common ratio 10), and the other in the order of
arithmetical progression, (the first term of which shall be zero (0),
and the common difference 1), these two series will constitute a
basis, or the ground work for forming a table of logarithmic num-
bers, like those most generally used.

§ d. By arranging the two series of numbers, the one in geome-
trical, the other in arithmetical order of progression, thus,

Geometrical progression.

Arithmetical progression.*

1

•

10

-

1

100

.

8

1000

.

3

10000

.

4

100000

-

5

1000000

-

6

10000000

.

7

100000000

-

8

1000000000

.

10000000000

-

- - 10, *

H

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50 LOGARITHMS.

the relation and connexion between the orders of the two series
will appear. The terms in the order of arithmetical progression,
are the logarithms of those in the geometrical order : each the loga-
rithm of the term to which it is opposite ; t. e., is the logarithm
of 1 ; the logarithm of 10 is 1, of 100 is 2, of 1000 is 3, of 10,000
is 4, and of 100,000 it is 5, etc.

$ e. By observing the different relations which exist between the
terms in the two series above, it appears, —

1st. That the logarithm of any whole number has for its index
the figure, which expresses the number of digits, minus one, that
are in the natural number for which the logarithm is required.
Thus 1 is the logarithmic index of 10, which has two digits;

2 is the logarithmic index of 100, which has 3 digits ; 3 of 1000;
4 of 10,000, etc.

- 2d. That the logarithm of any number intervening between 1
and 10, must be a fraction less than unity ; for the logarithm of I is
0, and of 10 it is 1 ; then the logarithm of any number between 1
and 10 must be less than 1. The logarithm that corresponds to
any number between 10 and 100, must be greater than 1 and less
than 2, etc.

3d. That the product of any two numbers in the series of geo-
metrical progression, answers to the sum of the two numbers in the
arithmetical order, which correspond to them. Thus ; 100,000,000
in the geometrical, answers to 8 in the arithmetical order, and
100,000,000 is the product of 1000x100,000, which two numbers
correspond with 3 and 5 in the arithmetical series ; and in this
series 3 + 5, or 8 = 100,000,000 in the geometrical series. Now

3 is the logarithm of 1000, and 5 of 100,000. Hence, therefore, if
the logarithms of any quantities whose product is required, be added
together, this sum will be the logarithm of the required product.

4th. That the difference (4) between any two numbers (2 — 6)
in the series of the arithmetical order of progression, is the loga-
rithm of the quotient (10,000) which arises from dividing by each
other, the two numbers (1,000,000 -f- 100) that correspond, in the
geometrical order, to the two said arithmetical terms (2 and 6).

5th. That logarithms of numbers are a series of terms in the order
of arithmetical progression, which series corresponds to another
series of numbers in the order of geometrical progression.

$/. The logarithm of 10 is not necessarily J. Any other num-
ber as well as 1 may be assumed as the logarithm of 10. But if
any other number were taken as the logarithm of 10, it would
establish another base for calculating a table of logarithms, and this
change would effect the logarithm of every other number in the
same ratio. From this it appears that the value of a logarithm is
entirely conventional. But in order to simplify, and facilitate their
application to practice, mathematicians have assumed, in the order
of geometrical progression, a series of numbers in the ratio of 1,
10, 100, 1000, 10,000, etc. ; and as logarithms to these, they have
assigned, in arithmetical progression, a series of numbers in the

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LOGARITHMS. 51

order of 0, 1, 2, 3, 4, etc., having 1 for the common difference.
The latter numbers, taken in the order in which they stand, are
called the logarithmic indices, or characteristics, of the former, taken
in the same order. Whence (§ e. 1st) the logarithmic index of every
whole number, is made always to express one less than the number
of digits contained in the whole number. Thus, the characteristic
of 100 is 2, as the number of digits or figures in 100, is 3. The
number of digits in 1000 is 4, and the logarithmic characteristic of
1000, is 3.

$ g. The logarithmic signs and tangents, co-sines and co-tangents,
etc. of degrees, minutes, and seconds, are also of conventional value.
They are expressed in parts of the radius of a circle ; the value of
which radius is assumed to be equal to 10,000,000, etc., with as
many ciphers affixed as the compiler intends the mantissa shall
consist of digits.

$ h. The mantissa is the decimal part of a logarithm ; or the part
which is not the index. The logarithm of 250 is 2 .397940 ; 2 is
the index, and .397940 is the mantissa.

§ 81. The characteristic of a logarithm depends upon the number
of digits which are contained in its geometrical* number; being
one less than the number of digits, it is known by counting the
figures of the geometrical number, and writing down one less than
their number for the index.

$ 82. The mantissas have been previously calculated and arranged
in a tabular form, and in such a manner that the mantissa for any
geometric number, (however great or small), may be determined
with readiness.

$ 83. The first column of the tables (I.) is marked N. (Numbers.)
It contains the geometrical numbers, from 1 to 999. The mantissa
of each term in this series, is in a line with its geometrical number,
and in the next column which is marked 0. The nine other columns,
headed with the cardinal numbers 1, 2, 3, etc., to 9, also contain
mantissae ; but these are of geometrical numbers that have four
digits ; the last figure of which, being found at the head of one of
these columns, and the three first in the first column (N), show at
their angle of meeting, the proper mantissa. Thus, the logarithm
of 3681 is 3.565966 ; 368 being found in the first column, N., the
mantissa for 8681 is found in a line with 368, and in the column, at
the top and bottom of which, stands 1.

$ 84. To take from the table (I.) the mantissa for any geometrical
number less than 1000.

$ a. Find the given number in the first column (N.); its mantissa
is opposite to it in the next column (0).

$6. The mantissa for 9 is .954243; for 14 it is .146128; and
for 964 it is .984077.

• The natural numbers, which correspond to, er are represented by, a loga-
rithm, are here called geometrical numbers, from their being terms in the
geometrical series 1 of progression. Thus 419, of which 2.622214 is the logs*
rithm, is the geometrical number of log. 2<622214.

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52 L0GAEITHM8.

$ c. These mantissa?, with the proper characteristics prefixed, con-
stitute the logarithms of those numbers ; thus, logarithm of 9 as
0.054243 ; of 14 *= 1.146128 ; and of 064 = 2.084077.

§ 85. To take from the tables, the mantissa for any geometrical
number that is greater than 1000, but less than 10,000.

$ a. Find, in the first column (N.) of the tables, the three first
figures of the proposed number ; in a line with them, and in the
column over which the fourth figure stands, is the proper man-
tissa.

$ b. The mantissa (.973500) for 9410 is found in the column
marked (0), and opposite to 941 in the first column; and the man-
tissa (.954918) for 9014, is found in the column marked (4), and
opposite to 001 in the first column (N.).

$ c. The geometrical numbers in this case have four digits ; 3,
then, is the characteristic of their logarithms ; which are, 3.973590
— 9410 ; and 3.954918 « 9014.

$ 86. To take from the tables the mantissa of any geometrical
number that has more than four digits, or that is greater than
10,000; say of 43568.

$ a. Take out the mantissa for the four first digits ($ 85. $ &•), and
subtract it from the mantissa next in order. Then say,— If unity
(i), prefixed to as many ciphers as there are digits remaining of the
proposed geometrical number, give this mantissa! difference, what
mantissal difference will the remaining digits of the geometrical
number, give ? This last mantissal difference being added to the
mantissa for the four first digits, produces the required mantissa.
{i $ o. The mantissa for the four first digits (4356) of 43568 is
\ 639088 ; the mantissal difference between this and the next
(.639188) in order, is .000100 ; one digit (8) of the proposed geo-
metrical number remains ; and unity (1) ($ a.) must be prefixed to
one cipher (0), which makes (10) ; then the oider and terms
of the proportion are, 10 : .000100 : : 8 : .000080. This last
term, plus the former mantissa, (or .639088 -f. 000080) =.639 168,
the mantissa required.

$ c. The geometrical number (43568) in this example has Jivt
digits ; then (4) being prefixed as an index to the mantissa (.639168)
makes the logarithm of 43568 » 4.639168.

$ d. The mantissa for the four first digits (7419) of 741946 is
.870345 ; the difference between the mantissa of 7419, and of 7420,
is .000059. And the order and terms of the proportion are,
100 : 000059 : : 46 : .0Q0027. This last term feeing added, and
the characteristic (5) being prefixed to the mantissa (.870845),
makes the logarithm of 741946 « 5. 870372.

$ e. The mantissa for the four first digits (5941) of 594106734
is .773860 ; the mantissal difference «a .000073 ; the remaining
digits (06734) are Jive in number. Prefixing 1 to Jive ciphers, the
proportion is 100000 : 000073 : : 06734 : 000004. Prefixing the
characteristic (8) to .773860 + 000004, makes the logarithm of
594106734 - 8.773864.

$/. The mantissa for 124, and the mantissa for 1240000000 are

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LOGARITHMS. W

the same (.093422). But the logarithm of 124 = 2.093422 ; and
the logarithm of 1240000000 = 9.093422. Hence, j

$ g\ The effect of final ciphers in a geometrical number, on its I
logarithm, is confined to the characteristic of the logarithm, for the I
logs. ($/) of 124 and 1240000000 have the same mantissa.

$ 87. To find the logarithm of a geometrical number that is mixed
with a decimal fraction.

$ a. The mantissa is found as it would be, were the proposed
number integral ; but the characteristic of the logarithm must express
one less than the number of digits in the integral part of the number
proposed. Thus ;

Mixed Geometrical Nob.

Their Logarithms.

9414.5

= 3.973797

941.45 -

= 2.973797

94.145 -

= 1.973797

9.4145

— 0.973797

' § 88. As the logarithm of unity, or (1), is 0, it follows that the
logarithm of any quantity less than unity, or the logarithm of a frac-
tion, must be less than ; that is, it is a negative quantity. Thus, —

Logarithm of 1=0 .000000
Logarithm of 0.1 = — 1 +.000000
Logarithm of 80 = 1 .903090
Logarithm of 0.08 = — 2 +.903090
§ a. But the use of negative and positive quantities (such as the
logarithms of integers and of fractions, as just quoted above), in the
same operation, has a tendency sometimes to perplex the calcula-
tor. It is better, therefore, that all quantities in the same sum
should be affected with like signs. This may be effected with the
logarithms of fractions by means of a little artifice, by which negative
quantities are made to change their signs, or become as positive
quantities.

$ b. This artifice consists in borrowing and applying 10, or 20,
or 100, etc., to the logarithmic index of the fraction, and restoring
the borrowed quantity again during the progress of the calculation.
Thus, in the summing up of the logarithms of 80, and of 0.08, in-
stead of adding the mantissa and subtracting the characteristics of
the two logarithms ( 1.903090 — 2 +.903090 = 0.806180% the
operation may be performed entirely by addition, if the arithmeti-
cal complement of — 2, which is 8, be used as the index ; thus,
1.903090+8+.903090=*10.806180 ; rejecting the 10, which was
borrowed in the logarithmic index of 0.08, there remains 0.806180.
$ c. Any quantity with the sign (— ) prefixed, may be commuted
into a positive quantity by substituting for that quantity its arith-
metical complement. In this way subtraction may be performed
by addition, as we have seen (§6.) in the case of the negative
index — 2.
$ d. The arithmetical complement of a number, is the difference

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64 LOGARITHMS.

between that number, and the first number that ends with a cipher,
and is one scale higher in the decimal order of notation. The arith-
metical complement of 6, is 4, (10 — 6 = 4) ; of 77, is 23, (100 —
77 = 23) ; of 115, is 885, (1000— 115 =885), etc.

$ e. If it be required to subtract 6 from 9, the result is the same,
whether we say 9 — 6 = 3, or take the arithmetical complement
(4) of 6, add it to 9, (4 + 9 =13), and reject the borrowed (10)
from their sum (13 — 10 = 3).

§ 89. To find the arithmetical complement of a logarithm.

$ a. Prefix (1) to as many ciphers as there are digits in the pro-
posed logarithm; then from this number subtract the proposed
logarithm, and the remainder will be the required arithmetical com-
plement.

§ 6. Thus, the arithmetical complement of 1.903090 is 8.096910 ;
for 10000000.
1.903090

8.096910

§ c. The same result may be obtained by beginning at the left,
and subtracting from 9, every figure except the last significant one,
which must be subtracted from 10.

Thus— 9.9999(10)0
1.9030 9

8.0969 1

§ d. If the index be greater than 9, it must be subtracted from 19.

§ 90. To find the logarithm of a decimal fraction.

5 a. The mantissa for the proposed fraction is taken from the
table, as for an integral geometrical number of the same significant
figures which the fraction has. »

$ b. The logarithm for the same figures expressed both integrally
and fractionally, differs only in the characteristic ; the mantissa is
the same in both cases. Therefore we have only to teach how to
determine a positive characteristic, or the logarithm of a decimal
fraction.

$ c. The characteristic is determined by means of the number of
ciphers which precede the first significant figure of the decimal, for
the number of these subtracted from (9), gives the required charac-
teristic.

Thus, logarithm of the decimal .301 = 9.478567
Logarithm of the decimal .0301 = 8.478567
Logarithm of the decimal .000301 =» 6.478567

$ 91. To find the logarithm of a vulgar fraction that is less than
unity.

$ a. Take from the tables the mantissa for the numerator, and
for the denominator, separately, and as if each were a whole num-
ber expressed by the same figures. Then, from the logarithm of
the numerator, plus 10 to the index, subtracting the logarithm of
the denominator, leaves the logarithm of the proposed fraction. Thus,

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LOGARITHMS. 55

$ 6. To find the logarithm, of ■£&.

Logarithm of 25 = 1.397940
Logarithm of 100 = 2.000000

Logarithm of j£ — 9.397940

To find the logarithm of f .

Logarithm of 3 — 0.477121
9 Logarithm of 8 = 0.903090

Logarithm of | = 9.574031

$ 92. To find the logarithm of a mixed number.

$ a. If the number proposed (9.05) consist of a whole and a dV
cimal fraction, take from the tables the mantissa for the mixed
number as if it were a whole, and then prefix the index, which ex-
presses one lea than the number of digits in the integral part of
the mixed number proposed. Thus, the index of the logarithm for
9.05 is 0, and for 90.5 it is 1 ; the mantissa (.956649) is the same
for each number ; because it is taken out for the figures (905) a*
though they stood for a whole number.

Then, the logarithm of 9.05 = 0.956649
The logarithm of 90.5 = 1.956649

$ b. But if the proposed mixed number consist partly of a vulgar
fraction, let it be reduced to an improper fraction ; then find the
logarithm of the numerator, and of the denominator, as if they were
separately whole numbers ; and the remainder of logarithm of the
numerator, minus logarithm of the denominator, is the required
logarithm.

Thus, logarithm 121 = 1.102663 ; for 12f reduced to an impro-
per fraction is V ; and

Logarithm 38 = 1.579784
Logarithm 3 = 0.477121

1.102663 = 12|

r-

§ c. The index to the logarithm of an improper fraction, is always
a proper index ; for the numerator, being greater than the deno-
minator of such a fraction, the logarithm of the latter can be sub-
tracted from that of the former, without borrowing 10 in the index.

$ 93. To find the geometrical number which corresponds to a
logarithm.

$ a. The number of digits in the geometrical number for a loga-
rithm, is known by the characteristic of the logarithm ; for 1 added
to the characteristic tells the number of digits.

$ b. Find, in the table, the mantissa of the proposed logarithm.
The figures in the first column (N.) being prefixed to the figure
that stands at the top of the column in which the mantissa is found,
compose the geometrical number that is required.

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ft LOGABITHMS.

$ c. The geometrical number of log. 3.750314 = 5744.

The geometrical number of log. 2.759214 *= 574.4.

The geometrical number of log. 1.750214 = 57.44.

The geometrical number of log. 0.759214 = 5.744.
This mantissa is found in the column (4) of the table, and opposite
to 574 in the first column (N.).

§ d. Had the index of the logarithm been greater than (3), the
number of digits for the geometrical number required, would have
been made up by affixing ciphers to the numbers (574 and 4) found
above, and opposite to, the mantissa. Thus, the geometrical num-
ber for 4.976854 is known ($ a.) to consist of five digits, because of
the index (4). The mantissa (.976854) is found opposite to 948
(in the column N.), and under 1 ; making up with ciphers, the proper
number of digits.

The geometrical number for 4.976854 =- 94810
The geometrical number for 5.976854 « 948100
The geometrical number for 6.976854 «* 9481000
$ e. If the proposed mantissa cannot be found in the tables, take
that mantissa in the tables, which, being less, comes nearest to it,
and affixing any number of ciphers to the difference between these
two mantissa?, divide it by the difference between said tabular man*
tissa and the one next in order after it ; the quotient, being affixed to
the numbers opposite to, and at the head of the column of, said
least mantissa, comprise the figures of the required geometrical
number. The digits for the integral part of the geometrical num-
ber are determined by the index of the proposed logarithm ($<*«)>
and the figures which are to the right of these digits, constitute the
fractional part of the geometrical number. Thus,
$/. Geometrical number for 4.979744 = 95443.

Geometrical number for 2.979744 = 954.43.

Geometrical number for 0.979744 == 9.5443.

The proposed mantissa (979744) cannot be found in the tables.
The difference between the next less (.979730) and the next greater
(979776) found in the tables, is 46 ; and the difference (14) between
the less of these two and the proper mantissa being prefixed to any
number of ciphers, (14000 ----), and divided by the tabular dif-
ference (46), gives 3 + to be affixed to the four figures 9544, from
which the required digits for the integral part of the geometrical
number, are to be separated according to the index of the logarithm
proposed.

$ g. If the index of the logarithm proposed be an improper cha-
racteristic, the geometrical number of the logarithm is a decimal
fraction. And the difference between this characteristic and (9),
tells how many ciphers must intervene between the decimal point
( . ) and the first significant figure of the fraction. Thus (the in-
dices being improper),

The geometrical number for 7.911424 — 0.008155.

The geometrical number for 9.897027 «* 0.79.

The geometrical number for 5.698970 =« 0.00005.

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LOGARITHMS. 5?

$ 94. To perforin multiplication by logarithms.
$ a. Add the logarithms of the factors together; the sum of these
is the logarithm of the product. Thus,
$ b. To multiply 3 by 4 ; 20 by 2.5; and .25 by .30.
Logarithm 3 = 0.477121 Logarithm 20 = 1.301030

4 = 0.602060 " 2;5 = 0.397940

1.079181=12 1.698970=50

Logarithm 0*25 = 9.397940
„ 0.30 a 9.477121

8.875061 = 0.075

§ 95. To perform division by logarithms; \

$ a. Subtract the logarithm of die divisor from the logarithm of

the dividend, the remainder will be the logarithm of the quotient.
$bi To divide 36 by 3; 8941 by 19; 50 by .05; and 82.7 by

70.9f.

Log. 36 » 1.556302
„ 8 = 0.477121

Log. 8941 = 3.951386
„ 19 = 1.278754

1.079181 = 12.

2.672632 = 470.57+

Log. 50 — 1.698970
„ .05 = 8.698970

Log. 82.7 = 1.917506
„ 70.91 = 1.850708

3.000000 = 1000

0.066798 = 1.16+/^

§ 96. To perform involution by logarithms.

$ a. Multiply the logarithm of the proposed geometrical number
by the index (§ X.) of the power to be involved ; this product will be*
the logarithm of the required power.

$&. Raise 8* 17 s 112* 516».

Log. 8 = 0.903090 Log. 17 = 1.230449

2 3

1.806180 = 64 3.691347 = 4913

__ 11

Log. 112 = 2.049218

4

8.196872 = 157351936

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58 LOGARITHMS.

Log. 516 = 2.712650
9

24.413850 = 2593267484132083176308736

§ c. If the geometrical number that is proposed to be raised,
be a decimal fraction, the difference between the characteristic
of the logarithm of the power, and the product of 10 by the index
of the power involved, expresses one more than the number of
ciphers, that must precede the first significant figure of the required
power.

$ d. To raise .17 s .05* .064*.

Log. 0.17=9.230449 Log. 0.05=8.698970

3 2

27.691347=0.00491 + 17.397940=0.0025

Log. 0.064 = 8.806180
6

44.030900 = 0.000001073+

§ 97. Evolution is the converse of involution.

$ a. To perform evolution by logarithms.

\b. The quotient, obtained by dividing the logarithm of the
power proposed, by the exponent (§ XII.) of the root to be extracted,
is the logarithm of the root required.

§ c. To evolve the square root of 196, the <& of 81, and the
1/ of 128.

Log. 196 = 2.292256 -*- 2 = 1.146128 = 14.
Log. 81 = 1.908485 -*- 3 = 0.636161 + = 4.326+.
Log. 128 = 2.107210 -5- 7 = 0.301030 = 2.
# § d. If fiie geometrical number whose root is to be evolved, be a
decimal fraction, in order to obtain the logarithm of the root required,
the exponent, less one, of the root to be extracted, must be prefixed
to the characteristic of the logarithm of the number proposed ; and
then the quotient, that arises from dividing this quantity by the
exponent of the root to be evolved, will be the logarithm of the
power required. Thus, to find the cube root of .27. The loga-
rithmic index of 0.27 is 9 ; now, to 9, prefixing one less (2), than
the exponent (3) of the root to be evolved, makes the logarithm of
0.27 = 29.431364 -s- 3 = 9.810454 + = 0.646+. To find the
square root of 0.00776, the logarithmic index of which is (7) ; pre-
fixing to 7 one less (1) than the exponent (2), makes the loga-
rithm of 0.00776 = 17.889862 -+• 2 = 8.944931 = 0.088 +.

$ 98. In the tables (II.) containing logarithms for sines and co-sines,

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LOGARITHMS.

59

/

I / \

tangents and co-tangents, the logarithms are calculated from the
ratio between the radius and the sine, or the tangent of an arc.

$ a. In these tables, as in all others, the logarithmic value of a
sine or a tangent is conventional, as are the logarithms of geometri-
cal numbers.

$ b. In forming these tables, radius is taken as the base, or the
ground work for the calculations ; and its value is assumed to be
10.000000. The logarithmic value of the sine of 90°, or of the
tangent of 45° (§ 62.), shows that this is the value with which
radius enters this system. Wherefore, in every calculation, ra-
dius SB 10.

§ 99. If an arc of 90°, or if a right
angle, be divided into any two parts,
the sine or the tangent of either part,
is the co-sine or co-tangent of the other.
Thus, in the quadrant AB C ; AB=30°,
B C = 60°, A a = 35°, and a C = 55° ;
P B is the sine of A B = 30°, and co-
sine of B C = 60° ; B S is the sine of
60° (B C), and co-sine of 30° (A B);
therefore the co-sine of A B, is B S, which
is sine of B C, and the co-sine of B C is
B/7, which is sine of A B. In the same
manner A H and C D are reciprocally the
tangents and co-tangents of A a and a C.
Wherefore the tables, by being calculated
m * as far as 45°, answer for the whole circle,

or for two right angles.
§ 100. The logarithmic value of the sec. of an arc, or an angle,
is the arithmetical complement of the logarithmic co-sine of the
same arc or angle.

. % a. The logarithmic co-sec. is the arithmetical co. of the loga-
rithmic sine, and the logarithmic co-tangent is the arithmetical co.
of the logarithmic tangent, and vice versa (Vide § 109. § A.).

$ b. Wherefore in logarithmic tables of the trigonometric func-
tions, it is only essential that the logarithms of the sine, co-sine,
and tangent, or of the sec, co-sec, and co-tangent, should be given,
for the first three are the arithmetical co. of the latter, and vice
versd ; but to facilitate beginners in their calculations, the logarith-
mic sine, co-sec, co-sine, secant, tangent, and co-tangent, for every
degree (°) and minute ('), are given in the tables (II.), as well as for
every hour (h), minute (m), and four seconds (*), of the day.

§ 101. In trigonometrical calculations, when any logarithm is to
be subtracted, the proper result may be obtained by adding the
arithmetical co. of such logarithm. ,

§ a. This method of changing subtraction into addition is found
very convenient in practice. After a little exercise in it, the arith-
metical co. of a logarithm may be read from the logarithm, as readily
as the logarithm itself is read from the tables.
§ b. Wherefore in the solution of trigonometrical problems, sub-

1/

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K

00 LOGARITHMS.

traction need never be performed, for (§ 101.) instead of subtracting
a logarithm, the same result is obtained by substituting, for this loga-
rithm, its arithmetical co., and adding this arithmetical co. in the
calculation ; therefore (§ 100. § a.),

Ck logarithmic sine of an arc or an angle is to be subtracted,
I add its co-sec, and vice versd.
WhenJ ^ logarithmic co-sine of an arc or an angle is to be sub*
™ | tracted, add its sec, and vice versa.

I A logarithmic tangent of an arc or an angle is to be subtract-*
[^ ed, add its co-tangent, and vice versa.

§ 102, To find in the tables (II.), the logarithmic value of the sine
pf an arc or an angle.

§ d. If the proposed arc or angle be an extreme one, the number
at degrees contained in it, is at the top of the page. But if the one
proposed be a mean arc or angle, the degrees contained in it, are to
Be found at the bottom of the page.

§ o. An extreme arc or angle is one that contains less than 45°,
pr more than 135°. A mean arc or angle contains more than 45°
but less than 135°.

§ c If the proposed arc contain less than 45° or 3 h, or having
— than 00° or 6 h, contain less than 135° or h, the odd mi-
to be found in the proper minute column at the left
pf the page. TJfrHfit be greater than 135° or h, or being greater
than 45° or 3 h, contain tess than 90° or 6 b, the odd minutes of \t
are contained in the minute column, that is at the right hand side
pf the page.

§ d. The columns marked at the bottom, Cos., Sec, Sin., Co-
sec, Go-tang., and Tang., contain the logarithmic co-sine, secant,
sine, co-sec, co-tang., and tang., of the degrees, etc, that are at the
bottom of the page. And those marked Sin., Co-sec, Cos., Sec,
Tang,, and Co-tang., at the top, contain the logarithmic sine, co-sec,
co-jnne, secant, tangent, and co-tangent of the degrees,etc, at the top
of the page.

§ e. To take out of the tables the logarithmic sine of an extreme
arc, that is less than 45°, say of 34° 40'. The degrees (34°) of
the arc proposed being found at the top of the page, and the minutes
(40') in the minute (') column at the left of the page, the logarithm
(0.754060) which, in the column of Sin. stands opposite to the
minutes (40'), is the required logarithm. In the same way the lo-
garithmic co-sine, or tangent, etc, is found in its proper column,
and opposite to the given minutes.

Co-sine 21° 14' logarithm =* 9.969469.

Tangent 19° 56' logarithm =* 9.659491.

$/. If the arc (179° 7') proposed be extreme, and greater than
135 , the odd minutes (7') must be found in the last minute (') co-
lumn on the page ; the required logarithm (8.187985) is then found
ppposite, and in the column marked (Sin.) at the top.

§g. The odd minutes of a mean arc, say 61° 12', that is less
than 90°, are found in the last minute (') column ; and those of one,
say 114° 59', that is greater than 90°, are found in the first minute

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LOGARITHMS.

61

(') column of the page ; and the character (Sin,, Cos,, Co-tang.)
of the column is taken from the bottom of the page.
Sine 61° 12' = 9.942656.
Co-tangent 114° 59' = 9.068343.
Tangent 114° 59' = 10.331657.

§ h. The trigonometric functions of hours, minutes, and seconds,
are taken from the tables in the same way.
$ 103. To find the degs. etc. for a log. sine.
$ a. The minutes, in the minute (') column, which are found op-
posite to the proposed logarithm, and the degrees, which stand where
the column in which the proposed logarithm is found, is marked
Sin., are the degrees and minutes required. The same directions
answer for taking out the tangent, secant, co-sine, etc., of a loga-
rithm. Thus,

Logarithm 9.627300 sine = 25° 5' or 154° 55'.
„ 9.956566 cos. = 25° 12' or 154° 48'.

„ 9.831709 tang. = 34° 10' or 145° 50'.

„ 9.745494 cos. = 56° 11' or 123° 49'.

t , 10.377793 tang. = 67° 16' or 112° 44'.

§ 104. An arc (A B) and its
supplement (B C) have the
same sine (B H) ; therefore
the same logarithm answers
either to the sine of AB or
B C. But, for the most part,
in calculations, there is some
circumstance connected in the
operation, which determines
whether an arc or its supplement be required. If not, the case or

the solution is called doubtful.
sines, secants, tangeants, etc.

The same remarks apply to co-

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PLANE TRIGONOMETRY.

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PLANE TRIGONOMETRY.

§ 105. Plane trigonometry is that branch of mathematics by
which, with certain data, the unknown parts of triangles are de-
termined.

§ a. Plane trigonometry is divided into right, and oblique, angled
trigonometry. The solution of right angled triangles pertains to
the former ; and that of oblique angled triangles to the latter. The
methods of solving problems in either, in most cases are similar*

§ 106. Every plane triangle consists of three sides and three
angles ; which, in a general term, are' called the six parts of a
triangle.

§a. If the value of any three of these six parts (the three angles
excepted), be known, the value of every one of the remaining parts
is determinable by trigonometrical operations.

§ b. When the angles constitute the only data, the sides cannot
b£ determined (except in species) ; because there may be any num-
ber of triangles which are equiangular to each other, and their
homologous sides may all be unequal, as the sides of the two tri-
angles under § 73. are.

§ 107. In order to solve a trigonometric problem, the value of
three of five parts, viz., two of the angles and the three sides, must
be given, and at least one of these three parts must be a side*.

§ 108. The several combinations of three, which can be formed
of five parts, comprise all the cases that are necessary for solving
trigonometric problems.

§ a. To one of these every problem in trigonometry resolves
itself for solution.

§ b. These several combinations are reduced to Jive cases ; viz.,

1st, When two sides and the angle which is opposite to either
of them, are given :

2d, When two angles and the side which subtends either of
them, are given :

3d, When two sides and the angle which they contain, are given :

4th, When two angles and the side that is between them, are
given:

5th, When the three sides are given.

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66 , PLANE TRIGONOMETRY.

CASK I.

§ 109. Given the sides (c & b)
and the angle (B), that is opposite
to (b) one of them:

4 _ 450 yards.
b — 600 yards.
B — 74° 49'
Required the remaining parts (A,
C, * «)!

§ a. First, to find the value of C
By § 74., b : sine B : : c : sine C ;
and substituting for these given parts (6, c, and C), their values,
the proportion is, 600 : sine 74° 49' : : 450 : sine C.
Log. sine of 74° 49' — 9.984669
Logarithm of 450 — 2.653213

12.637782
Logarithm of 600—2.778151

Log. sine C « 9.859631 « 46° 22' 15".

$6. The sum of the two angles B and C (§ 32. $6.) being sub-
tracted from 180°, gives the remaining angle (A) of the proposed
triangle ABC. Tims, B + C - 121° 11' 15"— 180° « 58°
48' 45" —A.

$c. The logarithmic process ($«.) of finding the value of C,
might have been abbreviated, and the operation might have been
performed entirely by addition. This abbreviation (§ 101.) con-
sists in using the arithmetical complement of the logarithm of the
first term in the order of proportion, in the place of said logarithm.
This arithmetical complement, added to the logarithm of the second
and third terms, gives the logarithm, (rejecting 10 from the index),
of the fourth or required term.

$ d. When one or more logarithms are to be subtracted in the
process of a calculation, the operation may be simplyfied ($ 88. $ c.)
by substituting for such logarithms, their arithmetical complements
(§88. § e.), and adding these arithmetical complements in the calcu-
lation, and then rejecting the borrowed 10 from the index of the
sum thus obtained, the remainder is that which would have resulted
by adding together the logarithms which were to be added, and then
subtracting from their sum, the logarithms which were to be sub-
tracted.

J i e. Hereafter, instead of the logarithm of the first term in the
er of proportion being used, its arithmetical complement will be
adopted m calculation. This substitution of the arithmetical com-
plement renders the arrangement for calculation uniform ; for it
changes the process of addition and subtraction, into the simple
operation of addition.

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PLANS TRIGONOMETRY. 67

$ f. Let it be borne in mind that the arithmetical complement ($ 88.
$ d.) of any numerical expression, is obtained, by subtracting the
expression proposed from the number which is made by prefixing
unity (1) to as many cyphers, as there are figures in sand expres-
sion. Thiif the aritnmetical complement of 89 is 11, for 89 — 100
■■ 1 1 . Or, the arithmetical complement is obtained by beginning at
the left, and subtracting each figure from 9 (§ 79. $ c), except the
last significant figure, which must be subtracted from 10. Thus the
arithmetical complement of 9,984569 is 0.015431 (what the former
wants of 10.000000), and is obtained thus :

9.9 9 9 99(10)
Logarithm 9.984 5 6 9

Ar. co. 0.0 15 4 3 1

$£. And the arithmetical complement of 3.450000 ($/.) is
6.550000, and is obtained thus :

9.9(10)0 00
Logarithm 3.4 5 000

Ar. co. 6,5 5 000

$ h. When the logarithmic index of any of the trigonometric
functions exceeds 9, the arithmetical complement of such index is
the difference between itself and 19, ($ 89. $ d.) and consequently
the arithmetical complement is found by subtracting said index from
19. Thus the logarithmic tangent of 74° is 10.542504, and its
arithmetical complement is 9.457496, which is the co-tang, of 74°.

19.9 9 99 9(10)
74° log. tang. 10.5 42 5 4

Ar. co. 9.4 57 49 6

$ »• To find the value of the side a; according to § 74, sin. B :
b : : sin. A : a ; or, substituting their Tables, sin. 74° 49' : 60Q
: : sin. 58° 48' 45" : a.

Log. sine 74° 49' ar. co. = 0.015431 — co-sec. B (§ 100. §a.

Logarithm 600 - = 2.778151

Log. sine 58° 48' 45" * 9.982208

Log. a wm 2.725790 = 531.8 yards.

The 10, which was borrowed for the arithmetical complement of
the logarithmic sine of 74° 49', being rejected in the sum, gives 2
for the index of the logarithmic value of a.

%j. In the solution of all trigonometric problems by calculation,
the logarithmic values of the function* employed are always used ;

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68 PLANE TRIGONOMETRY.

therefore the repetition of the word log. before sine, tangent, etc.,
will be omitted ; and the arithmetical complement of the logarithm
of a function or number, is that which is to be understood, and not
the arithmetical complement of the number itself.

case n.

§ 110. Given two angles (A & C) and the si4e (c) that subtends
one of them.

c A = 23° 56'

y^\ C = 76° 4'

*f \ c = 1760 feet.

*f' \ m To find the other parts (B, a

\ and b),
^^ \ $ a. The process of solution in

A ■* c m m ^" B this case is a repetition of the me-
thods shown § 109. §6. & §».; for A + C subtracted from 180°
(§ 32. $6.) gives B=80° ; and(§ 74.) sine C : e : : sin. A : a : : sin.
B:6.

§ b. To find a. Sin. 76° 4' : 1760 : : sin. 23° 56' : a.

Sine 76° 4' ar. co. = 0.012970— co-sec. C($ 101. $6.)
Log. 1760 = 3.245513

Sine 23° 56* « 9.608177

Log. a = 2.866660=735.6 feet.

$ c. To find b. Sin. 76° 4' : 1760 : : sin. 80° : b.

Sine 76° 4' ar. co. = 0.012970=co-sec. G
Log. 1760 = 3.245513

Sine 80° = 9.993352

Log. b = 3.251835—1785.8 feet.

$<f. If, with similar data, the proposed, be a right angled trian-
gle, the problem may be solved as the one above is, by means of

the proportion between the sides
and sines of opposite angles ; or it
may be solved by means of the
principles involved ($ 75.^ in the
proportion between the sides and
trigonometric functions of a right
angled triangle. Thus, the data

c

^1

'' . \

**£. ■ 1a being,

° C=67° 19' '

c =1760 feet.
The value of A is known by the rectangularity of the proposed tri-
angle (D A C). To find the other parts (D, d & a),

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PLANE TRIGONOMETRY. 69

$ e. The third angle (D) is always known ($ 32. $ b-)by sub-
tracting the sum of the two other angles from 180° ; A+C — 180°
=D, or 22° 41'.

§^. To find the two sides a and d; making D A radius, the fol-
lowing proportions (§ 75. § e.) are deduced, viz.:
Rad. : c : : tang. D : d\ and Rad-: c : : sec. D : a.
Radius (always, $ 98. $ k)* 10 ' 00000 **
c=1760 log. = 3.245513

D=22° 41' tang. =9.621142

Log. d = 2.866655=735.6 feet.

Radius =10.

c=1760 log. = 3.245513

D=22° 41' sec. = 0.034963

Log. a = 3.280476=: 1907.5 feet.

case in.

$ 111. Given two sides (b & a) and the angle (G) they contain ;
6=479 fathoms.

^\ a=419.8 "

b^y^ >va 0=119° 41'.

^S^ >^ To find the other

^S ^ >* parts (A, B & c),

a s^ ^v j § a. The solution of

& this problem is founded

upon $77.

§ b. One angle (G) of the proposed triangle being known, the
sum of the two other angles (A & ; B) (§ 32. § b.) is obtained by sub-
tracting the given angle from 180°. Thus, 180°— 119° 41 '=60°
19', the sum of A & B.
$ c. To find the values of A and B ; according to § 77., (b+d) :

A+B AeoB

tang. — §— : : (6od a) : tang. — g . Therefore, by substitution,

A tr> "11

898.8 : tang. 30° 9' 30" : : 59.2 : tang. —^-.

( b+a )=898.8 log. ar. co. =7.046337
(A+B)-*-2=30°9'30" tang. =9.764207
(&* a)=59.2 log. =1.772322.

A<» B
Tang. —£~ =8.582866=2° 11/ 30".

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70 PLANE TRIGONOMETRY.

Now half the difference (2° 11' 30") between A and B, being added
to half their sum (30° 9' 30") (§ 78. § A.), gives B the greater angle; and
being subtracted from said half sum, gives A the less angle. Thua
30° V 30"+2° 11' 30"=32°21',orB; and 30° 9> 30" — 2° 11' 36"
=27° 58% or A. A is known to be the less angle ($ 63.), because
its subtending side (a) is the less of the sides b & a.

$ d. The angles and two sides being known, the value of the
third side (c) is deducible by means of calculations conducted upon
the proportion ($74.) between the aides and sines of the angles of
plane triangles.
$ e. To find c. Sin. B : b : : sin. C : c.

B=32° 21' sin. ar. co. — 0.271 573 =* co-sec. B ($100. $ a.)

6=479 faths. log. =2.680336

0=119° 41' sin. =0.988908 ($ 102. %g.)

Log. € =2.890817=777.7 faths.

$/. If the problem comprise a right angle triangle, with similar
data, the process of solving it is much more simple. For the two
legs and the right angle being the given parts, the calculations for
determining the unknown parts, are founded upon the relations of
the sides and trigonometric functions of a right angled triangle, as
they are shown $ 75.

$ g. In the prOpoeed triangle (D B
JP C), right angled at B, the values of
jf the legs, are \

* c=479 faths.; andrf =419.8 laths.; to

find the other parts.

$A. Making either leg (c\ radius,
indordei

/

*A

\

$ u To find D.

c—479. Log. ar. co.= 7.319664

Radius - - =10.

<*= 419.8 log. = 2.023042

($ 75. $ c.) the terms and order of the
proportion for finding the value of D,
are, c : rad. : : d : tang. D. And for
finding the hypothenuse (6) they are,
rad. : c : : sec D : b.

Tang. D = 9.942706=41° 13' 53"

$j. To find 6.

Radius - =10.

c—479 log. = 2.680386

D=41° W 53" sec. = 0.123777

Log. b = 2.804113=636.9 faths.

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FLAWS TMGOWOMBTBY.

71

CASK IT.

$112. Given two angle* (A &
C) and the side (b) between them;
A— 26° 1'
C=38° 49'
6« 179 chains,
To find the values of the other
parts (B, c & a) of the proposed
triangla(ABC). -
$a. Two angles being known, the third (B) (§ 32. $ b.) is also
known; it is B=*115° ltf.

§ b. The remaining unknown parts (a & c) of this triangle, are de-
termined by means of the principles involved in the solution of the
problem under Case 2. The process of operation in the solution
of this, is precisely similar to that of $ 109. § t. under Case 1. In
the method of their solution, Cases 1, 2, and 4, are only repetitions
of each other. For whenever two angles of a triangle are given,
the third is also known (§ 32. § 6.) ; and, whenever an angle and
its subtending side are two of the known parts, the required parts
are determinable through calculations conducted upon the proportion
(5 74.) which exists between the sides and sines of angles in all
triangles.

$ c. To find the value of a. Sin B : b : : sin. A : a.

B=115° 10', sin. ar. co.=0.043316=co-sec. B(§101.$6.)
6—179 chains log. =2.252853
A=26° 1', sin. =9.642101

Log.

To find the value of
= 115° 10' co-sec.
= 179 chains log.
z 38° 49' sin. -

Log. c

a
c.

=1.938270

=86.7 chains.

B =

b =
C =

Sin. B : b
0.043316
2.252853
9.797150

: : sin. C : c.

2.093319 =

123.9 chains.

§ e. Transferring the con-
ditions of this problem to one
in right angled trigonometry,
the use of sines may be re-
tained in the calculation for
determining the unknown
parts (c & d) ; for by making
the given side (6), radius to-
an arc (C H), the two legs of
the proposed triangle (D B C)
become (§ 75. § a.) sine and co-sine to the acute angles (D & C) ;

->n

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72 PLANE TRIGONOMETRY.

and ($75. §6.) rad. : b : : sin. Did:: co-sine D : e* The
parte given axe D = 26° 1', and £ = 170 chains.

$/. To find the values of e and d.

Radios - - * 10.

6 = 179. log. = 2.252853

D = 26° 1' sin. = 0.642101

Log. d = 1.804054=78.5 chs.

Radius = 10.
6=170 log. = 2.252853

D=20° 1' cos. = 0.053500

Log. c = 2.206452 = 160.8 chains.

§ £. The same result may be obtained by making a radius of
either leg (c), and introducing different trigonometric functions into
calculation, upon the principles shown under § 75. $ c. Thus ; sec
D : b : : radius : c : : tang. D : rf.

$ A. To find c and d by this method of calculation :

D = 26° 1' sec. (ar. co.) = 0.053500=cos. D ($101. $6.)

b = 170 chains log. = 2.252853 .

Radius - - - = 10.

Log. c = 2.206452=160.8 chains.

D=*26° 1' cos. (§ 101. § b.) = 0.053500
6=170 chains log. = 2.252853

D=26° V tang. = 0.688502

Log. d = 1.804054 =78.6 chains.

§ t. The use of sine and co-sine as functions in plane trigonome-
trical calculations, being more convenient than that of other func-
tions, the former (§/.) is generally preferable to the latter method
(§ A.) of solving problems.

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PLANE TRIGONOMETRY. 73

CASE V.

^

§ 113. Given the
three sides (B C, b, e)
B C = 144 chains.
\ b = 220 chains.

\ c = 103 chains.

\ To find the angles

\ (C, A & B)
\ § a. Here is the ap-
j plication to practice of
j the principles involved
/ in §79. And in order
...... /j| to determine the value

/ of any one (C) of these
/ angles by calculation,

f the distance (a a') from

the middle (a) of the
base (C B), to the per-
pendicular (A a') upon
it, must be found by means of the proportion, (§ 79.) the base B G :
b cdc : : b+c : 2 a a'. Then either of the segments (G a'-) is found
(§ 79. §tf.) by applying (C a) half the base (C B) to (aa!>) the
distance of the perpendicular at of from the middle of the base.
This segment (G a'), the perpendicular (A a'), and the side (6),
adjacent to the required angle (G), form a right angled triangle
(G A of) ; of which are known the hypothenuse (6) and a leg
(Ca').

§ b. The value of the required angle (G) is then found according
to the principles established under § 75. For by making the hypo-
thenuse (G A) radius, the segment (G a!) becomes co-sine to the
required angle (G) ; the value of which may then be determined
(§ 75. § b) by means of the proportion, b : radius : : Co 7 :
cos. G.

§ c. Knowing the value of any one (G) of the angles, either of
the others (A) may then be determined by means of calculations
conducted upon the proportion (§74.) of the sides of triangles
to the sines of their opposite angles. And the remaining angle (B)
(§32. § 6.) is determinable by subtraction.
§ d. To find the value of the angle C,

1st. Gall C B the base, in order to find twice the distance (a of)
from the middle of the base, to a perpendicular drawn upon the base,
from its opposite angle (A) (§79.). Twice the length of this die*
tance(a a') is evolved by the proportion; C B : b & c : : b+c : % a a' .
C Bt= 144, log. ar. co.=*7.841837
6o»c«=117. log. =2.068186
&+c«>38&. log. =.2.509208

Log. 2 a of =2.419026=262.4
L

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74 PLANE TRIGONOMETRY.

2d. To find the greater segment C a'. One half of 262.4 (2 a a')
is equal to the distance (a a') of the perpendicular from the middle

B f
of the base ; and (§ 79. § d.) a cr + -£-= c «'• Therefore, 131.2

+72«C a* =203.2.

§ e. If twice the distance (a a') of the perpendicular from the
middle of the base be greater than the base, the perpendicular
falls without the triangle, as in the case before us. But if it be
less than the base, the perpendicular falls within the triangle.

To find the value of G. According to § b. we have, b : rad. : :
C of : cos. C.

6=220. log. ar. co.= 7.657577

Radius =10.

G o / =203.2 log. = 2.307924

Cos. 0=9.965501=22° 32' 11"

§/. To find the value of A (§ c.) we have, c : sin. C : : B C
sin. A. x

c=103. log. ar. co.=7.987l63

C=22° 32' ll"8ine= s 9.583504
BC=144. log. =2.168363

Sine A =9.729030=32° 24' 2"

§ g. To find the value of B. According to § 32. $6. C+A—
180°»B; therefore B=125° 3' 47".

$ A. If, when the perpendicular falls without the triangle, the
obtuse angle (B) be the first part required, the side (c) adjacent to
it becomes hypothenuse to the right angled triangle, of which, the
perpendicular to the base and the smaller segment (B a!) of the
base, are legs, and the supplement (A B a!) of the required angle

) is that which is evolved from the proportion ($6.), c : rad. : :
\ a' : cos. A B a\

EXAMPLE I.

ffi 1

§ 114. A rock is seven miles from a light-house, and a ship,
having the light-house to bear E. by N., is nine miles 8. W. of
the rock ; What is the distance between the ship and the light-
house ?

Ans. 12.4 miles.

§ a. The solution of this problem is conducted after the methods
shown under § a. and § t. (§ 109.) The parts given are the sides
a and c, and the angle (A) contained at the ship, between the bear-
ing of the rock and light-house. The value of A is known by con-
verting into degrees, etc., the points contained between the bear-

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PLANE TRIGONOMETRY.

75

nock

ing of the rock and light-house, from the ship. The rock hears
N. E., and the light-house E. by N. from the ship. Three points
intervene between E. by N. and N. E., and each point contains
11° 15' ; therefore the angle (A) contained between the lines of
these bearings is 11° 15'x3, or 33° 45'.

§ b. Unless the problem were embraced in a right angled tri-
angle, the value of C must be found, and B known (§ 32. § b.) before
that of b is determined. C is found (§ 109. § b.) from the proportion
(§74.) which exists between certain parts of every triangle ; viz.,
a : sin. A : : c : sin. G. Then C+A — 180°=B; and then, sin. G
: c : : sin. B : 6, affords the solution required.

EXAMPLE II.

§ 115. A ship was steering N. and sailing 10 knots per hour ; she
made a point of land bearing N. E. by N. ; two hours afterwards
the same point bears E .by S. What is the distance of it from the
ship?

Ans. 12.+ miles.

§ a. The solution of this problem depends upon the principles
which Example 1 involves. The angle (A) between the point of
land and the ship's course is three points, or 33° 45'. The angle
(C) between the bearing of the ship from the point of land at each
station, is six points, or 67° 30* . And the angle (B) between the
second bearing of the land and the line upon which the ship sailed,

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T6

PLANE TRIGONOMETRY.

'£Xq

ta

W.

4»

is seven points, or 78° 45* ; or it is A +C— 180=78° 45'. Then
the proportion of sin. C : c : : sin. A : a, determines the value of
the part required.

EXAMPLE III.

§ 116. Several boats, intending to cut a ship out from under
the guns of a castle, wish to know how far she is off from the castle,
which bears N. i E. 917 fathoms from a watch-tower, that is
491 fathoms N. W. from the ship. '
Ans. 1285.7 fathoms.
N

$ a. The solution of this
problem involves in it the
principles demonstrated § 77,
by which either angle (B or
C) is found ; and then the
length of the required side
(a) is determined by the
ratio between the sides and
sines of opposite angles.

§ b. The parts given are,
the distances (6 & c) of
the watch-tower (A) from
the ship and castle, and
the angle (A) contained at
the watch-tower between
the lines of bearing from
it to the castle and ship.
The angle (A) at the watch-

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PLANE TRIGONOMETRY. 77

towr contains 111 points, or 129° 22' 30". The sum of the angles
B and C is found by subtracting A (§32.) from 180°. And half
their difference is determined by the proportion (§77.), b+c :

tangent Z :: ba> c : tang. _ — _.

§ c. The value of B and of C is then known by § 77. § h. ; and
the solution completed by evolving the value of a from the propor-
tion, — sin. C : c : : sin. A : a.

EXAMPLE IV.

§ 117. Wishing to know the height of a Chinese pagoda that
stood in a plane, I took the altitude of the sun from an artificial
horizon, and at the same time the length of the shadow cast by the
pagoda was marked and measured*

The sun's corrected altitude was 21° 10'.
The length of the shadow 347 feet.
Ans. Height of pagoda 134.3 feet.

§a. The pagoda is supposed
c to stand upright. The angle (A)
.^.-*"f1 which is formed at the end of
the shadow by an imaginary line
drawn thence to the top of the
pagoda, is 21° W, or whatever
be the altitude of the body which

causes the shadow. The solution of this problem may be conducted
according to the principles shown § 75. § c.

§ b. The parts given are the length (c) of the shadow, the angles
A and B. The latter is known from the primitive condition of the
triangle, which is right angled.

§ c. Making the given leg radius, the other leg (a) or height of
the pagoda, becomes tangent to the angle (A) of altitude. The value
of tne required part (a) is evolved by means of the proportion,
rad. : c : : tang. A : a.

example v.

§ 118. A road crosses a mountain, the base of which is nine miles
across. The distance from the foot to the top of the mountain, is
4.5 miles on one side, and 5.3 on the other.

What is the angular ascent of the rbad on each side of the moun-
tain, and how high is the top above the base ? „
Ans. Angle of ascent on one side=^H? 3tfc\ «2 ' • - ' • ' ^
Angle of ascent on the other=2fl? 1W»'-Z $7 -V i . 3 c
Height of the mountain 1.94 miles, -f.^?: .
§ a. This problem involves in the solution of it, the principles

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78 PLANE TRIGONOMETRY.

demonstrated § 79. and § 75. The parts given are the three sides

^ a ' ' §&. The operation for

finding the ascent of the

^^T^^^ road, or the angles A and

2 - j ^^^-^ C, is a repetition of the me-

^s^ \ ^*^^ thod shown in Case 5, §11 3,

a-^- jTft ° for finding C and B. Bj>,

the perpendicular upon the
base (b) of the mountain, shows its height ; the value of it is deter-
mined (§ 75.) after the angles A& C are determined. AC : cm a

i:c+a : 2pb; and (§79. §d.)p b +^-=pC; and (§75. §6.)

2

a : rad. : : p C : cos. C ; and (§ 74.) c : sin. C : : a : sin. A ;

and rad. : a : : sin. C : B p.

EXAMPLE VI.

Wishing to ascertain the breadth of a river, I observed that a
cocoa-nut tree, which stood at the water's edge on the opposite
bank, bore west of me ; then after having walked due south 194
yards, the same tree bore N. W.

How wide was the river ?

Ans. 194 yards.

EXAMPLE VII.

Wishing to know the height of a mountain in South America, on
the morning of the 22d December, 1832, between eight and nine
o'clock, I measured with a sextant the altitude of the sun above
the top of the mountain; it was 19° 40' 15". The apparent alti-
tude of the sun, measured at the same time from an artificial hori-
zon, was 41° 47' 45". Then going from the mountain, I measured
4420 fathoms, and the next day, when the top of the mountain was
again in the plane of the sun s azimuth circle, the sun's altitude
above the top of the mountain, and from an artificial horizon taken
as before, was, from the horizon = 40° 21' 30", above the moun-
tain — 25° 14'

What is its height?

Ans. 3564.1 fathoms, or 21384 feet 7 inches.

EXAMPLE VIII.

Supposing that a ray of light is not refrangible, that the earth is
a perfect sphere, that its diameter is 7924 miles, and that the height
of Chimborazo is 21384 feet, How far can the top of this mountain
be seen from the surface of the earth ?

* The mean diameter of the earth is about 7923.57 miles.

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SPHERICS.

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SPHERICS.

$ 119. Spherical Trigonometry is one of the principal branches
upon which the science of navigation is founded. The solution of
all problems for finding latitude, longitude, variation, etc., from data
obtained by means of observations made upon celestial objects,
depends upon the principles of spherical trigonometry.

§ a. As the figure of the earth is an approximation to that of a
sphere, the whole science of navigation, properly speaking, is based
upon the doctrines of the sphere.

$ 120. The figure of a sphere may be generated by the revolution
of a semicircle about its diameter, as an axis. A perfectly round
globe or ball is a sphere.

$ 121. Trigonometiically speaking, all lines upon a sphere are
curved lines. These lines are either circles, or arcs of circles. Cir-
cles are either great or small.

$ 122. A great circle is concentric with the sphere.

$ a. The shortest distance between two points on a plane ($ 8.)
is the straight line that joins them ; and the shortest distance on a
sphere, between two points, is the arc of a great circle, that joins
them.

$ b. Every great circle has two poles.

§ 123. The poles of a great circle are two points on the surface
of a sphere, that are diametrically opposite to each other, and are
equidistant from all points at the circumference of their circle.
Consequently each pole is 90° from the circumference of its circle.

$ a. The north and south poles are the poles of the equator. The
equator is 90° from either pole.

§ 124. The straight line, which, passing through the centre of a
great circle, joins its poles, is the axis of that circle.

§ 125. All great circles cross each other in points diametrically
opposite. Therefore, •

$ a. Two great circles cannot be parallel ; but they divide each
other into arcs of 180° each.

$ b. The space contained between the intersecting halves of two
great circles, is called a lune.

§ 126. One great circle is perpendicular to another, when the two
cross each other at right angles.

§ a. When two great circles cut each other at right angles, the
axis of either lies along the plane of the other; and they pass
through the poles of each other.

M

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82 SPHERICS.

«$ 127. A secondary is a great circle that crosses another, perpen-
dicularly.

$ a* A meridian of longitude is secondary to the equator.

§ 128. Every great circle divides its sphere into two equal parts,
as the Equator does the earth.

$ 129. A small circle divides the sphere into two unequal parts ;
as the Tropic of Cancer, or of Capricorn, or as a parallel of latitude,
does the earth.

§ a. If every point at the circumference of a email circle be equi-
distant from either pole of a great circle, the small is parallel to the
great circle.

§ b. The centre of a small circle is in the axis of th« great circle
to which it is parallel.

$ 180. The radius of a small circle, is the sine of the arc inter-
cepted between its circumference and the pole of the great circle,
to which it is parallel.

$ a. The radius of a parallel of latitude is the co-sine of its own
latitude.

$ 131. All spheric, like all rectilineal, angles, are either acute
right, or obtuse ; and their values are expressed under the same de-
nominations of degrees ( ° ), minutes ( ' ), and seconds ( " )•

§ 182. Two arcs of great circles, like two straight lines, (§ 25 &
§ 27.) that cross each other, make the vertically opposite angles
equal to each other, the two angles that are on the same side of
either, equal to two right angles, and the four angles together equal
to four right angles.

§ 133. A spherical angle is at the pole of the circle, upon which
it is measured.

$ a. And the arc of this circle, which is intercepted between the
two arcs forming the angle, is its measure.

$ b. The two circles which form an angle are secondaries ($ 127.)
to that upon which the angle is measured.

§e. This circle ($ 126. $a.) passes through the poles of its se-
condaries.

§ d. The distance between the poles of two great circles that
form an angle with each other, measures that angle.

$ 134. Every spherical, like every plane triangle, has three sides
and three angles, and any two of the former are greater than the
third.

§ 135. The sides of a spherical triangle are arcs of circles. Their
values are always expressed in degrees ( ° ), minutes ( ' ), and se-
conds (").

§ 136. Plane triangles are unlimited as to the value of their sides ;
the sum of the three sides of a spherical triangle is less than 360°.
For if two arcs that form an angle be each 180°, they will intersect
each other ($ 125.) in another point, that is diametrically opposite
to their angular point; then, if a third arc cross these two, it will
divide the lune (§ 125. § b.) into two triangles, and be a side to each.
This third side is less (§ 134.) than the sum of the two other sides
of either triangle, and the sum of these four sides, by supposition,

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8PHERIC8. 83

b two semicircles, or equal to 360° ; therefore the sum of the two
sides of either triangle plus the third side is less than 300° ; and so
with any spherical triangle.

§ a. The minimum of the sides of a spherical triangle is without
limits*

$ 137. The sum of the angles of a plane triangle is equal to two
right angles ; the sum of the angles of a spherical triangle, in all
cases, is greater than two, but less than six, right angles.

§ a. The exterior angle of a spherical triangle (§ 132.), therefore
is not, as in a plane triangle, necessarily equal to the sum of the
two remote interior angles of the triangle.

§ b. Most of the other propositions of elementary geometry which
•how the ratio, relations, or equality between the parts of plane tri-
angles, are likewise applicable to spherical triangles.

$138. Two angles, or two legs, of a spherical triangle, are alike,
or of the same affectum, when both of them are either acute or
obtuse.

$ 139. In spherical, as in plane triangles (§ 63.), the greatest side
and angle subtend each other ; so also do the mean and least.

$ a. An angle and its opposite side are not always of the same
affection.

$ 140. Spherical, besides being, as plane triangles, divided into
equilateral, isosceles, and scalene, are either right angled, quadrantal,
or oblique.

$ 141. A right angled spherical triangle has one right angle.

$ a. The side subtending the right angle In the hypotheouse ; but
it is not, as in a plane triangle, necessarily the greatest side.

$ b. If the hypothenuse be less than 00°, the legs are of the same
affection ; and the oblique angles also.

$ c. If the hypothenuse be greater then 00° the legs are unlike,
and the oblique angles also.

$ J. A leg of a right angled spherical triangle, and its opposite
angle, are always of the same affection.

$ 142. A quadrantal triangle has a side that is a quadrant, or
90°.

$ 143. An oblique spherical triangle has neither a side, nor an
angle of 90°.

$ 144. In plane triangles, the sides, and the sines of their opposite
angles, have the same ratio to each other; in spherical triangles, the
sines of the opposite sides and angles are proportional,

$ 145. In plane triangles ($ 1(J7.), one side, at least, must be among
the three parts that constitute the data necessary for finding the other
parts ; in spherical triangles any three parts are data sufficient for
determining the other parts. Therefore—

§ 146. In spherical trigonometry, the three sides of a triangle can
be determined by having the three angles as data ; and the reverse.

$ 147. For the purpose of facilitating, by logarithmic calculation,
the solution of problems in spherical trigonometry that are included
in oblique triangles, spherical trigonometry is here divided into right
and oblique angled trigonometry.

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84 SPHERICS.

$ a. All the problems that come under the several cases, two ex-
cepted, of oblique spherical trigonometry, may be solved by reduc-
. ing them to analagous cases in right angled trigonometry ; this is
done by means of drawing a perpendicular from an angle upon its
subtending side, of the proposed triangle, and from such an angle,
that the proposed oblique triangle will be divided into two right
angled triangles, one of which shall contain two of the parts given in
the problem.

§ 148. Every problem in oblique trigonometry may be solved
without the direct intervention of a perpendicular, and of right an-
gled triangles.

§ a. The process of calculation is shortened by solving the pro-
blems without conducting the operation upon the principles of right
angled triangles. For this reason spherical trigonometry is here
divided into right and oblique trigonometry.

$ 149. Problems comprising as data the three angles, or the three
sides of an oblique triangle, do not admit of the intervention of right
angles for determining the unknown parts ; because, in neither case
can a perpendicular be so drawn in the proposed triangle, that two
of the given parts will fall in either of the right angled triangles, or
on either side of the perpendicular.

§ 150. By some, problems that depend upon quadrantal triangles,
are included in a separate division of spherical trigonometry, called
" quadrantal trigonometry." But knowing three parts, the other
three of a quadrantal triangle may be determined by the rules for
finding with similar data, like parts that are unknown in a right an-
gled triangle.

§ a. In arranging for calculation the trigonometric functions of
the parts in a problem included in a quadrantal triangle, the less
must be called angles, and their adjacent angles, legs; the supple-
ment of the angle that subtends the quadrantal side, must be called
the hypothenuse, and the quadrantal side, radius.

$ 151. The side and angle that are opposite in a quadrantal trian-
gle, are of the same affection.

$ 152. The relations between the parts of a triangle in plane, and
the parts of a triangle in spherical, trigonometry, are similar; and
the formula) for the solution of problems in the one, are analogous
to the formula) for the solution of similar problems in the other.

$ a. As much, with regard to the sides, can be determined from
the three angles of a plane triangle, as can be determined from the
three angles of a spherical triangle. In the former case, the trian-
gle can be determined in species; and in the latter, the sides are de-
termined in degrees and minutes of arcs, which arcs, in absolute
length, may be infinitely great or small. And the number of miles,
or of any units of a positive quantity, contained by any one of these
arcs, depends upon the relations of other quantities to it, and not
at all upon the relations between it and the other parts of its tri-
angle.

f ft. In order then to arrive at absolute values for the sides and
angles of a triangle which are submitted for calculation, a third

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8PHERIC8. S5

quantity, that will answer the purpose of a common measure for
lines and angles, or arcs, must be brought into consideration.

$ c. And this common measure is the radius of a circle. When
we say that a side of a spherical triangle is an arc containing a cer-
tain number of degrees and minutes, we have no reference to the
absolute length of this arc, or to the number of inches, or fathoms,
or miles, contained in it. It may be an arc whose radius is in-
finitely great or small. But if the length of its radius in inches, or
miles, be known, the absolute length of the arc also in the denomi-
nations of the same dimensions is determinable. But this belongs
to another branch of mathematics, which is not relevant to the pur-
poses for which this treatise is designed.

$ 153. The analogy between the formulae for the solution of pro-
blems of like data and quaesita in plane and spherical trigonometry,
grows out of the resemblance between the parts of a plane and
a spherical triangle.

$ a. Suppose that the sides of a spherical triangle should remain
of the same absolute length, and that the radius of their sphere be
increased ad infinitum. The angles at the centre of the sphere,
which these sides subtend, and the number of degrees contained in
these sides, or arcs, will be decreased until they reach the ultima-
tum, when the surface included by these three sides becomes a
plane, the triangle a plane triangle, and a finite portion of infinity.

RIGHT ANGLED SPHERICAL TRIGONOMETRY.

§ 154. In the trigonometrical solution of right angled spherical
triangles, there are five parts, (three sides and two angles,) any
one of which may be the unknown and required part of the pro-
blem.

§ a. The right angle is ever known from the condition of the
triangle. In calculation the right angle is not called zpart.

$ 155. Any two of the five parts of a right angled spherical tri-
angle being known, the rest are determinable by means of trigono-
metrical calculations.

$ 156. In right angled spherical trigonometry, there can be six
cases of two different parts as data, viz. :

1. The hypothenuse and a leg.

2. The hypothenuse and an angle.
a. The two legs.

4. The two angles.

5. An angle and its opposite leg.

6. A leg and its adjacent angle.

§ 157. Lord Napier has given two rules, by which every problem
that can occur in any of these several cases, may be solved. But
he has not made known the process of reasoning, by which he ar-
rived at the conclusions, whence he deduced these rules ; nor have
mathematicians been able to follow the steps of this bold reasoner.

$ a The truth of his, called the Catholic Proposition, is sufficiently

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86 RIGHT ANGLED TRIGONOMETRY.

established by practice, and by its utility, and may therefore be ad-
mitted as an axiom or received truth.

$ 158. In his Analytical Treatise on Plane and Spherical Trigo-
nometry, Dr. Lardner observes, " We have thus established Napier 1 a
rules, by proving separately all the several cases which they in-
clude. There is no independent or general demonstration of these
remarkable theorems, nor is it easy to conceive the process of mind
by which their illustrious inventor arrived at them. Professor
Woodhouse justly observes, that there are not, perhaps in the whole
compass of mathematical science, rules which more completely at^
tain that which is the proper object of rules, namely, brevity and
facility of computation. He might have added, that few, or per-
haps no theorems equally general, make such an immediate and
permanent impression on the memory."*

§ 150. The Jive parts of a right angled triangle, are called, in
Napier's rules, the circular parts.

$ a. The circular parts are the two legs, and the complements of
the hypothenuse and its adjacent angles, instead of these three parte
themselves.

• $ 160. The right angle ($ 154. $ a.) is thrown out of considera-
tion ; and the five circular parts join each other.

§ a. The complement of the hypothenuse joins the complement .
of each of the two angles; each of which angular complements
joins its adjacent leg, and the two legs join each other.

$ 161. In every problem two of these parts are given and a third
is sought They are named from their relative position with regard
to each other. *

§ a. One of them is the middle part; the two others are extremes ,
either conjunct or disjunct.

§ 162. If the parts given and the part sought join each other in
circular order, the first and last in this order are called extremes
conjunct; and the part between them connects them together, and
is called the middle part.

$ a. The triangle ABC, represents a right angled spherical tri-
angle, right angled at A ; its hypothenuse is a, and b &, c are its two
legs. The circular parts are the leg 6, the complement of C,
do. of a, do. of B, and the other leg c.

§ ft. The three parts, a, B, c, join in circular order. The com-
plement of B is the connecting part between c and the complement
of a. The complement of B is the middle part; c, and the comple-
ment of a, are extremes conjunct.

§ c. Of the parts

B, c, b.
e, by C.
6, C, a.

C, a, B.
a, B, c.

Middle part is,

c.

b.
Complement of C.
Complement of a*
Complement of B.

Extremes conjunct are,
6, and complement of B.
c, and complement of C.
b 9 and complement of a.
Complements of C and B.
c, and complement of a.

• See Analytical Treatise on Plain and Spherical Trigonometry, by the Rev.
Dfonyatue Lardner.

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8PHBEIC8.

«

"\\ Complement of C/

SL

■qttq

/

^o^'-~

§ 163. If the parts given, and the parts sought, be not adjacent to
each other in the circular order, that part which stands alone, or is
not adjacent to either of the two other parts, is the middle part, and
the two others are extremes disjunct

$ a. The three parts, a, c, b, do not join each other in the circu-
lar order; the complement of a is not adjacent to either of the two
other parts, being separated from them by the complement of the two
acute angles, C & B; it stands alone, and is therefore (§ 163.) the
middle part, and the two other parts that do join each other, are the
extremes disjunct.

Middle part is,
Complement of B.

§6. Of the parts
B, 6, C.
c, C, a.
c, C, B.
6, a, B.
b.

a, c,

Extremes disjunct are,
6, and complement of C.
Complements of C and a.

and complement of B.
Complements of B and a.
b and c.

Complement ofC.

b.
Complement of a.
§ 164. Napier's rules establish an equality of ratio between ra-
dius, the sine of the middle part, and certain trigonometric functions
of the extremes. They are,

$ 165. The product of radius and sine of the middle part, is equal
to the product of the tangents of extremes conjunct. And,

§ 166. The product of the co-sines of extremes disjunct, is equal
to the product of radius and sine of the middle part.

§ 167. Every problem in right angled spherical trigonometry can
be solved either by the one or the other of these two rules.
§ a. They have been put in the following mnemonic form ;
" The product of radius and middle part's sine
Equals that of the tangents of parts that combine ;
Or, that of the co-sines of those that disjoin."

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RIGHT ANGLED TRIGONOMETRY.

§ 168. If the data and quesita be
opposite sides and angles, the problem
proposed may be solved by the pro-
portion (§ 144.) between the sines of
opposite sides and angles.

§ 169. ABC represents a right an-

ged spherical triangle upon a plane;
is the right angle, and b the hypo-
thenuse.

§ a. The circular parts, commencing
with c, and naming them in the order
around to the right, in which they
join; are c, a, cos. C, cos. 6, and

[jl cos. A

CASE I.

§ 170. Oiven the hypothenuse (6) and a leg (c).
The hypothenuse b « 74° 40'.
The leff c — 60° 14'.

To find the other parts.
§ a. 1st. To find the other leg (a).

The three parts that, in this case, comprise the data and qussitum
of the problem, are the two legs (c, a), and the hypothenuse (6).
The complement of 6, (§ 163. § a.) is not adjacent to either of the
other parts, (c, a.); therefore (§ 163.) it is the middle part, and c
and a are extremes disjunct; and ($ 166.) cos. e x cos. a ■■
Sin. 6°xRad.;* then (§ 71.) cos. c : Rad. : : sin. b° : cos. a.
c » 60° J 4'. cos. ar. co. =■ 0.304108 (=« sec. c.)

Radius 10.

b a 74° 40'. cos. (- sin. ft ) — 0.422318

Cos. a a 9.726426 — 57° 40'.

$ c. In looking in the tables for the degrees, etc., which corres-
pond to the logarithmic co-sine, 9.726426, it will be seen that
122° 1 T, the supplement of 57° 49', also corresponds to it. Which
of the two to take, is known ($ 141. $6.) by the affection of the
two legs.

$ d. The sine, tangent, secant, etc., of an arc or angle, ($ 52. § d.)
are the co-sine, co-tangent, co-secant, etc., of the complement of
that arc, and ($ 104,) the sine, tangent, secant, etc., of the supple-
ment of the same arc. Therefore,

§ c. Cor. The logarithmic value of any trigonometric function

* To obviate the necessity of writing complement, the circular ports of the
hypothenuse end two angles will be denoted in the rest of the work by writing
en (°) after the letter which standi for any one of those parts. Thus, o°, C°,
B°, stands for the compleipents of a, C, B.

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SPHERICS. SO

corresponds to two arcs or angles ; viz., an angle and its supplement.
Thus, log. sine 9.611576 = 24° 8', or 155° 52'.

§/. There is generally some circumstance connected with the
conditions of the problem, or triangle, under solution, which deter-
mines the affection of the required part, and thence the arc required ;
for every one of the sides and of the angles of a spherical triangle
(§ 136. & $ 137.), may be either greater or less than a right angle, or

§ g. To find the value of a. In this case A is the middle part.
It is between, and joins, the two other parts 6° and c. Therefore
($ 162.) 6° and c are extremes conjunct And (§ 165.) rad. x sin.
A° = tang, c x tang. 6°. Then (§ 71.), rad. : tang. o° : : tang, c :
sin. A . :

Radius =10.

03-74° 40' co-tang. = 9.438059 (= tang. 6°, $ 52. $ d.)

c=60° 14' tang. 10.242655

Sin. A°(=co-ein. A. (§ 52. § d.) = 9.680714=61° 21' 9".

§ A. 61° 21' 9", and not its supplement, is known ($ 141. $6. &
§ d.) to be the required value of the angle A.

$ t. If the angle (C) opposite to (c) one of the given sides be re-
quired from the data of this case, the' ratio ($ 144.) between the
sines of opposite sides and angles, will determine the value of it,
viz. : Sin. 6 : sin. B (= ($ 62.) rad.) : : sin. c : sin. C.

$/. Or the value of C may be determined by means ($ 166.) of
the Catholic Proposition ; c ($ 163.) is the middle part, and b° and
O are extremes disjunct; and ($ 166.) cos. o° x cos. C° = sin. c
Xrad.$ wherefore (§71.) cos. b° : rad. : ; sin. c : cos. C°;
which is the proportion under § *., stated under different denomi-
nations, but the same ultimately ; for the co-sine 6° and cos. G°
(§ 52. § d.) are the same as sin. 6 and sin. C.

6=74° 40 sin. ar. co. = 0.01 5741 =co-sec. (§ 101. § b.)
Radius =10.

c=60° 14' sine = 9,938547

Sine C « 9.954288=64° 10' 14" (§ 141. § d.)

N

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00

RIGHT ANGLED TRIGONOMETRY.

CASE II.

§ 171. Given the hypothenuae (ft)
and an angle (C),

$—119° 39'

C— 104° 01'
To find the other parts.

§ a. 1st. To find the other angle (A).
The two angles (A, C,) are adjacent to
the hypothenuae (ft), which is therefore
($ 162.) the middle part, and the two
angles (A, C,) are extremes conjunct;
then ($ 170. § g.) tang. C° : rad. : :
sin. 6° : tang. A° ; and ($ 52. $ d.) co-
tang. C : rad. : : cos. ft : cotang. A.

C— 104° 1' co-tang. ar. co. — 0.602601— tang. ($ 100. § a.)
Radius —10.

6=119° 89' cos. - 0.604842

Co-tang. A— 10.207088— 26° 46' 86".

§ ft. The value of A is known to be less than 00° (§ 141. $ e«),
because C is greater than a right angle.

$ e. 2d. To find a, which ($ 141. § d.) is also less than a right
angle. The three parts, a, C°, and 6°, join in circular order; and
($ 162. $ c.) C° is the middle part, and 6° and a are conjunct ex-
tremes. Then (§ 170. §g.) tang. 6° : rad. : : sin. C° : tang, a; and
($ 52. § d.) co-tang. ft : rad. : : cos. C : tang. a.

ft— 110° 80' co-tang. ar. co.— 0.244700— tang. ($ 101. $6.)

Radius =10.

C— 104° 1' cos. « 0.384182

Tang, a = 0.628801-28° 2' 68'

$rf. 3d. To find e; c(§ 141. $rf.) is greater than a quadrant The
method for finding the value of e in this case, is similar to that
($ 170. $ j.) of fining C, Case 1.

Rad. : sin. ft : : sin. C : sin. c.

Radius —10.

ft— 110° 30' sin.— 0.030052

C— 104° V sin. — 0.086873

Sin. c- 0.025025-122° 31' 18".

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SPHERICS.

01

case ni.

$ 172. Given the two legs (a, c),

<t»81° 16'

c—74° 14'
To find the other parts (A, C, b.)

5 a. 1st To find the hypothenuse (6);
it is less than a quadrant, because ($ 141.
$ 6.) the legs are of the same affection.
The three parts, a, b°, and c, do not join
each other in the circular order ; 6° stands
alone: then ($ 163. $a.) a and e are dis-
junct extremes, and b° is the middle part ;
and ($ 170. $ a. & $ XLIX. Alg.J radius :
cos. e : : cos. a : sin. 6°=>(§ 52. $ a.) cos. 6.

Radius »10.

e—74° 14' cos.= 0.434122

a— 31° 16' coe.^- 0.031845

Cos. b — 0.365067=76° 84' 12'

$ b. 2d. To find an angle (C); it is less (§ 141. $ d.) than 00°.
The three parts C°, a, c, join in circular order; and (§ 162.) C° &
c, are conjunct extremes, and a is the middle part : then (§ 165. &
$71.) tang, e : rad. : : sin. a : tang. C°=co-tang. C.

c=»74° 14' tang. ar. co.= 0.450777=co-tang.

Radius =10.

<i»31° 16' sine — 0.715186

Co-tang. C= 9.165963=81° 30' 47".

$ c. 8d. To find the other angle (A) ; it is also less than 00°, and
it is an extreme conjoined to a by c, which (c) is the middle part;
then ($ 6.) tang, a : rad. : : sin. c : tang. A°— co-tang. A.

a— 31° 16' tang. ar. co.= 0.2 16659= co-tang. ($ 101. $6.)

Radius =10.

c— 74° 14' sin. = 0.088345

Co-tang. A= 10.200004=32° 15'.

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92 RIGHT ANGLED TRIGONOMETRY.

CASK IT.

§ 173. Given the two angles (A, C),

A=31° 66'

C=lll° 11'
To find the sides a, b, and c.

§ a. 1st. To find the hypothenuse
(6); it is (§ 141. §c.) greater than a
quadrant; it connects A° and 0° in the
circular order, which are conjunct ex-
tremes ; and (§ 171. $ a. § XLIX. Alg.)
rad. : tang. A° : : tang. C° : sin. b° ; or,
rad. : co-tang. A : : co-tang. C : cos. b.

Radius =10

A=31° 56' co-tang. =10.205336

C=lll°ll' co-tang.= 0.588316

Cos. 6= 9.793652=128° 26' 53".

§ b. 2d. To find the leg c ; it and A° are extremes disjunct ; and
($ 166. & § 71.) cos. A° : rad. : : sin. C° : cos. e ; or,
Sin. A : rad. : : cos. C : cos. c.

A=31° 56' sin. ar. co*= 0.276600=co-sec. ($ 101. $6.)
Radius =10.

C=lll° 11' cos. = 0.557032

Cos. c= 0.834532=183° 5' 32" ($ 180.)

G and c ($ 141. § d.) are of the same affection.

$ c. 3d. To find the other leg a ; it is less than a quadrant, and a
disjunct extreme ; A is the middle part, and (§ 6.) cos. C° : rad. : :
sin. A° : cos. a ; or,

Sin. C : rad : : cos. A : cos. a.

C=lll° 11' sin. ar. co.= 0.080384=co-eec. ($ 101. $*.)

Radius =10.

A=31° 56' cos. = 9.928786

Cos. a= 0.050120=24° 28' 18".

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SPHERIC8.

98

CASK V.

§ 174. Given an angle (C) and its x
opposite leg (c),

c=45° 19'

C— 54° 49'
To find the other parts (a, b f A).

$ a. This is called the " doubtful
ca*e," because there is nothing in
the condition of any problem that
falls under this case, which will de-
termine the affection of the part
whose value is sought.

§&. 1st. To find the other leg
(a) ; it is the middle part, connect-
ing the conjunct extremes, C° and
e; and ($ 172. $6.) rad. : tang. C°
: : tang, c : sin. a: or,

Rad. : co-tang. C : : tang, e : sin. a.

Radius 10.

C— 54° 4V co-tang.— 9.848181

c— 45° 19' tang. —10.004801

Sin. a— 9.852982=45° 37' 54".

$ c. 2d. To find the hypothenuse (b) ; it is a disjoined extreme,
and is found ($ 170. §/.) cos. C° : rad. : : sin. c : cos. b° ; or,
Sin. C : rad. : : sin. e : sin. b.

C=54° 49' sin. ar. co.= 0.087612= co-sec. (§101. $6.)
c=45° 19' sin. — 9.851872

Radius =10.

Sin. 6— 9.939484*60° 27' 2".

$ d. 3d. To find the other angle (A) ; it is a disjunct extreme ;

*

and (§ 173. § b.) cos. e : rad. : : sin. C° : cos. A° ; or,
Cos. e : rad. : : cos. G : sin. A.
c=45° 19' cos. ar. co.= 0.152929— sec. ($ 101. $ b.)
Radius —10.

€—54° 49 / cos. — 9.760569

Sin. A- 9.913498-56° V 60".

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04 RIGHT ANGLED TRIGONOMETRY.

CASK VI.

$ 175. Given a leg (c) and its adjacent
angle (A),

c=35° 15' 7"

A=74° 47'
To find the values of the other parts

(6, C, a).

$ a. 1st. To find the value of the hypo-
thennse (b) ; it is ($ 141. § 6. & $ d.) less
than a quadrant; and the problem is the
converse of $g. 9 Case I, (§ 170.). Tang.
c : rad. : : sin. A° : tang. 0° ; or,

Tang, e : rad. : : cos. A : co-tang. b.

c— 35° 15' 7" tang. ar. co.— 0.150715— cot. ($101. $6.)

Radius —10.

A— 74° 47' cos. = 0.419080

Go-tang. 6— 9.569795—69° 37' 37".

$ b. 2d. To find the value of the angle G. This is the converse
of $ b. 9 Gase IV. ($ 173.), Radius : cos. A° : : cos. c : sin. C° ;
or,

Rad. : sin. A : : cos. e : cos. G.

Radius —10.

A»74° 47' sin. — 0.984500

C— 85° 15' 7" cos. — 0.912022

Cos. C— 9.896522=38° W 7".

§ e. 3d. To find the other leg a. The calculation for this is the
converse of § c, Gase III., (§ 172.)

Tang. A° : rad. : : sin. c : tang, a ; and tang. A°—

Co-tang. A=74° 47' ar. co.— 0.565421— tang. ($ 101. § b.)

Radius =10.

c— 35° 15' 7" sin. — 0.761305

Tang, a— 10.326726—64° 46' ($141. $<*.)

$ 176. The rules which have been given for the solution of any
problem that can occur in right angled spherical trigonometry, are
good in theory ; but instances may occur in practice, wherein they
will not yield the most accurate result. This apparent defect arises
from the trigonometric functions by which certain arcs, or angles,
that enter into calculation, are denominated. For instance : If the
required part be a very small arc or angle, and the result give its log.

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SPHERICS. 95

co-sine, the solution is not a good one ; because a small error in the
log. cos. will produce a much larger one in the arc or angle. This
arises from the different manner in which an arc and its sine in-
crease*

$ a. By referring to a logarithmic table, even of six digits in the
mantissas, it will be observed that an arc of 89° 51' may increase
18*, or to 00° 9', and that the value of its sine will only vary
.000001.

§ 6. The same thing is true of the co-sine of the complement of
such an arc.

$ c. For these reasons, the sine of an arc near 90°, or the cos. of
one near 1', should not be used in trigonometrical calculation.

$ 177. The value of sine and co-sine of such arcs may be sub-
stituted by their equivalents in other trigonometric functions.

$ 178. The principles from which such equivalents are obtained,
are involved in $ 75.

$ a. The elements of trigonometry are chiefly contained in the
principles of $ 43. and $ 75.

$ 179. An equivalent for the sine, co-sine, etc., of any angle, may
be found by means of the principles developed under $ 75.

$a. Let the proposed angle (C) be
made an oblique angle in a right an-
gled triangle (ABC); and from it,
with its adjacent siae (G B) as ra-
dius, describe the arc H B, and draw
the sine (H^p) of the angle (G) pro-
posed; Cp is its co-sine, A B its tan-
gent, and C A its secant.

§b. Now, instead of naming these
sides by letters, denominate them by
the trigonometric functions whose
offices they fill ; thus, calling Ho, gj 4
sin. C ; C p 9 cos. C ; A B, tang. C ;
and so on; we have ($75. and §73.),

Sin. O : cos. C : : tang. O : radius ; or, sin. C ^ Co *-Cxtang.C

radius.

$c.Co8.C:sm.C::rad.:tang.C;or,eos.C« Sk ' Cxi ^*

tang. G

$ d. Equivalents for other trigonometric functions may be found
after a similar manner; and the same relations exist between the
trigonometric functions of plane and spherical triangles. There-
fore—

§ 180. The logarithmic sine of any angle is equal to its log. co-
sine and tangent, minus radius, or 10. And —

§ 181. The log. cos. of any angle is equal to its log. sine and rar
dius, minus the log. tang, of the same angle.

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OBLIQUE SPHERICS.

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OBLIQUE SPHERICS.

§ 182. In every oblique spherical triangle, there are six parts,
any three of which constitute sufficient data (§ 145.) for determin-
ing the value of any of the remaining parts by means of trigonome-
trical calculation.

$ 183. Every problem in oblique spherical trigonometry is com-
prised in one of the six following cases ; in which the data are,

1st, Two sides and angle opposite to one of them.

2d, Two angles and the side subtending either.

3d, Two angles and the side between them.

4th, Two side* and the angle they include.

5th, The three sides.

6th, The three angles.

§ 184. When any one of the four first cases, comprises the data
of the problem, it may be solved by right angled trigonometrical
operations; for this purpose the proposed triangle (§147. § a.)
must be divided into two right angled triangles.

$ a. The perpendicular arc must be so drawn, that two of the
given parts must be on the same side of it, and consequently, in
one of the right angled triangles, into which this arc divides the
triangle of the problem.

$ 185. The rules in right angled trigonometry, for determining in
species an unknown side or angle, (i. e. whether it be greater or
less than 90°), do not hold good in all cases in oblique trigo-
nometry.

§ 186. The opposite sides and angles of oblique spherical tri-
angles, are not necessarily ,of the same affection.

$ a. But if the greatest side be greater than a quadrant, it and its
opposite angle are of the same affection.

§ 187. The converse of § 186. § a. does not hold good ; for the
sides (§ 136. § a.) of a spherical triangle may be indefinitely small,
and the sum of the angles (§ 137.) being greater than two right
angles, an obtuse angled spherical triangle may be formed of these
indefinitely small sides ; in which case an obtuse angle and its sub-
tending side will be unlike.

§ 188. In most cases though, there are circumstances connected
either with the condition of the problem, or of the triangle it in-
volves, by which the species of the part required may be determined.
Some of these circumstances are mentioned below.

§ 180. If an angle of a triangle be greater than 90°, an arc drawn
from it, perpendicular to the subtending side, will fall within such
triangle.

§ a. If the angles adjacent to the base, be of the same affection,
the perpendicular falls upon it, also within the triangle.

§ 190. If all the angles of a triangle be not of the same affection,
and a perpendicular arc be drawn from one of the angles which is less
than 90° upon the subtending side, it will fall without the triangle. ,

Digitized by VjOOQIC

100 TRIGONOMETRY.

$ 101. An opposite side and angle of a spherical triangle cannot
be unlike, unless the sine of this side be greater than the sines of the
other sides.

$ a. The sine of the angle opposite to such a side is also greater
than the sines of the two other angles.

$ 6. In an isosceles triangle, each side and its opposite angle are
alike.

$ 102. If an obtuse side be opposite to an acute angle in any tri-
angle, the two other sides are unlike; and ($ 101.) they and their
opposite angles are of the same affection; consequently, the two
angles are unlike.

| a. If an obtuse angle subtend an acute side, the other sides
and angles are all alike.

§ 103. Whether the perpendicular fall within or without the tri-
angle, the less segment of the base is next to that angle which, of
the two adjacent to the base, is the nearer 00°.

$ a. When the perpendicular falls within the triangle, the ram of
the segments equals the base.

$ 6. And when it falls without the triangle, the base equals the
difference of the segments.

$ 104. If a side and its subtending angle be each less than 00°,
a smaller side and its opposite angle, of the same triangle, must be
(§ 130.) of the same affection.

§ 105. If the sum of two angles of a triangle be less than 90°, the
third angle ($ 137.) is greater than a right angle.

$ 196. By writing co-tang, for the arithmetical complement
(§ 101. $ b.) of the tangent; sec. for the arithmetical complement of
cos.; cosec. for the arithmetical complement of sine, etc., greater
simplicity and uniformity of calculation will be effected.

§ a. The learner will have observed by the reference ($101. $6.)
appended to the arithmetical complement of sines, etc., in some of
the calculations in right angled spherical trigonometry, that by
taking from the tables the cosec. of an arc, he obtains at once the
arithmetical complement of the logarithmic sine of the same arc.
So, co-tang, for arithmetical complement of the tangent, etc.

case 1.

$107. Given two
sides (b & c) and
an angle (C) oppo-
site to one of them,

6*21° 17'

c«74° 14'

C— 69° 10'
To find the other
parts (A, B and a) by
right angular opera-
tions.

$ a. If the third
side (a) or its oppo-

Digitized by VjOOQIC

OBLIQUE SPHERICS. 101

lite angle (A.) be computed, the two other remaining parte are de-
terminable by means of the ratio (§ 144.) between the sines of op*
posite sides and angles.

§ b. In order that two (b & C) of the given parts, may be on the
same side (§ 184. § «.) of the perpendicular kp, it is drawn from
A, upon its opposite side a, so as to divide the triangle of the pro-
blem into two right angled triangles kp C and kp B ; the former of
which contains two of the given parts (G & b).

$c. G & B are less than 90° (§ 104.); therefore (§ 141. §6.)
G p & kp are each less than 00° ; the perpendicular upon the side
(a) adjacent to C & B (§ 189. §«.) falls within the triangle; and
makes the sum of the segments of the angle A, equal to A.

$ d. The distance (C p) of the given angle (C) from the inter-
secting point (p) of the perpendicular and the base, is called
auxiliary a'.

$ e. To compute the value of the auxiliary arc a'.

The method of doing it, is shown under Case II. ($ 171. § «.).

Tang. o'=^!l^— B (C 196.) tang, b cos. C.
cot. b x J °

$/. £=.21° IT. Tang.— 9.590562
C— 69° 10'. Cos. =9.551024

Tang. auxl. a 1 — 9.141 586—7° 52' 16"*

$ g. The value of B is calculated from the ratio (§ 144.) between
the sines of sides and of angles. Sin. c : sin. C : : sin. b : sin. B ;
sin. e ar. co. (§ 196.)— cosec. c.
$ h. e— 74° 14' cosec— 0.016655
C=69° 10' sin. —9.970635
6—21° IT sin. —9.559883

Sin. B-9.547173-20 38' 27"

$t. Now, in the other right angled triangle kp B, the value
both of c and of B is known, and the other segment (p B) of the
base can be determined; for (§ e.) tang. B />— cos. B tang. c.

& j. c— 74° 14' tang.— 0.549223
B— 20° 38' 27" cos.— 9.971 187

Tang. B p»0.520410— 73° 12' 39"

* Had. (§ 98. % £.), being always equal to 1 or to 10 in logarithmic calcula-
tions, is not written down in these formula ; its value is brought into account
by applying H to the log. index of the result, as in the case above. By omitting
rad. the formula) for calculation are rendered more compact and convenient. The
proper value will always be given rad. in calculation, by having the fog. index
of the result to consist only of one figure, as in the case above, where radius ia
taken into the account by simply writing 9 instead of 19 for the log. index of
auxl. a'.

Digitized by VjOOQIC

10* TRIGONOMETRY.

$ k. The perpendicular falling within the triangle, makes the sum
of the segments (§ 193. $ a.) equal to the base a; therefore (§/. &
jy.) a«81° 5' 55".

$ /. To find the value of A.

Sin. c : sin. C : : sin. a : sin. A.

e =74° 14' cosec. =0.0 16655

C»69° 10' sin. =9.970636

a =81° 5' 55" sin. =9.994738

Sin. A=9.982028=106° 22' 12"; A is greater

($ 195.) than 90°.

$ m. The value of A may also be determined bv the Catholic Pro-
position; for (§ 171. $ a.) cotang. C A/>«=^~=(§ 196.) cos. b

tang. C. Also co-tang. B A p=cos. c tang B.

$ n. B Ap+C Ap«A, according as the perpendicular arc Ap
falls within or without the triangle.

$o. C— 69° 10' tang. =0.4 19611
ft«21°17' cos. »9.96932l

Co-tang. C Ap=0.388934—22° 12' 51"

B— 20° 88' 27" tang.»9.575983
c— 74° 14' cos. -9.434122

Co-tang. B Ap=9.010105— 84° 9' 21"

BAp+CAp(§c.)«=A 106° 22' 12"

5 p. The process of solution by such methods of calculation as
the above, is circuitous. But cases sometimes occur when they
may be used with advantage. And in order that the process, by
which the required result is obtained from calculations conducted
upon the principles of right angled spherical trigonometry, may be
made familiar to the learner, the calculations are carried out.

§ q. By analysis and the use of a little artifice, rules are deduced,
and formula constructed, for obtaining the same result from the ap-
plication of the same principles to calculation, but by less tedious
operations. The auxiliary arcs and angles used in such methods,
are derived from the principles of the Catholic Proposition ; and
the methods themselves are nothing more than right angled sphe-
rical calculations, rendered less circuitous in execution by previous
combinations, eliminations, and substitutions of the parts that are
contained in the two right angled triangles, into which the oblique
one of the problem is divided.

$ r. To find the value of A by the help of auxiliaries. Let
the angle B A p be called auxiliary A" ; and let auxiliary A', be the
angle (C Ap) which is in the right angled triangle in which two
(5, C,) of the given parts of the primitive triangle are contained.

Digitized by VjOOQIC

OBLIQUE SPHERICS. 103

§ *. According to § m. co-tang. A'— cos. b tang. C.
$ t. Now, (§ 165. & $ XLV. Alg.) jI^Af,.

Cos. A' , Tang. As Cos. A" . ' Cos. A"

—t- ; also 2-j — - — — - ; wherefore — : —

cot b rad. cot. c cot. c

Cos. A' , , A -. ~ A „ Cos. A' Cot. c

r- ; and by transposition, Cos. A — r—

cot. b J r cot. b

& ($ 101. § 6.) C ° 8 ^' ^ 0t C -Cos. A' Cot. e tang. b. Therefore,

Cos. auxiliary A"— cos. A' cot. c tang. 6. Whence the general
rale for finding the value of (A) the angle opposite to the unknown
side.

$ u. The product (§ «.) of tang, of the given angle, and cos. of
the given side that is adjacent to it, is co-tang, of auxiliary A'. And,

$ v. The product of tang, of the same side, and co-tang, of the
other given side, multiplied by cos. of auxl. A' (§*.), is cos. of the
other auxiliary A". And the sum (f 193. § a.), or difference (§ 193.
§ &.), of the two auxiliaries, gives the required angle.

$w. c— 74°14' cot. —9.450777

b —21° 17' cos. —9.969321 - - - tang.— 9.590562
0=0:69° 10' tang.— 0.419611

Cot. A'— 9.388932— 22° 12' 51" - cos. —9.966506

84° 9' 21" - Cos. A"— 9.007845

A— 106° 22' 12" ($ o.)

§x. The formula for the calculation (§w.) l * arranged in the
most convenient order for operation. The value of auxl. A' is
evolved during the process of finding that of the other A" ; and the
log. tang, and cos. of 6, are taken out at one opening of the .tables ;
so, also is the value of A' and its cos. System and method in cal-
culation should by no means be neglected. They promote accuracy
and facilitate practice. It is therefore the business of every calcu-
lator to introduce, system in his operations. The habit of arrang-
ing the several quantities in the most convenient order for calcula-
tion, contributes to accuracy, and makes verification more ready.

§y. To find the value of a, by the help of auxiliaries. Call
the segment C /», auxl. a', and the segment B/>, auxl. a".

$ z. From a train of reasoning analogous to that under $m., it ia
shown that, tang. auxl. a'— tang, b cos. C.

Rad. Cos. a'

%za. Also ( 166. & $ XLV. Alg.) r-- r ;and

* v 9 oy cos. Ap cos. b

Rad. Cos. a" 4 . r Cos. a" Cos. a' , ♦*.«„«_

i ■ — • ; therefore — r- ; and by transpo-

cos. Ap cos. c cos. c cos. b

Digitized by VjOOQIC

104 TRIGONOMETRY.

,i Cos. a cos. c , e 1A , - . N , .

sition cos. a"= r —($101. $6.) cos. a' cos. c sec. ft;

then a n a" (§ c. & §n>)=a. Whence the general role for finding
the third side.

§ * ft. The product of cos. of the given angle and tang, of its ad-
jacent side, is (§*.) tang, of auxl. a'. And,

$ z c. See. of the same side, multiplied by the product of cos. of
the other side and cos. of auxl. a' ($za.), is cos. of the other
auxl. a".

%zd. If the two angles adjacent to the base, be each less than
60°, the perpendicular falls within the triangle, and (f 193. $ a.) the
sum of the auxiliaries is the required part.

%z e. But if these two angles be unlike, the perpendicular falls
without the triangle; and then (§ 193. $6.) the difference of the
auxiliaries is the required part

$ */. c— 74° 14' cos. -9.434122

ft =21° 17' tang,— 9.590502 - - - sec— 0.080679
C— 69* 10' cos. —9.551024

Tang. cr*- 9.141586— 7° 53' 16'' - cos. =9.995872
73° 12' 89" - Cos. «"— 9.4*8679

a— 81 p 5'55"(§*<J.)

%*g. The value of the third side (a), or of its opposite angle
(A), may also be determined by another method ; but in this, the
value of the other unknown anrie (B) is necessary.
$ z A. To determine the third side (a),

Cos. of i the difference of the two angles, (B, G) ;
Is to the cos. of 4 their sum ;
As tang, of i the sum of the two sides (ft, t) ;
Is to tang, of 4 the required side.
J*t. 1 (C«B)-24°15'46" sec. =0.040162
4 (C+B)« 44°54'13" cos. -9.850215
i (ft -f c)— 47»45'30" tang. —0.041880

Tang, i a— 9.932257— 40* 82' 68"

2

«— 81° 8' 56"

$ zj. To find the value of A by a similar method
Cos. of 4 the difference of the two sides (ft, c,) ;
Is to the cos. of 4 their sum;
As tang, of 4 the sum of the two angles (A, B);
b to co-tang, of 4 the angle required.

Digitized by VjOOQIC

OBUQUB 8PHBBICS. {(ft

$ z k. i (c m 6)«26° 38' 80" sec. =»0.04S1 14
| (e + o)=47? 45' 30" cos. =9.827537
i (C+BJ-44 54' 13'' tang,-9.99S538

Co-lang. $ A=?9.874189*53° 11' 7"

■ ■' r

2

A«106° 22> 14''

These two last methods are this most common in practice.

CASE II.

->*

B*
C=

=29° Iff
i74° 46'
-150° 51' 50"

co-8ec.>

sin.

sin.

.0.810862
*9.984466
=9.687427

§ 198. Given two
angles and the side
that subtends one of
them,

B=o=29° 16'

C=150° 51' SO''

6*=74° W
To find the value of
each of the other
parts (A, c & a.)

§ a. The value of
c (§ 144.) is deter-
minable by a direct
calculation ; Sin. B :
sin. b : : sin. C :
sin. e.

Sin. c«9.982695=106° 4' 2"

§ 6. Suppose the perpendicular (A p) from A, upon the side op-
posite to A, to fall without the triangle. Then the two known
parts in the right angled triangle Ap C, are b, and (§ 132.) the
supplement of U. The unknown parts of the proposed oblique tri-
angle may be computed by right angled trigonometric operations.

§ c. In the triangle Ap C, (§ 197. § e.) tang. C p«=cos. C tang.
b. Also in the triangle pAB, tang, p B*=cos. B tang. c. Then
C$193. S*0 C P*P B—a-

^ (J. The value of A also, may be determined by rules of right an-
gled trigonometry.

§ 6. When the perpendicular falls without the triangle, the dif-
erence of the two auxiliary angles B Ap, and C A/7, is the an-
gle A.

P

Digitized by VjOOQIC

106 TRIGONOMETRY.

$ /. Bat if it falls within the triangle, B kp+C A j»— A, ($ 197.

$n.)

$ g. Co-tang, p A Cs«cos. 6 tang. C ; and co-tang. B A/>-*cot.
e tang B (§ 107. § m.). Then ($ e.) A==B A pnp A C.
$A. C«150°51'50" cos. — 0.041246 ($ c.)
6s 74o 46* tang.= 0.564022

Tang. Gp= 0.506168«72° 41' 3"

B— 20° 16' cos. « 0.040603

c«-106° 4' 2" tang.= 0.540585

Tang, jo B= 0.481278— 108° 16' 15"

Now (5 c.)pB—C />—««=. 35° 35' 12"

$t\ C«150°51'50" tang. -0.746181 (§£.)
&= 74° 46' cos. =0.410544

Co-tang. /> A C =0.165725= 81° 4C 4''

c=106°4'»" cos. =0.442111
B=20° 16 7 tang.=0.748505

Co-tang. B A />= 9. 190010=98° 49' 0"

Now ($ e.) B kp—p A C=A=17° 8' 56"

$j'. To determine the third angle (A) by the help of auxilia-
ries. The angle p A C is auxiliary A'; and the angle B kp is
auxiliary A".

$ *. Co-tang. A'=cos. b tang. C (§ g.).

$/. And ($ 166. and $ XLV. Alg.) ~ = SlP '^ ; also
* oy cos. Ap cos. C

Rad. Sin. A" . t Sin. A" Sin. A' . . _
r — = =- ; therefore =--= 7= ; and by trans-
cos, kp cos. B cos. B cos. C '

•^ . i» Sin. A' cos. B , t %M e . N ., _

position, sin. A"= = =(§ 101. § 6.)=sin. A' cos. B

see. C. Whence the general rale for finding the third angle.

$ m. The product of cos. of the given side, and tang, of its adja-
cent angte* (§ i.} is co-tang, of auxl. A'. And,

$ n. The product of sec. of the same angle, and sin. of auxl. A%
multiplied by cos. of the other angle, ($0.) is sin. of auxl. A".
And ($ 107. $ n.) A'+ A" is the required angle.

Digitized by VjOOQIC

OBLIQUE SPHERICS. 107

5 *• B=29° ltf cos.=9.940693

C=160° 61' 60" tang.=9.746181 - - sec.=0 068764

6= 74° 4C cos. =9.419544

Cot AW.165726=81°40' 4" sin.=9995391

98° 49 / Sin. A"=9.994838

A=17° 8'60"($e.)

$/>. To determine 'the value of the side (a) opposite to the un-
known angle with the help of auxiliaries* Let the segment p C be
auxiliary a 9 ; and the other segment p B, be auxiliary a",

$ q. Tang- <*' ($ c)=-cos. C tang. b.

$ r . And by a process of reasoning similar to that under Case I,
($ 197. § z a.), it is shown that sin. a"=sin. a' tang. C co-tang. B ;

forfflffg.) Tang ' A ^ Sin ' g/ - 3lfl0 Tang.A J> Sin.A- ,
,or u i*o.j_ — - __, also — -j -55^-.

.» - Sin. a" Sin. a'

tneretore, =» — — ; and by transposition, sin. a"—

Sin. A' cot. B ^ ^ ", , ,
— c «sin. a' cot- B f<ro#. C Whence the general rule

for finding the value of the side that subtends the unknown angle.

§ 9. The product of tang, the given side, and cos. its adjacent an-
gle ($ q.) is tang, of auxl. a'. And,

$ U Tang, the same angle multiplied by the product of co-tang,
the other angle, and sin. the auxl. a', is sin. the other auxiliary a".

$"» B»29° 16' cot. =0.251495

G— 160°61'50" cos.=*9.941246 - tang. =9.746181

$= 74°46' tan.=0.664922

Tang. a'=0.506168= 72° 41' 3" sin.=9.979867

108° 16' 16" Sin.a"=9.977533

a= 35° 36' 12'

Digitized by VjOOQIC

108

TKKTCNOMSTBY.

/

CASE m-

$ 109. Given two
angles* and the tide
between them.
Ail 9° 41'
B— 134° 17'
c— 36° W
To find the value of
the other parti (a, 6,
&C.)
$a. The perpendi-
__ cular (A j») being so

y^..-- * * "*"""****** Jp drawn that the two

v** - ^ given parts B, c,

be on the same side of it, falls without the triangle pit>[
and (( 198. $ «.) the auxiliary angle B A p+A, equals the auxiliary
tingle C Ap. Also the difference of the auxiliary arcs/) B &p O
($193. §&.)isthesjfea.

§ 6. If any one ot the unknown parts be determined, the two
others may be found from the proportion between the sines of op-
posite sides and angles.

$ c. To determine the value of C by the roles of right angled tri-
gono4Betry.

$ d. In the right angled triangle B p A, ($ 166. & $ 197. $ #*)#
co-tang. B Ap— cos. c tang. B. Also, ($ 196.) sin. Ap— sin. B
sin* e. And B Ap+ A=C A p.

$e. Then in the right angled triangle Cp A, the parts C Ap &
kp aft* known, and ($ 196.) cos. C— sin* O Ap cos. A p.
$/. c— 36° 19' cos. —9.906204
B— 184° 17' tang.— 0.010866

Co-tang. B Ap— 9.917070— 60° M 15"

c sin.- 9.772503
B sin.— 9.854850

Sin. Ap— 9.627353— 25° 5' 12"

B Ap+A— C Ap-70° 7' 15" (§ a.)

§£. CAp— 70° 7' 15" sin.— 9.973318
A p— 25° 5' 12" cos.— 9.956969

Cos. C— 9.930287— 31° 36" 10"

$ h. To find the value of the third angle (C), with the help of
auxiliaries. The angle B A p is auxiliary A'. Co-tang. A' (§ d.)
—cos. e tang. B.

Digitized by V^OOQlC

OBLIQUE 8PHBRIC*. 109

r • a j re ino * , v COS. A f> COS. B . COB. Afi

Coe.C Cos.C Cos. B . _ ^ ^

-r — / A , N ; therefore — : — ■- . 77? — - ' *; 5 •■* *y *««***•
■in. (A + A') sm. (A+A') sin A' J r ^

. . ^, « Cos. B sin. (A+A')

sitkra, Cos. O— -■ t v B ^cos. B sin. (A + A') co-
nn. A' CD

94c. A'.

$j. Cos. Cssscos. B co-sec. A ; sin* (A+A') when kp falls with-
out th* triangle. And,

$ ft. C&s. Csacos. B co-sec. A' sin. (Aoo A') when A p falls within
it Whence the rule.

$/. The product of cos. the given side, and tang, of either angle
(5 A.) 9 is co-tang, of auxl. A'. And, (5 1.\

§ m. Cos. of the same angle, multiplied by the product of co-sec.
of auxl. A' and sine of the sum ($,/.)» or difference (§ i.) of auxl.
A' and the other angle, is cos. of the required angle.

in.
c- 36° 10' cos. =9.906204
B-»184° 17' tang. ^0.010866 cos. =9.843984

Cot. A'=9.917070=50° 26' 15" cosec. =0.1 12985

(A+A')«70° 1' 15" sin. ~9.978318

31° 36' 10" Cos. C-9.930287

To find the value of the two sides (a and b) with the help of
auxiliaries.
$ o. According to a train of reasoning similar to that shown under

Case I, (§ 197. § t.) cot. &=cot. c cos. (A +A') sec. A' ; for
Bad. Cot. c . Rad. Cot. b

x — =■ 7-' > also : 1 — — tk . a/n 5 tne »

tang. A p cos. A tang. A p cos. (A -f A )

Cot b Cot. c » '

— -« r# 5 and by transposition, cot. 0a*

cos. (A + A') cos. A' J r

CD

Cot. c cos. (A + A') ,

I-; -sssCOt. C COS. (A+A') JCC. A'.

COM. A' v « y

$ j). By drawing the perpendicular from B upon its opposite Side,
and calling the angle A B b auxiliary B', we obtain by a similar
process of reasoning. Cot. a=cot. c cos. (B + B') sec. B'.

$q. Cot B'=cos. c tang. A. Whence the rule for finding the
value of either unknown side.

$ r . The product of cos. the given side and tang, of the angle
($ A. & § qJ) opposite to the required side, is co-tang, of the required
auxiliary which call A'. And ($ o. & $ ».),

$ s. The product of cot. the given side and sec. of this auxl. A',

Digitized by VjOOQIC

HO TRIGONOMETRY.

multiplied by cos. of the sum, or difference of A' and the angle ad-
jacent to the required side, is co-tang, of the required side.

§ t. To find the value of b by this rule.
B— 134° 17' tang.»0.010866
Cum 36° 19' cos. =9.906204 - - - cot.=0.133700

Cot. A , =9.917070=50° 26' 15" - sec.=0.195916

(A+A')«70<>7'15" cos.«9.581528

54°0'27" Cot. ft=9.861144

$ u. To find the value of a by the same rule.
A-19°41' tang.=9.553548
c= 36° 19' cos. «9.906204 - - cot.=0.133700

Cot. A'=9.459752=73° 55' 15" - sec. =0.557576

(B— A')=60° 21' 45" cos.=9.694175
22° 22' 82" Cot a— 0.886451

$v. A triangle that has a side or an angle greater than 180° most
never be used in the solution of trigonometrical problems.

$ t*. Therefore in the solution of all cases (§ ti.) where the sum,
or difference of an auxiliary, and an arc or angle, is to be brought
into calculation, if the sum would exceed 180°, the difference is the
proper quantity to be used.

§ x. The two unknown sides may also be determined by a me-
thod of calculation differing from that above, but depending on prin-
ciples analogous to those under § 77. for plane triangles. This
method is the most common in practice.

$y. The sine of i the sum of the two given angles (^, B) ;

Is to the sine of 4 their difference;

As the tang, of i the given side (c) ;

Is to the tang, of i the difference of the two required sides
(a, ft,). And,

$ z. Cos. of the same J sum;

Is to the cos. of the same J difference ;

As the tang, of J the given side ;

Is to the tang, of 4 the sum of the two required sides (a, b.)

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OBLIQUE SPHERICS. Ill

$za. i (A+B)=76° 59' co*ec. =0.01 1305
I (A— B)=57° 18' sin. =9.925060
i ea 18° 0' 30" tang. =9.515844

Tang, i (o o&)=9.452209=15° 48' 58"

i (A+B) sec. =0.647365
i (A— B) cos. =9.732587
| c - tang.=9.515844

Tang, i (a+6)=9.895796=38° 11' 29"

$z b. The greater side (6) is opposite (§ 139.) to the greater an-
gle ; and (§77. $g.) 38° 11' 29"+15° 48' 58"=54° 00' 27" or b ;
and a=(38° 11' 29"— 15° 48' 58"=), 22° 22' 31".

§ z c. To determine the value of the third angle (G) with the
help of an auxiliary arc a'.
%z d. The sine of I the turn > of the two given angles
Is to i the sum of the sines) (A & B) ;
As the sine of £ the given side (c) ; *

Is to the sine of auxl. a'. Then,
$ z t. The product of cos. of auxl. a* and sine of J the sum of
the two given angles, is cos. of £ the third angle (C).
$ zf. B=134° 17' sin. =9.854850
A= 19° 41' sin. =9.527400

2)19.382250

9.691 125=4 sum of the sines of the
two given angles.
i c =18° 9' 30" sin. =9.493659
i (A+B)«76° 59' co-sec. =0.01 1305 - sin. =9.988695

Sin. auxl. a'=9. 196089=9° 2' 12" cos.=9.994576

15° 48' 5" Cos. i 0=9.983271

2

C— 31° 36' 10'

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lie

TRIGONOMETRY.

CASE IV.

$ 200. Given two sides and the angle they include,

a=109 o 30'
6« 60° 00'
C— 27 p 06'
To find the other parts (A, B, c). This is the converse of Case
III., for we have to find here what is there given.

§ a. If the perpendicular A p fall within the triangle, the sum of
the segments (C p+p B) is equal to a ; and the sum of the auxiliary
angles, G kp+p AB=A.

§ ft. Any one of the unknown parts of the triangle proposed, be-
ing found, the two others are determinable from the relations (§ 144.)
of sides to their opposite angles.
£ c. To find B, by the Catholic Proposition. By Case II.,
Cos. C
(§ 171. § c.) tang. Cpss q . =(§ 101. § 6.) cos. C tang. 6; also

(§ 171. § d.) sin. A />=sin. b sin. C. Then (§ a.) *~ € p**p B ;
and the value of An, and of B/>, in the triangle kp B, being known,
that of B ($ 155.) is determinable ; cotang. B « ($ 172. § &.)«-
Sin./>B
f — j-» 8m .^B cotang. A p.

§d. C=27° 6' cos. =9.949494
6=00° 0' tang. =0.236661

Tang. C/>= 0.168065=57° 2' 4"

C sin.:
b sin.

=9.658531
=9.937531

Sin. A p =9.596062= 23° 14' 8'

109° 30'— 57° 2' 4" (§<*.)=/> B =52° 27' 56."

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OBLIQUE SPHERICS. 113

§ e. p B=52° 27' 56" Sin. =9.899267

Ap=23° 14' 8" Cotang. =0.367204

Cotang. B=10.266471=28° 25' 54"

§/. To find the value of each of the unknown angles (A, B), with
the help of an auxiliary arc (a'). A perpendicular drawn from
either of the unknown angles (A, B) fulfils the conditions of § a.
(§ 184.) ; suppose the perpendicular drawn from B to the fall with-
out the triangle.

§ g. Let the distance (Cp or C A) of the given angle from the per-
pendicular be auxiliary a\ Then whether the perpendicular fall
within or without the triangle, the difference (§ d.) between the
side upon which it falls and the auxl. a' is the other segment of
that side.

$ A. First, to find the value of B, the arc Cp being auxl. a! ;

tang, a' ($ d.)s=cos C tang. b.

Rad. Cot. C
§t\ According to Case II. ($ 198. ^O^^^-g^ -

Cot. B , t . . ^ . Cot. C Sin. (am a')

Sin. (a Ma') ; md ** *™W»««* Cot - B : SihT*.

sscot. C sin. (a co a') cosec. a'.

§j. To find the value of A. In the triangle C A B, C A is auxl.
a'; and tang. a'=cos. C tang. a.

^, Rad. Cot. C Cot. A

**' lb0 * Tang. B A^SinT^Sin. (b « a') ; ^ V transposi-
tion, €ot A S5 Cot.CSin.(6«) a 9 ) mmeoL c sin< (6w a , } ^^ fl ,

Sin. a'
Whence the general rule for finding either angle.

§/. The product of cos. of the given angle (§ A. and §j\) and
tang, of the side opposite to the required angle, is tang, of auxl. a'.
And,

§ rn. The sin. of the difference between auxl. a! and the side ad-
jacent to the required angle, multiplied by the product of the cosec.
pf.jauxl. a 1 and cotang. of the given angle (§ t. and § A.), is cotang.
of the required angle.

§». J=60° 0' tang.==*0.238561

C=27° 6' cos.=9.949494 - - cot.=0.290963

Tang. auxl. a'=0.188055=*57°2'4" cosec. «0.076240

(a' cc fl)=52° 27' 56" sin.=9.899268

28° 25' 54" Cot. BsO.266471

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114 TRIGONOMETRY.

§ o. Or, to find the value of A ;
a«100°'30 / tang=0.450851
C= 27° 6' cos. ==9.949494 - - - cot.=0.290963

Tang. ^=0.400345=111° 41' 32" cosec. =0.031898

(<!'«>&)= 51° 41' 32" sin.«9.894699

148° 47' 10" Cot A=0.217560

%p. The value of the third side (c) can be found with the help
of the same auxiliary a'.

Cos. A ©
§ q. When the perpendicular falls within the triangle, p A
, N Cos. 6 Cos. c .. *■*•

($ 198. § ^)-c5s7? g Cos.(aa a') ; ^ *™*?°* l *<>» Co8 - «■ -
Ooi.tCoi.(q»«0 d ( }

Cos. a' v '

$ r . And when the perpendicular (B A) falls without the triangle,
Cos. B A Cos. a Cos. c
T^"c5S~Co..(6 »«') ; b 7 *"■***». cos. c=

Cos. a Cos. (&w a') ,* fV

" \ i«cos. a cos. (6 w a ') sec. tf.

Cos. a' v J

N. B. (am «0—p B, and (6 »a') s =A A.

Hence the general rule for finding the third side,

§ *. The prQduct of cos. of the given angle, and tangent of either
given side, (§ A. & $y.), is tang, of auxl. a'.

§ U Then (§?.&§ r.) cos. of the same given side, multiplied
by the product of sec. of auxl. a', and cos. of the difference between
auxl. <?, and the of Air given side, is cos. of the required side.

§u. C=27°6' cos.=9.949494

6=60° 0' tang. =0.238561 - - cos.=9.698970

Tang. a'=0.188055=57° 2' 4" sec. =0.264293

(aaoa')=52° 27' 56" cos.=9.784786

55° 57' 24" Cos.c=9.748049

§ v. Or, taking the tang, and cos. of a;
C= 27° O' cob. =9.949494

«=102° 30' tang. =0.450851 - - - cos. =9.523495

Tang. a , =0.400345=lll° 41' 32" sec. =0.432244

(6«a')= 51° 41' 32" cos.=9.792312

55° 6Y 24" Cos. c= 9.748051

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OBLIQUE 8PHERICS. 115

§to. The difference between the two log. co-sines of e arises
from fractions of a second ("), which are in some of the parts
operated upon. These fractions may sometimes cause an error of
a few seconds (") in the value of the part required.

% x. There is a method for finding the value of the two angles
(A & B) analogous to that'under §y. and § z. (§ 199.), for finding two
sides. In drawing up formulae, and selecting methods, for trigo-
nometrical calculations, attention should be paid to what is given,
as well as to what is required, in the problem ; and that method of
solution should be adopted, which equally as correct as others, leads
most directly to the result required.

§ y. To find A and B without the help of auxiliaries.

$ z. The cos. of £ the sum, > f . .. .

Is to cos. of £ the difference, 5 ot me tw0 Sldes '

As co-tang, of £ the given angle ;

Is to tang, of £ the sum of the required angles.

$ za. The sine of £ the sum, > f , ., .

Is to the sine of £ the difference, 5 ot me two 8iaes '
As co-tang, of £ the given angle ;
Is to tang, of £ the difference of the required angles.
§ zb. £ (a+&y=84° 45' sec.=s 1.038571

£ (a— b)=*24° 45' cos.s 9.958154

£ C =13° 33' cot.= 0.617980

\

Tang. £ sum (A & B) =1.614705=88° 36' 32"

£ (a+b), co-sec.= 0.001826
£ (a—*), sin.= 9.621861
£ C cot.= 0.617980

Tang. £ diff. (A & B>=0.241 667=60° 10' 38'

§ x c. 88° 36' 32"+60° 10' 38"=A=148° 47' 10" > ,*„ c , x

And, 88° 36' 32"— 60° 10' 38"=B= 28° 25' 54" } U".j*.;

%z d. The following method of finding the value of the third
side (c) with the help of an auxl. a', is useful and of frequent oc-
currence in nautical calculations.

§ z c. Half the product of the sines of the given sides, and twice
the sine of half the given angle, multiplied by the co-sec. of £ the
difference between the two sides, is the tang, of auxl. a'.

% zf. Then the product of co-sec. of a' and said half product of
the sines of the three said quanttties, is the sine of half the required
side.

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116

TKIGONOMETRY.

10-

109° 30' sin. » 9.974347
60° 00' sin. = 9.937531
13<> 33' sin.x2=»18.739522

2 ) 38.651400

19.325700
i(a»&)«24 45'co-sec.= 0.378139

19.329700

Tang. auxl. a'= 9.703839=26 o 49'22"cosec.:=0.345600

27° 58' 42"
2
c=55° 57' 24"

Sin. i c« 9.671300

CASE T.

§201. Given the
three sides
a«66°
6«=44°
c«30°
To find the angles.
§ a. Neither in
this, nor in the pre-
ceding case, can
g the value of any
one of the unknown
parts he determined by the Catholic Proposition; for the proposed
triangle cannot be so divided into two right angled triangles, that
two of the given parts shall be contained in either of them.

§ 6. This, and the case that immediately precedes it, are particu-
larly useful to the navigator. Some of the most important problems,
and those which are of most frequent occurrence in navigation,
come under one or both of these cases for solution. The problems,
for finding azimuths and the time of day at sea, fall under this
case. This and the preceding case are both involved in the calcula-
tions for finding the true, from the observed, lunar distance. Both
cases are also involved in finding the latitude by " double altitudes."
And by Case IV, the lunar tables in the nautical almanac, are cal-
culated.

§ c. In this, as under the other cases, there are several methods
for finding the value of the required part In all cases the methods,
which are the best adapted to practical purposes, are given.

§ rf. To find one of the angles (A.). The product of the co-sec.
of the sides that contain the required angle, multiplied by the pro-

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OfiLfQUft ftPHBATCS. Uf

duct of the sine of half the sum of the three sides, and sine of .that
half turn, less the side opposite the required angle, is double the
cos. of £ the required angle.

$ e. 7tatce the tang, of i the required angle, is the product of
co-sec. of i the sum of the three sides, and cc~*ec. of said i sum,
less the side opposite the required angle, multiplied by the product
of the sines of the difference between said half sum and each of the
sides that cdntain the required angle.

$/. «« w(s<io

6= 44°co-8ec.= 0.158220
c=* 80° co-sec, = 0.301030

2)140
(a+b+c)

2

(a-t-6+0— a)

70° sin.= 9.072986
4°flin.= 8.843585

Twice cos. J A= 2)10.275830

Cos. | A=« 9.637915= 64° 15' 5"

2

A=128° 30' 10'

(a+b+c)

2
(a+b+c

§ jT. v 2 J =70° c<Hsec.= 0.027014 (§ e.)

-a) « 4° co-sec.=» 1.156416

(a+b+c

2

-6) =26° sin. = 0.641842

-c) =40° sin. = 0.808068
Twice tang, i A=2)20.633339

Tang, i A= 10.316660=* 64° 15' 5"

A=128° 30^ 10"

§ h. In all cases § g. is a good solution. But when the required
angle (A) is near 180°, (§ 176. § c.) solution §/. will not carry great
accuracy into the result. But this seldom occurs in practice, and
in most cases either solution may be adopted with success.

§ i. One of the angles being found, the two others (§ 144.) are
determinable.

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118 TRIGONOMETRY.

./

■/

CASE VI.

▲ § 202. Given

> *»*"" s v the three angles,

' x s v A=114°

N. B= 39°

n / \* C— 49°

/ X To find the value

/ v of the sides.

\ § a « This pro-

\ blem is but of lit-

c£- ——————— y..,^^ "•— -*^ \ ^ e Poetical utility

*^"^""*«*,A to the navigator,
B for problems in
which the three angles of a triangle are the data, seldom occur.
But there are several methods by which the side required may be
found ; one of which, being general in its application, is thought
sufficient.

§ 6. The product of cos. of the difference between half the sum
of the three angles, and each angle adjacent to the required side,
multiplied by the product of co-sec. of each of said angles, is twice
the cos of i the required side.

§ c. To find the value of a.
(A+ B+C) B _ 620c(H , ne

= 9.671609

(A+B+C)
2

-C

=s 52° co-sine

= 9.789342

B =39° co-sec.
C =49° co-sec.

wice cos. of £ a=

= 0.201128
» 0.122220

T

:2)19.784299

Cos. i a

1— 9.892149=

*38°

43'

50"

a=

2

=77°

27'

40"

§rf. The value of one side being known, that of the others is
determinable.

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NAUTICAL ASTRONOMY.

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1

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NAUTICAL ASTRONOMY.

§ 203. That part of astronomy which treats of the motions and
ef the positions of the heavenly bodies, is an important branch of
navigation. A knowledge of these motions and positions is highly
essential to the navigator ; for it is by understanding them, that
methods have been devised for determining latitude' and longitude at
sea, by means of observations made upon the heavenly bodies.

$ 204. The figure of the earth is that of an oblate spheroid. It
resembles that which would be described by the revolution of a
semi-ellipse about its minor axis. It is flattened in at the poles, and
elevated towards the equator.

$ 205. To suit the common purposes of navigation, the earth may
be considered as a perfect sphere ; the sun as the centre of the uni-
verse, and the centre of motion in the planetary system ; the fixed
stars may be considered to be almost stationary, and immeasurably
distant from the earth and from each other; from every part of the
earth's orbit, they are seen in the same relative positions, with re-
gard to each other.

$206. The earth has two rotary motions ; one about its axis,
which produces day and night; the other in its orbit, and around
the sun, which causes the seasons.

$ a. The latter is called the earth's animal, and the former its
diurnal, motion.

$ 207. In its annual revolution around the sun, the earth describes
the periphery of an ellipse.

$ a. The centre of the sun is in the plane, and at one of the foci
of this ellipse, and the centre of the earth moves on its circumfer-
ence.

$ 208. The axis of the earth is inclined, from a perpendicular to
the plane of its orbit, nearly at an angle of 23° 28'.

$ a. If the earth's axis were perpendicular to this plane, there
would be no change of seasons, or variation in the length of day
and night; and at either pole there would be continual day.

$ b. It is this angle of inclination which causes the declination
of the sun.

$ c. In a northern winter, the earth is nearer to the sun than it is
in summer ; but owing to the sun's declination, or the inclination of
the earth's axis to the plane of the earth's orbit, the south pole is
turned towards the sun daring the former season, and the sun's rays
striking the northern hemisphere more obliquely than they do in

R

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122 NAUTICAL ASTRONOMY.

summer, are in consequence, spread over a greater surface, and are
therefore less effective in producing heat.

$ 209. The earth revolves from west to east. It completes one
revolution on its axis in a day, and one around the sun in a year.

$ 210. The periphery of the earth's orbit, like the circumference
of every re-entering curve, contains 360°.

$ a. The earth completes one revolution in its orbit, in 365(Z, 5A,
48m, and 48*.

§ b. Therefore, the motion of the earth in its orbit, in one day,
is, as 365<Z, 5A, 48m, 48* is to 360° ; the ratio of which is 59' 8".

$211. Aja astronomical, sea or civil, day, is the time between
two consecutive noons, or midnights.

$ 212. A sidereal day is the time between two consecutive tran-
sits of a star across the same meridian.

$ 213. The intervals of time from the passage of the sun and a star
across, until their return to, the same meridian, are unequal. The dif-
ference between these intervals, results from the combined effect of
the earth's motion about her axis, and in her orbit ; for, during the in-
terval between two consecutive transits of a star across the same
meridian, the earth advances in its orbit ($ 210. § b.) nearly 69' 8" ;
so that the transit meridian has to go 59' 8" more than one com-
plete revolution around the axis of the earth, before it returns to the
sun.

$ a. Thus, during the time in which the earth is making one
complete revolution in its orbit, a star has one transit more than the
sun has, across any proposed meridian.

$ 6. Sidereal accelerates upon mean solar time, 3m, 56*.55 in
24A, or about 9|s in an hour.

$ 214. The motion of the earth about its axis, is uniform ; but
the velocity of the earth in its orbit, is irregular. This irregularity
is caused by the variation in the mutual action of the centrifugal and
centripetal forces upon tha earth.

$ a. The sidereal day is consequently of a uniform length ; and
the solar day varies with the velocity of the earth in its orbit.

$ b. This variation produces equation of time.

$215. Apparent time is the time which is deduced by calcula-
tion, fiom the bearing, or the altitude of the sun.

$ a. The time shown by a dial, is apparent time.

$ 216. Mean time is the average of apparent time.

$ a. Watches, etc., are designed to keep mean time.

$ 217. Equation of time is the difference between mean and ap-
parent time.

$ 218. If the motion of the earth in its orbit, were uniform, de-
scribing equal arcs in equal times, there would always be the same
length of time from one transit of the sun to another, across the
same meridian ; and apparent and mean time would always agree.
But from noon to the succeeding noon, there is sometimes more,
and sometimes less, than 24 hours.

$ a. Twenty-four hours is the average, or mean of the time, in

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NAUTICAL ASTRONOMY. 121

whieh, during one complete revolution of the earth in its orbit, the
sun performs every one of its 365 transits across the same meridian.

$ o. This is the time which all good time pieces show. The
time shown by them and the sun agrees only four times during a
year ; when this agreement takes place, then there is no equation of
time.

$ 219. All astronomical calculations, such as those in the nautical
almanacs, etc., are computed for astronomical time.

$ a. Of those in the ephemeris, some are for mean, and some for
apparent astronomical time.

§ 220. An astronomical day commences when the centre of the
sun is on the meridian (say of Greenwich), and ends, when it re-
turns to the same meridian.

$ a. An astronomical day is divided into 24 hours, which are
reckoned in succession, from 1 up to 24.

$ 221. A civil day commences at the midnight, that precedes the
beginning of the astronomical day.

$ a. A civil day is divided into two equal portions ; the first is
A.M.; it ends at noon, when the astronomical day begins : the latter
is P.M.; it ends at midnight, when the succeeding day begins.
Eaeh division consists of 12 hours.

$ b. Consequently, the day civil, is always 12 hours in advance
of the day astronomical. Thus, Dec. 11th, 9A 34m 44s A.M. is,
according to astronomical time, Dec. 10th, 21 A 84w» 44*; and 16
hours of civil time, or Dec. 11th, 4A P.M., is Dec. 11th, 4A, ac-
cording to the astronomical time. Therefore, when the civil time
is A.M. 12 hours added to it makes it astronomical time.

$ 222. The ecliptic is a great circle, the centre of whieh is that
of the earth, and the plane of which coincides with the plane of the
earth's orbit. Therefore these planes cut the axis of the earth at
the same angle.

$ 223. The sun is always in the ecliptic.

$a. Considering the earth to be stationary in its orbit, the sun
appears to perform an annual revolution, moving around the earth,
on the periphery of the ecliptic, and crossing the equator twice dur-
ing the year.

$ 224. The ecliptic is divided into 12 ares, of 30° each. These
divisions, named and marked, are called the Signs of the Zodiac.
They are—

$ a. Aries (V), Tauius(b), Gemini (n> Cancer (25), Leo (aj,
Virgo (nj), Libra (a), Scorpio (m,), Sagittarius (/), Capricornus
(yfy, Aquarius (»), and Pisces (x).

§ b. The first six being north of the equator, are called northern
eigne.

$ c. The other six are south of the equator, and are called eouth-
ern eigne.

$ 225. The points in which the ecliptic ($ 223. $ a.) crosses the
equator, are called the equinoctial points; because, when the sun
passes through these points, the days ind nights are equal.

$ 226. The sun crosses the equator, and enters the first point of

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124 NAUTICAL ASTRONOMY.

Aries, about the 21st of March. The son then pastes through Aries*
Taurus, and Gemini, towards the north ; and about the 21st of June, it
reaches its greatest northern declination at the first point of Cancer,
where it appears to be stationary for a while ; it is then said to be in
the summer solstice.

$ a. The first point of Cancer, is a solsticial point.

$ b. Returning thence, towards the south, the sun passes through
Cancer, Leo, and Virgo, and completing its tour, or the north side of
the equator, it arrives, about the 23d of September, at the intersection
Qf the ecliptic with the equator, when the day and night are again
equal, and the sun is in the autumnal equinox.

§ c. Recrossing the equator then, the sun enters the first point of
Libra, and continuing on towards the south, it descends through
Libra, and the succeeding signs, and re?.ches its greatest southern
declination, about the 22d of December; then it is at the first point
of Capricorn, and again appears to stand still.

§ d. The sun is now in its winter solstice ; and returning towards
the north, it ascends through Capricorn, Aquarius, and Pisces, and
entering the first point of Aries, completes one annual revolution,
and goes on to renew the seasons.

$ 227. The time from the sun's passing the first point of Aries,
until its return to that point again, is about ($210. § a.) 365<Z 5A
48m 48*.

$ a. This is called a solar, or a tropical, year, and it is the year
by which the seasons are regulated.

$ b. A solar, differs from a sidereal, year, about 20m 28s.

§ 228. A sidereal year is the time from the sun's leaving, until
its return to, the same part of the heavens, or to the same fixed
star.

$ a. A sidereal, is longer ($ 227. $ b.) than a solar, year, on ac-
count of the precession of the equinoxes, which is a motion con-
trary to that of the sun through the signs.

§ b. While the sun is completing a revolution from left to right
through the signs in successive order, they are moving in the con-
trary direction, or from right to left, and by the time the sun has
returned to the first point of Aries, for instance, this point will have
retrograded a little more than 51" ; and, on account of this motion,
the sun comes to this point sooner than it would have done, had
the first point of Aries remained stationary.

$ 220. The zodiac is a belt in the heavens that extends 8° on
each side of the ecliptic.

§ a. The orbits of all the primary planets are within the zodiac.

§ 230. The primary planets are Mercury, Venus, the Earth,
Mars, Jiipiter, and Saturn.

$ a. There are five other planets, but they cannot be seen with
the naked eye. They are called telescopic planets.

§ 281. The points in which the orbit of a planet cuts the ecliptic,
are called nodes.

$ a. That node is called the ascending node (&), through which

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NAUTICAL ASTRONOMY. 125

the planet passes, as it crosses the ecliptic, going from the south to
the north side of it.

%b. The other point of intersection is called the descending
node (SI).

$ 232. The equator, or equinoctial line, is a great circle, whose
plane divides the earth into two equal parts.

%a. These parts are called the northern and southern hemi-
spheres.

§ b. The centre of the earth is the centre of the equator, and of
all the great circles used in nautical astronomy.

$ 233. The poles ($ 123.) of the equator, also called the Poles,
are two points on the earth, that are diametrically opposite to each
other; each is 90° from th#» 4*prattjr.

§ a. The on* on the north side of the equator, is the north pole;
and thai en the south side, is the south pole.

$ 234. A straight line from one pole to the other, passes through
the centre of the earth, and (§ 124.) is called the axis of the earth.

% a. Around this axis, the earth performs its diurnal revolutions,

$ 235. From the best measurements, the equitorial appears to be
about 26 miles greater than the polar diameter of the earth. And
the degrees of latitude increase in length from the equator towards
the poles.

% a. The surface of the earth ($ 204.) being flattened in at the
poles, and elevated at the equator, its meridianal curvature is less
near the former than it is near the latter ; consequently a degree of
latitude near the poles contains more fathoms, feet, etc., than one at
or near the equator.

$ 236. Suppose the plane of the equator to be extended to the
heavens ; it there forms the celestial equator, and its poles, are the
poles of the world.

§ 237- The latitude of places on the earth, is measured from the
terrestrial equator ; and their longitude is measured on it.

$ a. In the heavens, the declination of the bodies are measured
from the celestial equator, and their ascension on it.

§ b. Declination and ascension are, in the heavens, what latitude
and longitude are on the earth.

$ 238. The sun crosses the equator twice in a year.

$ a. While the sun is on the north side of the equator, the sun is
constantly visible from the north pole, and daylight continues there
until after the sun recrosses the equator, when the sun goes below
the horizon, and does not rise again, until (§ 226. $ d.) completing
its southern tour, it is approaching the vernal equinox.

§ 239. The year, instead of being divided into seasons at the
poles, may, more properly, be divided into day and night ; for there
is but one day and one night at each pole during the year.

§ a. At the north pole, the day commences about the 21st of
March, when the sun crosses the equator, and continues until the
sun recrosses the equator, which happens about the 23d of Septem-
ber, when night succeeds the day, and day begins at the south
pole, and lasts until the sun again returns to die first point of Aries.

/

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126 NAUTICAL ASTRONOMY.

$ 0. Owing to atmospheric refraction, the sun may be seen at
either pole for several days previously to its rising above, and after
it has gone below, the natural horizon.

§ 240. The morning and evening twilights at either pole, are to*
gether about two months and a half long.

§ a. When the sun crosses the crepusculum, the twilight begin*
and ends.

§ 241. Crepusculum is a small circle parallel to the horizon, and
18° below it.

$ 242. All small circles that are parallel to the horizon, are alma*
canthers.

§ a. To an observer at the north, or south pole, the horizon and
the equator coincide, and the parallels of declination are almaean*
thers.

§ b. At the north pole, the crepusculum coincides witn tH« paral-
lei of the 18th degree of southern declination, which the sun crosses
about the 29th of January, (near 51 days before it rises;) and on
the 13th of November, (near 50 days after it has gone down).

§ c. On account of such long twilights, the winter nights, in high
southern or northern latitudes, are rendered less gloomy than they
otherwise would be.

$ 243. The earth is divided into five zones $ two frigid, two tem-
perate, and one torrid.

$ a. The torrid zone is the largest; it extends to the parallels of
23° 28 / north and sooth latitude; consequently it is 40° 50' broad.

$ b. The small circles which limit it, pass through the soLstkial
points, and are called tropics, from trepho, (Gr.) ; because, when the
sun reaches the parallel of declination for 28° 28' ($ 226. $ o. & § d.)
h appears to tern its course, and to recede from the pole which it
was approaching, and to retrograde towards the direction wheaee
it came.

$ c. That parallel which is north of the equator, (§ 228.) is the
tropic of Cancer.

$ d. And that on the south side of the equator, ($ 226. $ c), is
the tropic of Capricorn.

§ e. The sun is vertical twice a year, at every place within the
tropics.

$ 244. Each of the temperate zones extends over 48° 4' of lati-
tude.

$ a. That on the north side of the equator, called the north tem-
perate zone, extends from the tropic of Cancer to the arctic circle*

§ b. And the south temperate zone extends from the tropic of Ca-
pricorn to the antarctic circle.

$ 246. The arctic and antarctic are small circles, parallel to the
equator, and 23° 28* from the poles.

$ a. The arctic ie about the north, and the antarctic about the
south pole.

§ b. They are sometimes called the polar circles.

$ 246. The/rigid zones are between the polar circles and the
poles.

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NAUTICAL ASTRONOMY. 127

$ 247. Latitude is distance on the earthy measured north or
south, from the equator.

§ 248. Meridians are great circles, that are secondaries to the
equator.

5 a. Therefore (§ 126. §a. & $ 127.) they cut it perpendicularly,
and intersect each other at the poles.

$ b. The arc of a meridian that is contained between a place and
the equator, measures the latitude of that place.

§ c. The meridian of an observer passes through his zenith.

$249. Parallels of latitude are small circles, parallel to the
equator.

§ a. All places upon the same parallel have the same latitude.

§ b. And all places that are on the same meridian, are in the
same longitude.

$ 250. Circlet of longitude are meridians.

$ 251. Elevation of the pole is the distance of either pole above
the horizon of any observer.

§ a. This distance is measured on the meridian of the observer ;
it is equal to his latitude.

$ 252. Longitude is the distance, expressed in degrees, etc., be*
tween two meridians. It is measured on the equator, and east or
west from a meridian.

§ a. A meridian from which longitude is measured, is called the
prime meridian.

§ b. The location of this meridian is optional with topographers.
The French reckon. the longitude of all other places from the me-
ridian of Paris ; the Spaniards from Cadiz. The English use the
meridian of the Royal Observatory at Greenwich for their prime i
meridian : and the Americans generally construct their charts from I
the meridian of Washington City. |

§ c. But as long as most of the charts and the tables of the Nau-
tical Almanac which we use, shall be constructed and calculated to
the meridian of the Greenwich observatory, it will be found more
expedient to reckon longitude from this, as the prime meridian.

$ 253. It is to be wished that all nations would fix upon one
common prime meridian. One might be established from celestial
phenomena, by which all that is arbitrary in its locality, might be
made to disappear. La Place recommends the adoption of a uni-
versal first meridian, and suggests the propriety of selecting for this
purpose, that meridian, upon which it was 12 o'clock when the sun
entered the point of the vernal equinox in the year (1250), in which
the apogee of the earth's orbit coincided with the solstitial point in
Cancer. Such a universal meridian would pass about 6 miles west
of Cape Mesurada on the Coast of Africa.

$ 254. The longitude of a place is measured on the arc of the
equator, which is contained between the meridian of that place, and
the prime meridian.

$ a. The angle at either pole, which these two meridians make
with each other, is equal to the longitude.

§ o. Consequently the arc of a parallel of latitude ($ 249.), which

12S NAUTICAL ASTRONOMY.

is intercepted between that place and the meridian of any other
place, contains, in degrees, the difference of longitude between these
two places.

$ c. The difference of latitude between two places, is the arc
which is contained on the meridian of either place, between the pa-
rallels of latitude of these two places.

$ 255. Declination is the distance of a heavenly body from the
equitorial plane.

§ a. Declination is north or south, according as the body is
north or south of the equator. It is measured in the heavens, as
latitude ($ 248. $6.) is on the earth.

§ b. The arc of the circle of declination or of right ascension, that
is contained between the equator and the centre of the body, mea-
sures its declination.

$ 256. Circles of right ascension or of declination in the heavens,
correspond with meridians of terrestrial longitude.

§ a. They cut the equator at right angles, and intersect each other
m the poles of the world.

§ 257. Parallels of declination are small circles in the heavens,
that are parallel to the equator.
$ a. They correspond with parallels of latitude on the earth.
$ b. The tropics (§ 243. § c. & % d.) are the parallels of the sun's
greatest declination.

$ 258. Polar distance is the distance of any celestial body from
either pole.

$ a. It is the arc of the circle of declination that is contained be-
tween the body and either pole.

5 6. The complement of the declination of a heavenly body, ex-
presses its polar distance.

§ 259. The right ascension of a body is the arc of the equator,
which is between the first point of Aries and the circle of the decli-
nation of that body.

$ a. This arc is always counted in the order (§ 224. $ a.) of the
signs, and its dimensions are expressed in hours, minutes, and
seconds.

$ 260. Right ascension in the heavens, is what longitude is on
the earth. But the manner of reckoning them is different.

§ a. Longitude is reckoned from the prime meridian as far as
180° both east and west.

$ 261. Right ascension commences to be reckoned from the first
point of Aries, and goes entirely around, in the direction in which
the sun passes through the signs*

$ 262. The ascensional difference of the sun, is the difference be*
tween its right and oblique ascension, or between sunrise and 6
o'clock.

$ 268. Horary angles are those which meridians, or circles of
declination, make with each other at the poles.

§ a. The 6 o'clock hour circle, is that circle of declination ( $ 256.
$«•) which cuts the equator and the horizon in the east ana west
points.

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NAUTICAL ASTRONOMY. 129

1 6. It is secondary to the equator and to the meridian of the ob-
server.

S c. Any circle of declination is an hour circle.

$ 264. Celestial latitude is distance between the ecliptic and any
body in the heavens ; it is measured from the ecliptic, and upon a
secondary to it.

§ 265. The secondaries of the ecliptic, are circles of celestial
longitude.

§ a. The arc of the circle of celestial longitude, that lies between
the ecliptic, and a body in the heavens, measures in degrees, etc.,
the latitude of that body.

$ 267. The longitude of heavenly bodies is reckoned on the
ecliptic, as right ascension is on the equator (§ 259.).

$ a. It commences at the first point of Aries, and is reckoned
around in the direction in which the sun passes through the signs.

§ 268. The right ascension and longitude of an object are never
the same, except when the body is on the solstitial colure, which
cuts both the ecliptic and the equator at right angles, and passes
through their poles.

$ 269. The solsticial colure is that circle of declination which
passes through the solsticial points (§ 226. § a. & c).

§ 270. The equinoctial colure is the circle of declination, which
passes through the equinoctial points ($ 226. § b.).

271. The equinoctial and the solsticial points are the four cardi-
nal points of the heavens.

§ 272. The four cardinal points of the horizon are the east and
west, and the north and south points.

§ a. The east and west are the points in which the equator and
the horizon intersect each other.

§ b. The north and south are the points in which the meridian
of the place cuts the horizon.

$ 273. Every observer has two horizons, one rational, the other
sensible.

$ a. The eye is the centre of the latter, and is on the axis of the
former.

$ 274. The sensible horizon is that circle which terminates t 1 e
view upon an uninterrupted plane.

§ a. It is formed by the apparent meeting of the plane upon
which we stand, with the concave expanse above.

$ b. It is a small circle, and is parallel to the rational horizon.

§ c. The circle which terminates our view at sea, is the sensible
horizon.

$ 275. The rational horizon is a great circle ; its plane passes
through the centre of the earth, and its axis through the eye of the
observer. Suppose H a F to be half of the earth, D Z C the con-
cave blue which bounds the vision, and D H C the plane of the ra-
tional horizon. If an observer were placed at the centre (6) of the
earth, the angle D G Z would show the altitude of an object in the
zenith, at Z, from the rational hprizon. If the observer were placed
at the circumference (at a) of the earth, p ap would be the plane

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130

NAUTICAL ASTRONOMY.

of his visible horizon, and the angle p a Z (=»D 6 Z § 30.) would
show the altitude of a body at Z, from the sensible horizon. But if
the observer were placed at A above the surface of the earth, the
angle BAZ(>DGZ) would show the altitude of Z from his sensi-
ble horizon.

— iC

§ a. When the eye is elevated above the level of the sea, (as at
A,) the line of vision A B to a point B in the sensible horizon, cuts
the plane C D of the rational horizon at an angle, DEB, which
angle is called the dip. Consequently, when the eye is above the
surface of the plane on which we stand, the sensible horizon is be-
low the rational horizon.

$ b. The poles of the rational horizon are the zenith and nadir.

$ 276. The zenith is the point in the heavens which is directly
over head.

$ a. Being a pole of the horizon, the zenith is 00° from the
horizon.

$ 277. The nadir is the other pole of the horizon ; it is in the
lower part of the heavens, directly under our feet, and diametrically
opposite to the zenith.

$ 278. Azimuth circles are secondaries ($ 127.) to the horizon;
cutting it perpendicularly, they intersect each other in the zenith and
nadir.

$ a. They are also called vertical circles.

$ 279. The azimuth circle which cuts the horizon in the east and
west points, is called the prime vertical*

$ a. The prime vertical of every observer, is secondary to his
meridian.

$ 280. The altitude, and zenith distance of a heavenly body, are
measured on its azimuth circle.

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NAUTICAL ASTRONOMY. 131

§281. The zenith distance of a heavenly body is its distance
from the zenith of the observer ; thus, Z s (§ 275.) is the zenith dis-
tance of the star at *.

§ a. It is measured on the arc of the azimuth circle, which lies
between the centre of that body and the zenith.

§ b. The zenith distance is north when the body is south of the
observer.

$ c. And south, when the body is north of the observer.

5 282. The altitude of a heavenly body, is its distance above the
horizon.

§ a. The arc of the azimuth circle which is contained between
the centre of a body and the horizon, measures the altitude of that
body ; thus, D s (§ 275.) is the altitude of the star at 8.

$ b. The complement of the altitude of a body, gives its zenith
distance.

§ 283. The altitude which is taken of the sun or moon, with a
quadrant or sextant, is the observed altitude of one of the edges,
called a limb.

§ a. The apparent altitude of the centre is found by applying the
corrections, (which are laid down in the Nautical Almanac,) for the
semidiameter of the body, to the apparent altitude of its limb.

$ b. And the true, is obtained from the apparent altitude of the
centre, by applying to the latter, corrections for the parallax and
refraction, which are also known by previous calculations.

$ c. The apparent and the true altitude of a body, are always
measured on the same azimuth circle.

§ 284. The rays of light coming from the heavenly bodies strike
the atmosphere obliquely, and entering from a rarer into a denser
medium, are refracted, or bent downwards ; this causes the body
whence they emanate to appear higher up in its azimuth circle than
it really is. Suppose # e (§ 275.) be a ray of light from *, and A o
to represent the atmosphere ; when this ray strikes the atmosphere,
it will be reft acted so as to reach the eye of the observer at a, in
the direction a e, which makes * appear at 8, above its true place.

§ a. Hence the apparent altitude of the sun or a star is always ^
greater than the true altitude, unless the body be in the zenith. •

$ 285. Parallax has a contrary effect ; it acts in a direction oppo-
site to that of refraction, and causes a body to appear lower down
in its azimuth circle than it really is.

$ a. But the effects of parallax and refraction, though acting in
contrary and opposite directions, seldom counterbalance each other,
so as to make an object appear in its true place.

$ 286. The parallax of the moon is always greater than the re-
fraction; and the moon always appears below its true place.

§ a. Hence the apparent, is always less than the true altitude of
the moon.

$ 287. The horizontal parallax of a body, is the difference be-
tween the true and apparent place of that body, (supposing there be
no refraction), when it is in the horizon.

$ a. The horizontal parallax is equal to the angle at the body,

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132 NAUTICAL ASTRONOMY.

which is subtended by the distance of the observer, from the centre
of the earth.

$ b. The angle which this distance, or semidiameter, subtends,
is greatest when the body is in the plane of the rational horizon ;
thus, the parallax of a body at D, (§ 275.) is the angle G D a; and
the parallax of the same body, when at s, is G * a.

§ c. As the object rises above the horizon, the angle which this
semidiameter subtends, is called parallax in altitude $ it gradually
decreases until the object reaches the zenith, when it vanishes.

§ d. The nearer the object is to the earth, the greater will this
angle be at the centre. Hence the moon's parallax is greater than
the sun's, and this greater than that of the fixed stars.

§ 288. The parallax of a body decreases from the horizon to the
zenith, in the proportion of sine of the zenith distance (J 281.) to
radius.

$ 289. Owing to the effects of parallax and refraction upon the
heavenly bodies($ 285. $ a.)* they are never seen in their true places,
except when in the zenith.

$ a. The places in which the heavenly bodies are seen, are called
their apparent places,

§ 290. The true place of a heavenly body, is that place in which
it would appear, if seen from the centre of the earth.

§ 291. The azimuth of a celestial body is the angle which is con-
tained at the zenith, between the meridian of the place, and an arc
of the azimuth circle, which passes through the centre of that body.

§a. In north latitudes, the angle on the north side, and in
south latitudes, the angle on the south side, of the arc of this azi-
muth circle, is called the azimuth.

§ b. Before a body crosses the meridian of the observer, its azi-
muth is east, and west afterwards.

$ c. Thus, in north latitude, an azimuth is said to be north, so
many degrees east, if the body be east of the meridian ; or N 9 so
many degrees west, if it have passed the meridian.

§ d. And in south latitude, the azimuth is reckoned in the same
manner from the south point, to the east or west.

§ 292. The amplitude of a celestial object, is the arc of the hori-
zon that lies between the east or west point, and the centre of that
object when it is rising or setting.

VARIATION OF THE COMPASS.

$ 293. The variation of the needle, is determined by means of
azimuths, or amplitudes.

§ 294. The needle does not always point to the north and south
poles. At some places it points to the east, and at others to the
west, of the true north and south points.

§ a. Even at the same place, the polarity of the needle does not
remain constant

$ b. As to the direction in which the needle points, it is subject
to certain periodical changes, which do not follow any known laws.

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NAUTICAL A8TR0N0MY. 133

At some places, after having pointed, for several years, to the east-
ward of the true north, it has gradually pointed nearer to the north
point, until its position lay due north and south ; then crossing the
direction of the meridian, the needle has continued to turn more
and more to the westward of the true north, until it has attained
the maximum of its deviation for that place, when, after having
remained stationary for a time, it commenced its return towards its
former position.

$ 295. Variation of the compass is the deviation of the needle,
from pointing to the north and south poles.

$ 296. The points to which the needle tends, are called the mag-
netic north and south points.

§ 297. When the northern point of the needle, or the N point on
the compass card, points to the eastward, or to the right, of the
true north, the variation is easterly.

§ a. And westerly when the same point points to the left, or to
the westward of the true north.

$ 298. The true direction of the magnetic north point, is found
by applying the variation, when it is easterly, to the right of the true
north ; and to the left of the true north, when the variation is westerly.

§ a. Thus, when the variation is one point easterly, the north
point of the needle, or the N point on the compass card, points N. by
E. And it points N.N. W. when the variation is two points westerly,

5 6. Wherefore, knowing the magnetic bearing of any object, its
true bearing may be determined, by applying the variation, when
easterly, to the right of its compass bearing; and to the left, when
the variation is westerly.

$ c. The true bearing of an object, that bears east per compass,
is E. by S. if the variation be one point easterly ; but E. by N. if
the variation be one point westerly.

$ 299. The cause of variation, as well as of the attraction of the
needle, towards the poles, is unknown.

$ 300. The needle is subject to the influence of another power
equally mysterious in its nature ; it is called local attraction.

§ a. This attraction operates on ship-board, and with different
effects in different latitudes, as well as in the different directions in
which the vessel may be heading,

$ 6. Its effects upon the needle become obvious by taking the
bearing of a fixed point on shore, then swinging the ship entirely
around, and observing at several different points of her heading, the
bearing of said fixed point.

§ c. The effect of local attraction upon the compass, is not often .
taken into consideration by navigators, although on board of vessels I J\rQ'
in some latitudes, (as in the English channel), it is said to cause 1
the needle to deviate several degrees. The loss of fleets has been '
ascribed to the neglect of this attraction, on the part of navigators.

% d. In conducting surveys particular attention should be paid to
the effect of local attraction upon the needle.

$ 301. A magnetic meridian is a great circle that passes through
the magnetic north and south points, and through the zenith of the
observer.

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134

VARIATION.

V:

§ a. The needle always lies in the direction of this meridian.

5 b. Magnetic meridians cross each other in the magnetic poles.

§ 302. The magnetic equator is a secondary to all magnetic meri-
dians.

§ 303. The magnetic prime vertical is a secondary to the mag-
netic meridian of the observer.

§ a. It passes through the zenith and the magnetic east and west
points of the horizon.

§ 304. The magnetic azimuth of a celestial body, is an angle at
the zenith, that is contained between the magnetic meridian of the
observer and the zenith distance of the object, when its bearing is
taken.

§ a. The magnetic azimuth should always be reckoned from the
nearest pole, around towards the east, when the object is on the
east side of the meridian of the observer ; and to the west, after the
object has crossed the meridian.

§ b. The advantage of reckoning the magnetic azimuth in this
way, consists in having the true and magnetic azimuth always of
the same name ; i. e. either both east, or both west.

$ 305. The magnetic amplitude of a celestial body, is that arc of
the horizon, which lies between the centre of the body, when the
body is in the horizon, and the magnetic east or west point, accord-
ing as the body is rising or setting.

§ 306. Upon the magnetic equator, the needle assumes a horizon
tal position.

§ a. To the north or south of this equator, it points downwards 9
inclining towards the nearest magnetic pole.

§ 307. The angle of this inclination of the needle, below the plane
of the horizon, is called the dip of the needle.

§ a. The maximum of the dip is at the magnetic poles, and the
minimum at the equator.

§ b. The ratio of the increment in dip, from the equator, towaids
the poles, has never been satisfactorily established.

§ 308. The variation of the compass is found by ascertaining the
true and magnetic azimuths, or amplitudes, of any celestial object
at the same moment.

§ a. The difference between them is the variation.

§ 300. The magnetic azimuth, or amplitude of a celestial object,
is found by taking its bearing with an azimuth compass.

§ 310. The true azimuth or amplitude of a celestial object is de-
termined by trigonometrical calculations.

§ a. The usual data for this operation, are the co-latitude of the
observer, and the zenith, and polar, distances of the object

§311. In North Latitude ; the variation is easterly, if the magnetic,
be less than the true, azimuth, when the object is on the east side of
the meridian ; or the variation is easterly, if the magnetic be the
greater azimuth, when the object is west of the meridian.

§ a. And in South Latitude ; the variation is easterly, when the
magnetic is the greater azimuth in the former, and the less in the
latter case.

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tlatt V.

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NAUTICAL A8TR0N0MY. 185

$ 312. In each case, mutatis mutandis, the variation is westerly.

§ 313. Stereo graphic projection is the most useful, and being the
most natural, is the most simple mode of representing a sphere, or
circles of a sphere, upon a plaue.

$ 314. In stereographic projection the eye is supposed to be at
some point on the surface of the sphere, and to see one half of the
sphere.

§ a. The circle which terminates the vision is called, in projec-
tion, the primitive circle,

$ b. The eye is the centre of this circle, and is the projecting
point.

§ 315. The diagrams for the purposes of nautical astronomy, in
this treatise, are projected upon the plane of the horizon.

§ 317. This Fig. is a stereographic projection upon the 7
plane of the horizon, in lat. 40° north. 5 "

§ a. A, the projecting point, represents the centre of the horizon
W N E S, as well as the zenith and the eye of the observer ; P,
the pole of the observer; P N (§ 251.) the elevation of the pole ;
W Q E (§ 232.) the equator ; A Q (§ 248. § b.) the latitude of the
observer ; and A P the complement of his latitude ; W A E (§ 279.)
his prime vertical ; N P S (§ 248. § c.) his meridian ; A B / (§ 278.)
the azimuth circle of the body whose altitude is taken ; B / and
A B (§ 280.) its altitude and zenith distance ; dec (§257.) the paral-
lel of its declination ; P B (§ 258. § a.) its polar distance, and P B H
(§ 256. § a.) is an arc of its circle of declination ; A P B (§ 263.) is
the horary angle at which the body is ; B A P (§ 291 .) is its azimuth ;
N, E, S, and W, (§ 272.) are the cardinal points of the horizon ;
and W P E (§ 263. § a.) is the six o'clock hour circle.

§ 6. It is easily conceived how it is, that, of all circles which cut
the primitive circle of a projection, only that part of their circum-
ference which is above the primitive, can be seen from the project-
ing point ; and that 180° of every arc of these circles, that are great
circles, is above (§ 125. § a.) the horizon or plane of projection.

§ c. And consequently that all of these arcs will appear in the
projection to be more straightened out, or of less curvature than the
primitive circle ; and this curvature will be in proportion inverse to
the obliquity with which their planes cut the plane of the primitive.

§ d. The less obliquely these planes cut the plane of the horizon,
the more obliquely the observer at A looks upon them, and conse-
quently the less their arcs appear to be curved, until, looking upon
the edge of the planes of those which pass through his zenith, they
appear as straight lines.

§ 318. In these projections, every great circle (N P S, W A E,)
which passes through the zenith, is represented as a straight line.

§ a. And the length of every line (A N, A /, etc.) included be-
tween the projecting point (A) and the circumference of the primi-
tive, measures 90°, and is considered as the quadrant of a circle.

§ 319. All of those circles which cut the plane of the primitive
obliquely, and whose arcs (W Q E, P B, etc.) appear in the pro-

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136 VARIATION.

jection, to be of less curvature than the primitive, are called oblique
circles.

C § 320. And those circles, whose arcs (N P S, W A E,)
Plate 2. £ p^g t j ir0U gj 1 t h e ze nilh (§ 318.) are called right circles.

§ a. With the horizon as the primitive circle ; the meridian, (§ 248.
§ c.)t prime vertical, (§ 279. § a.), and azimuth circles (§ 278.), pass
through the zenith, and appear upon the plane of projection (§ 318.)
as right circles.

§ 321. In lat. 40 N., the sun's declination being 20° S. ; its mag-
netic azimuth was N. 119° 25' E. in the morning, when its true
altitude was 16° 48' 25"; required the sun's true azimuth, and the
variation of the compass.

§ 322. W N E S, (§ 316. § a.) represents the horizon of the ob-
server; A, the centre of it, his zenith, and the place at which he
stood to take the observation; n A *, the magnetic (§301.) me-
ridian ; and (§ 308. § a.) N A n is the variation.

§ a. The co-lat. (A P), the zenith, and the polar distance, (A B,
P B) (§ 317. § a.), are the sides of the triangle A P B, and (§ 321.)
are the given parts; and the azimuth P A B, is the required part;
the value of it is determined according to Case V, ($201. §rf.)

§ b. In the formula? for calculation,

P D, stands for polar distance.

Z D, " zenith distance.

Co-lat. " complement of the latitude of the place of obser-
vation.

§c. The complements of what is given (§321.) constitute the
data of the proposed triangle.

§ d. b= 50° - «Co-lat
p= 73° 11' 35"=Z D
a«110° - =PD

To find the azimuth P A B, (§ 201. §/.)
§e. a, orPD «110°

/>, or Z D — 73° 11' 35" co-sec.s 0.018959
6, or Co-lat. =* 50° - co-sec. «= 0.115746

Sum 2)233° 11' 35"

i Sum=116° 35' 47" sin.« 9.951426
i Sum qdP D= 6° 35' 47" sin.= 9.060224

2)19.146355

Cos. iPABa 9.573177«68° 1' 16"

P A B (the true azimuth, § 291. § c.)— N 136° 2' 32" E
B A n (the magnetic do. (§ 304. § a.)»N 119° 25' E

N A n» Variation ($ 311.) — 16° 3T 32" £.

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NAUTICAL ASTRONOMY. l$f

$/. The difference between the true and magnetic azi-> W1 _
muths (§ 808. § a.) (N A B— n A B-N A n) is the varia- 5 Phte *
tion.

$ g. The true, being greater than the magnetic azimuth, the varia-
tion ($311.) is easterly.

$ 323. Bearing in mind, that the co-sec. sine, etc., of an arc (§ 99.),
is the sec, cos., etc., of the complement of that arc, the process by
which the subjoined formula is deduced from (§ 322. § e.) the one
above, becomes manifest.

§ a. Calculation by this formula operates more directly upon the
data, and is, perhaps, preferable in practice on account of its greater
readiness.

$6. P.D.— 110°

Alt.— 16° 48' 25" sec— 0.018959
Lat.— 40° sec.— 0.115746

2)160° 48' 25"

i Sum— 83° 24' 12" cos.— 9.060242
(i S. ooP.D.)=~26° 35' 48" cos.— 9.951425

2)19.146372

Cos. i tr. azim.— 9.573186— 68° 1' 15'

Tr. azimuth— N 136° 2' 30" E.

$ c. This difference of 2" in the result of the two methods, arises
from the fraction of a second in i sum; which, in either case, is
not taken into computation.

$ d. Whence (§ 323.) the rule for calculating an azimuth. Take
the difference between the P.D., and i sum of the lat. alt. and P.D. ;
then the product of the cos. of this i sum and of this difference,
multiplied by the product of the sec. of the lat. and of the alt, is
double the cos. of i of the true azimuth.

AMPLITUDES.

$ 324. When the magnetic, and true amplitudes of a body, are
both north, or both south, the difference between them is the va-
riation.

§ a. But when one of the amplitudes is north and the other south,
they are of different names ; and their sum is the variation.

$ 325. If the amplitudes be of the same name (§ 324.), and the
true be the less northern, or the greater southern amplitude, the
variation is easterly when the object is rising ; and,

$ a. The variation is also easterly, when the object is setting, and
the true is north and the greater, or south and the less, of the two
amplitudes.

Digitized by VjOOQIC

138 VARIATION.

§ 326. If " magnetic 19 be read for " true 19 in the conditions above,
the variation becomes westerly.

§ 327. If the amplitudes be of different names (§ 324. § a.), and
the tnte be the northern amplitude, when the object is rising, or
the southern amplitude when the object is setting, the variation is
westerly.

§ a. The converse of these conditions makes the variation easterly.

§ 328. The true amplitude and declination of a body, are always
of the same name.

§ a. Therefore it is known by inspecting the declination, whether
the true amplitude be north or south.

§ 329. Every object in the heavens, always rises and sets to the
north, or to the south of the east and west points, according as its
declination is north or south ;

§ a. The equator (§ 272. § a.) cuts the horizon in the east and
west points; and a parallel of north of south declination, being
(§257.) parallel to the equator, must therefore cut the horizon, (if
at all), to the north or south of the east or west points.

§ 330. A body (say the sun), rises and sets in the points, in
which the parallel of its declination cuts the horizon.

C § a. Thus the sun's declination being 20° S., it rises
* I and sets in the points (c & d) in which the parallel (*rf e c)
of that declination cuts the horizon.

§331. To an observer who is on the equator, the sun's declina-
tion at the time of its rising or setting, is its amplitude ; for,

§ a. The equator is then the prime vertical, and that arc of the
horizon which (§202.) measures the amplitude, coincides with
the arc of the circle of right ascension, which (§ 255. § b.) mea-
sures the declination.

§ 332. The sun rises in the east, and sets in the west, point, onty
twice in the year.

§ a. This happens at the moments in which the sun enters the
first point of Aries, and the first point of Libra, for (§226. and
§ 226. § c.) the sun is then crossing the equator.

§ 333. To an observer at either pole, the horizon and equator
coincide, and the sun does not set, while it is on that side of the
equator, which is next to the observer. And,

§ 334. If the observer approach any point of the equator, that
point will rise above, and the one diametrically opposite, will sink
below i the horizon as many degrees as the observer advances from
the pole. And,

§ a. The observer will not see the sun go below the horizon,
until his distance from the pole is greater than the sun's declination.

§ 335. When the latitude and the sun's declination are of the
same name, the sun is not seen to set as long as its declination is
greater than the co-latitude.

§ 336. To find the sun's true amplitude, and thence the variation.

§ 337. At sun-rise in lat 40° N., the sun's declination being 20°
S. t its magnetic amplitude was E. 9° 53' 32" South ; required the
true amplitude, and the variation.

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NAUTICAL ASTRONOMY. 139

$ 338. The place of the sun at the time of its rising ($ 330. $ &)
is at the point (c) in which the parallel of its declination intersects
the horizon ; w A e (§ 303.) is the magnetic prime vertical ; >
the arc E c ($ 202.) is the true, and e c (§ 305.) the mag- 5
netic, amplitude of the sun ; and P A e is the triangle of the pro-
posed problem, of which the sides are given, and P A c is the part
required. The triangle (§ 142.) is quadrantal, of which the quad-
rental side is A c, the zenith distance.

§a. Now ($150. $ a.), calling the quadrantal side (Ac) radius,
and the legs (P A, P e) angles, and the angle P A c a leg, the pro-
blem is an example under Case IV. ($ 173. § &.).

$6. PAorco-lat.» 50° cosec. =0.1 15746
PcorZ.D. =110° cos. =9.534052

Cos. PAc«=9.649798=116° 31' 4'

§ c. Now ($ 133. § a.) the arc N E e is the measure of the angle
P A c ; and the arc E c (§ 52. § a.) is the complement of N E c, and
($ 292.) the sun's true amplitude.

§<*. E c, the sun's true amplitude«E 26° 31' 4" S
t c, the sun's magnetic do. »E 9° 53' 32" S

The variation«16° 37' 32" E (§ 324.)-

§ «• The amplitudes are both south, and the true is the greater ;
wherefore (§ 325.) the variation is easterly.

$ 339. By recurring to § 323. it is evident that the process of cal-
culation ($ 338. $ 6.) for determining the amplitude, may be simplified,
and rendered more convenient in practice, by a similar artifice.

§a. Lat. =40° sec. =0.1 15746
Dec.

=2G

)° sin. =9.534052

:26°

31'

Sin.

ampl.=9.649798=

4"

Magnetic ampl.=

i 9°

53'

32"

Variation=16°37'32"E

$ b. Whence the general rule in practice, for finding an ampli-
tude.

The sine of the amplitude is the product of sec. of the lat. and
sine of the dec. See Table VII.

SUNRISE.

$ 340. When the latitude of an observer, and the sun's declina-
tion, are either both north, or both south, the sun always rises be-
fore, and sets after, six o'clock.

§ 341. When the latitude and declination are of different .names,
the sun rises after, and sets before, six o'clock.

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140 TIMS.

$ 343. When the sun rises in the east, or sets in the west* point
of the horizon, the sun is ($272. $ a.) on the equator, and (§ 263.
put* a. 5 $ a * n * te s * x °' c l° c k h° ur circle ( W P E) ; and therefore
^"^ I rises and sets at six o'clock.

$ 343. The time of sunrise subtracted from 12 o'clock, gives the
time of sunset, when the points in which the sun rises and sets, are
equidistant from the point at which it crossed the meridian.

§ a. This happens when the declination at sunrise, and at sunset,
is the same.

§ 344. When the sun ($ 342.) rises and sets at six o'clock, it is
also in the first point of Aries, or of Libra;

§ a. As the sun's declination increases, its right ascension (§ 250.)
becomes greater, and the first point of Aries farther from the east
point of the horizon, when the sun is rising ; and,

§ b. The interval between sunrise and 6 o'clock, also increases.

TIME.

$ 345. The time between sunrise and 12 o'clock, in any latitude,
may be determined by knowing the sun's declination.

§ a. This operation consists in finding the value of the horary
angle A P c, which the circle (P c) of declination, in which the sun
rises, makes (§ 263.) with the meridian (N P S) of the place.

§ 340. The value of this angle, (A P c), like that of every other,
is expressed in degrees ; but may be converted into time, in the pro-
portion of 360° to 24A, which ($218. $ a.) is the mean time in
which the earth performs a diurnal revolution.

360°
§ a. — — =15° ; therefore the sun describes an horary angle of

15° in 1A ; — - or — — =4tn ; wherefore the sun describes an horary
15 10

angle of 1° in 4m, of 15' in \m, and of 15" in Is.

§ b. Whence the rule for converting longitude, degrees, etc., into
time, and the reverse.

§ 347. Divide the degrees by 15, for hours, the product of 4 and
the remainder, is minutes, (m) ; the quotient of the minutes (') by
15 is also minutes, (m) ; and the product of 4 and the remainder to
this quotient, is seconds («) ; also the quotient of the seconds (") by
15 is seconds («) of time.

§ 348. The product of hours by 15, and the quotient of the mi-
nutes (m) by 4, are degrees ; the product of the remainder of the
minutes (m) by 15, and the quotient of the seconds («) by 4, are
minutes (') ; and the product of the remaining seconds («) by 15, is
seconds (").

§ a. The two first and two last columns of Tables II., show
(with the hours at the bottom or top,) the value in time of the de-
grees and minutes which stand nearest to them.

$ b. And these columns may be used for finding the logarithmic
ralue of hours, minutes, and seconds, as well as for converting de-

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NAUTICAL ASTRONOMY. Hi

greet, etc., into hours, etc., and vice versa. Therefore in the solu-
tion of problems, in which a given or required part consists of
hours, etc., these need not be commuted in the process of solution,
into degrees, etc. ; but may be operated with in calculations, under
the denomination of time, and thus save the trouble of substituting
their value in degrees, etc.

§ c. Substituting hours, etc., for degrees, etc., the logarithmic va-
lue of a horary angle may be taken from Tables II., according to
the directions given under § 102. for degrees, etc. Thus, to take
from the tables the log. of 4A, 20m, 40* ; 4h is found at the right
hand bottom corner of the tables ; 20m is found in the right hand
(m) column ; and above 20m, 40* is found in the (*) column on the
same side ; opposite to 40*, and in the column which has the re*
quired precept at the bottom, is the required log. ; thus, log. sine
4A 20m 40*«9.957863.

$ d. The time corresponding to any log. is taken from the tables
according to the directions under § 103. given for taking out the de-
grees, etc., for a log. sine, etc. Thus, the value in time of log. sin. *
9.618004«1A 38m 4*. The log. 0.618004 is found in the column
marked (sin.) at the top. In the left hand column (*) of the page,
and opposite to 9.618004, is 4* ; above 4* in the (m) column is 38m,
and at the top of the page on the same side is \h.

$ e. The small columns marked (diff.) show the difference which
1*. or 15" make in the log. sine, etc. of an arc or angle.

§/. To convert 33° 10' into hours, etc.; in juxta-position with
33° is 2A ; and in the left hand columns, 10' is opposite to 40* ; and
above 40*, is 12m; then 33°10'»2A 12m 40*.

§£• To convert 5A 20m 28* into degrees, etc.; 5A is found at
the bottom, and 20m 28* in the right hand columns, In juxtapo-
sition with 5A is 80°, and above 20m, but opposite to 28*, stands
7' in the (') column ; theft 5A 20m 28*=80° T.

§ 349. To find the time of sunrise in lat. 40° N., the declination
being 20° south.

$ a. The triangle of the problem proposed, is the qua- > piate 2
drantal triangle A P c with the same data, and a similar)
process, which were used (§ 338. § b.) for finding the amplitude ;

$6. P c, or P. D.«110° cot.=9.561066
P A, or Co-lat.= 50° cot. =9.923813

Cos. A P c (S 348. § d.)=9.484879=4A 48m 62*.

§ c. 4A 48m 52* <» 12A (§ 341.)=7A 11m 8*, the apparent time
of day at sunrise.

§<f. If the declination were 20° N., Ah 48m 52* (§ 340.) would
be the apparent time of day at sunrise, and Ih 1 1m 8* at sunset,

$ e. The angle c P E, (§ 263. § 6.) is the complement of the ho-
rary angle A P c.

§/. And being the difference between sunrise and 6 o'clock,
($ 262.) it is the ascensional difference.

§ 350. To find the ascensional difference, we have only to take

Digitized by VjOOQIC

142 THE PLANETS, MOON, ETC.

C the tine, where cos. was taken above, in the operation
Plate 2. £ ^ j 34 g j ^ ^ £ or filing the time between sunrise and noon.
§a. P. Dist.- 110° co-tang.— 9.561066
Co-lat. = 50° co-tang.— 9.923813

Sin. c P E— 9.484879— M 11m 8*.

$ b. lh 11m 8* (§ 349. § e.) is the complement of Ah 48m 52s,
and (§ 349. §/.) is the ascensional difference.

§ 351. By using the trigonometric functions of the complements,
the calculation (§ 350. § a.) is rendered more convenient for prac-
tice.

§<z. Dec. 20° tang.— 9.561066
Lat. 40° tang.— 9.923813

Sin. c P E— 9.484879— lh 11m 8*.

Whence the rule for finding ascensional diff.

§ b. The product of the tang, of the lat. and dec, is the sine of
the ascensional difference.

§ 352. When the latitude and declination are of the name name,
the ascensional difference subtracted from 6 o'clock, gives the ap-
parent time of day at sunrise ; and,

$ 353. Added to 6 o'clock, when they are of different names, it
gives the apparent time of sunrise.

$ a. Hence Table VIII. also shows the apparent time at sunrise,
when the lat. and dec. are of the same name ; and the apparent time
of sunset when the lat. and dec. are of different names ; for ($ 343.)
the time at sunrise subtracted from 12A shows nearly the time at
sunset. In the Table, the time from sunrise to 12 o'clock is sup-
posed to be always equal to the time from noon till sunset.

§ 354. The length of the days and nights may be determined by
means of the problem (§351.); for doubling the time from sunrise
till noon, gives the time that the sun remains above the horizon ;
and,

§ a. Subtracting this time from 24, gives the length of the night,
or the time that the sun is below the horizon.

THE PLANETS, MOON, ETC.

$ 355.- A satellite is a body that revolves around a planet, as the
centre of one of its motions, and also revolves with that planet
around the sun.

$ a. The moon is the earth's satellite.

$ 356. Mercury and Venus, ($ 230.) are called inferior planets,
because their orbits are between the earth's and the sun.

$ 357. Mars, Jupiter, Saturn, Herschel, etc., (§ 230. § a.) are
superior planets.

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NAUTICAL ASTRONOMY. 143

$ a. Their orbits are further than the earth's is, from the sun.
§ 358. Jupiter has four satellites, which, revolving with different
radii around its centre, accompany it in its revolutions around the
sun.

§ a. As they revolve in their orbits from west to east, they pass
behind Jupiter ; and as they pass in front of it, they cross its disc.
From the frequency of these passages, (called eclipses,) and the
celerity of the motion of the satellites, their eclipses with Jupiter
furnish excellent opportunities for ascertaining longitude, when the
observer is on shore* The satellites cannot be seen with the naked
eye, and are of no use for ascertaining longitude at sea, because
telescopes for observing their immersions and emersions, cannot be
used on ship-board on account of the motion of the vessel.

$ b. When Jupiter is nearly in conjunction with the sun, the sa-
tellites cannot be seen at all.

§ 359. The moon moves from west to east in its orbit, around
the earth.

$ a. The average, or mean time in which the moon performs one
revolution in its orbit, is 27d 7 A 43m 5s ; this is called a periodical
month.

§ 6. The moon performs, in nearly the same time, one revolu-
tion on its axis, by which means the same side of the moon is al-
ways presented towards the earth.

§ 360. At the same time every day, the moon is seen about 13°
10' 30" further to the east than it was the day preceding.

§ a. This gives the moon an hourly motion with regard to the
sun, or fixed stars, of about 32* 56" ; and makes the lunar, about
40m longer than the solar day.

$361. When the moon presents a round, illuminated disc to-
wards the earth, the latter is between the sup and the moon.

§ a. The moon is then about the point in its orbit at which it at-
tains its greatest distance from the sun ; and is said to be in opposi-
tion with the sun.

§ 362. At new moon, the moon is nearest to the sun, being be-
tween it and the earth.

$ a. The moon is then in conjunction with the sun.
$ o. The interval between two consecutive new or full moons is
about 29d 12| h.

§ c. This interval of time is called a sypodical month, to distin-
guish it from a periodical month (§ 359. § a.).

§ d. The difference between the periodical and synodical months
is caused by the motion of the earth in its orbit around the sun ;
for by the time the moon returns in its orbit to the same place in
the heavens, and is again in the position with regard to the fixed
stars, which it had occupied 27 d 7h 43m 5s (§ 359. § a.) before,
the earth will have advanced in its own orbit, and the moon will
have to continue on in its orbit, about 2d and 5A, before it overtakes
the earth, and occupies the position, with regard to the earth and the
sun, and presents the same appearance, which it did when the two
months commenced together.

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144 THE PLANETS, MOON, ETC.

$ 363. The light of the moon is reflected from the sun.

§ a. And the different appearances of the moon's figure, called
phases , are caused by the enlightened part of the moon being turned
more or less towards the earth.

§ 364. When the moon, passing between the earth and the sun,
crosses the straight line which joins their centres, the whole of its
unenlightened side is turned towards the earth ; then it eclipses the
sun, and is seen, like an opake circular plane, passing over the disc
of the sun.

$ 365. If the plane of the moon's orbit coincided with the plane
of the ecliptic, the sun would be eclipsed at every new moon.

$ a. But as these planes are inclined at an angle of several de-
grees towards each other, the new moon passes sometimes be-
low, and sometimes above, the line of vision from the earth to the
sun.

§ 366. At full moon, when the earth intercepts the line of vision
from the sun to the moon, the latter passes into the shadow of the
earth caused by the sun, and is eclipsed.

§ a. The moon is not eclipsed at every opposition, on account
(§ 365. § a.) of the inclination of the plane of its orbit to that of the
ecliptic.

§ 367. Syzygies are the points in which the moon is, (§ 361 . $ a.
& § 362. § a.), at apposition and conjunction. Consequently, the
moon is in syzygy at every new or full moon.

§ 368. An eclipse of the sun or moon takes place only when the
moon is in syzygy, and at, or near, one of its nodes (§ 231.).

§ 360. When the moon is half way between the syzygies, it ap-
pears as a semicircular plane, with the round part turned towards
the sun.

§ a. Only half of the enlightened surface of the moon being
seen from the earth, produces the appearance of a half moon.

§ b. In this position, the moon's angular distance from the sun
is about 90°, and the moon is said to be in quadrature.

$ 370. A quadrature is marked (d) 9 opposition (#), and conjunc-
tion (<$).

$ a. There are two quadratures ; one between new and roll moon,
the other between full and new moon.

$ 371. Disc is the face of the sun, of a planet, or of a satellite.

$ 372. Digit is the twelfth part of the breadth of a disc.

§ 373. Apogee is that point in the orbit of a body which is fur-
thest from the earth.

$ 374. Perigee is that point in which, when a body is, it is
nearest to the earth.

§ 375. Aphelion and perihelion are positions of a body with re-
gard to the sun, similar to those of apogee and perigee with regard
to the earth.

§ 376. The motion of light is progressive.

$ a. During the time which a ray of light requires to come from
the sun to the earth, the latter advances in its orbit nearly 21" ; this

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rune s.

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NAUTICAL ASTRONOMY. 145

causes the sun to be seen about that much always in the rear of its
actual position in longitude.

§ 377. The sun's aberration, is the difference between the sun's
longitude, at the time of the emanation of a ray of light from it, and
its longitude, at the time when this ray reaches the earth.

$ a. The amount of aberration must always be added to the sun's
apparent longitude, in order to obtain the true longitude of the
sun.

§ b. The longitude of the sun, given in the ephemeris, is op*
parent.

4 378. Radius vector is a line supposed to join the centres of a
planet and the sun.

$ a. The logarithm of " the radius vector of the earth" is the lo-
garithmic value of the earth's radius vector ; the semi-axis major of
the earth's orbit being taken as the base, or unity, in the logarithmic
scale.

TIME OF DAY.

Plato 3.

S 379. In the triangle P A B, are comprised the ele-
ment* necessary to the operation of finding, by trigono-
metrical calculation, the apparent time of day.

$ a. B P A, (§ 263.), is an horary angle, and if any three parts of
the triangle in which it is, be given, the value of B P A is at once
determinable.

$ 381. The primitive data, in the problem for finding the time of
day by an observation, are usually the latitude of the observer, and
the altitude and declination of the body under observation.

§ a. The complements of which form the triangle (B A P) of the
problem.

$ 382. By inspecting the triangle of the problem proposed, it ap-
pears evident, that the horary angle (A P B) expresses the interval
of time that elapses between the taking of the observation, and the
transit of the body, (say the sun), across the meridian of the ob-
server.

§ a. The time thus deduced is apparent time.

$ 383. If the observation be made in the forenoon, the value of
the horary angle must be subtracted from 12A, in order to find the
apparent time of day.

$ a. But if the time be P. M. the value of the horary angle ex- /
presses the apparent time of day when the altitude was measured.

5 384. This problem, coming under Case V., (§201.), may be
solved by either of the methods of solution there shown.

$ a. But as it is a rule, in the application of theory to practice,
(and one too, which is highly important in navigation,) to combine
facility of design with readiness of execution, a solution of the pro*
blem after the formula, §/. (§ 201.), will be given, in order to show
thereby a process of solution less circuitous ; by which process the
problem under consideration may be solved.

$ 385. In lat. 20° N. ; the observed altitude of the sun's lower

U

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146

TIME OF DAY.

limb, was 29° 29' 50" ; and its declination was 10° 20' south. To
C find the time of day, the observation being made P. M.

Plate 3. | j fl> Y'mi to find the true altitude of the sun's centre.
i b. Obs. alt. Sun's L. L. -29° 29' 60"

Sun's semi-diam. ■■ 15' 47"

Sun's app. alt.— 29° 45' 37" ft 283. $ a.)

Refraction (§284.)= 1' 37" (Table IX.)

Dip - 4' ($275.$ a.)

Sun's true alt.— 29° 40' ($ 283. $ 6.)

$ c. To find the time of day.

B A, or Z, Dist.= 60° 20'

P A, orCo-lat. — 70° 00' co-sec— 0.027014 (§ 201. $/.)

P 8, or P. Dist— 100° 20' co-sec. —0.007 102

2)230° 40*

i sum— 115° 20' sin. —9.950089
(I sum" Z. Dist.)— 55° 00' sin. —9.913364

2)19.903569

Cos. i A P B-9.951784-1A 46m is

Horary angle A P B— Bh 32m U

This time ($ 383. $a.) is P. M.

$ 386. The data of the problem may be operated upon, more
directly, by using the lat. and alt., instead of their complements.
$ a. Alt. - 29o 40'

Lat — 20° 00' sec. —0.027014

P. D.— 100° 20' co-sec— 0.007102

2)150° 00'

i sum— 75° 00' cos. —9.412996
(i sum odAU.)— 45° 20' sin. —9.851997

2)19.299109

Sin. i A P B— 9.649554— 1 A 46m is

2

Horary angle A P B— 3A 32m 1*

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NAUTICAL ASTRONOMY. 147

$ 387. Whence the general rule for practice.

Take the difference between the true alt., and half the sum of the
lat, P. dist., and alt. ; then the product of the sine of this difference,
and cos. of said half sum, multiplied by the product of sec. of lat.>
and co-sec. of P. dist., is the sine of half the horary angle.

§ 388. The time thus found being apparent time, may be con-
verted into mean time (§ 217.), by applying to it the equation of
time, according to the precept given with the equation of time in
the Nautical Almanac.

$ a. App. time (* 385. $ c.) 3A 32m 1* P. M.
Equation of time — 13m 50*

Mean timess3A 18m lis

$ 380. By comparing the mean time thus found, with the time
shown by a chronometer, or other time-piece, when the altitude
was measured, the error of the time-piece is obtained.

$ a. If this time-piece be regulated for a prime meridian, which
is the case with chronometers; the difference between the true
chronometrical, and the mean, time found by observation, expresses
in time the difference in longitude between said prime meridian and
the place of observation.

$ 6. This difference of time being converted (§ 348. § g.) into
degrees (°), minutes ('), etc., expresses the longitude of the ob-
server.

LONGITUDE BY CHRONOMETER.

$ 390. The whole doctrine of determining longitude, consists in]
knowing the time of day at any two places at the same instant.

§ a. This is what is determined by every practicable method of
finding longitude ; whether it be by means of rockets, eclipses, oc-
cultations, lunar observations, or chronometers.

$ 391. The time of day at the prime meridian when an eclipse,
occultation, or distance, occurs, is set down in the ephemeris ; and
the time of day at any other place when the same eclipse, etc.,
occurs, is known, either by well-regulated time-pieces, or by ob-
servations.

$ a. And the difference between these times, (§ 389. $ 6.), gives »
the longitude of the observer.

S b. The longitude is west, when the prime meridian (Green-
wich, § 252. $ c.) time is in advance of the time of day at the ob- '
server.

$c. But if the observer's time of day be in advance of the
Greenwich time, his longitude is east ; for it is evident that the sun
must cross his meridian, before it does that of Greenwich, and con-,
sequently, that Greenwich must be to the westward.

i 392. Chronometers are generally regulated so as to show the
mean time of day at Greenwich.

§ a. But they are subject to variations from change of tempera-

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U& LONGITUDE BY CHRONOMETER.

ture, etc., and from other causes, and cannot be regulated so as UP
show at all times, the true time of day at Greenwich, or at any
other prime meridian.
r § 393. The difference between the time shown by the face of a
chronometer, and the mean time of day at Greenwich, is called
" the error of chronometer."

§ a. The daily variation of this error is " the rate of chrono-
meter."

§ 394. The rate of chronometer is found by noting the difference
between mean time, and the time shown by the face of the chro-
nometer, and after several days, noting again the difference between
i the mean and the chronometer time.

§ a. The difference between these two differences, (called " com-
parisons"), shows the time which the chronometer has gained, or
/ lost, from the first to the last comparison.

§ b. And the quotient of this gain or loss, by the number of days
that elapsed between the two comparisons, is the daily gain or loss,
or rate, of the chronometer.

§ 395. The error (§ 393.) being applied to the chronometer time,
with the precept of plus or minus, according as the chronometer
be slow or fast, of Greenwich time, gives the mean time of day at
Greenwich.

§ a. And the rate of the chronometer being applied to the chro-
nometer's error yesterday, with the precept +; or — , according as
the rate is increasing or decreasing, gives the chronometer's error
for to-day.

§ 396. Before the chronometer is rated, the mean time of day at
Greenwich for rating it, is found by turning the longitude of the ob-
server, (§ 347.) into time ; and,

§ a. Adding this time to the observer's mean time of day, if he
be in west long.

§ b. And subtracting it from his mean time of day, if he be in
east longitude.

$ 397. Ten or twelve days previous to the sailing of the vessel,
will generally serve for keeping her chronometer under comparison,
in order to find its error, and ascertain its rate.

§ a. But after sailing, the rate should be compared, (and correct-
ed if necessary), as often as opportunities for making comparisons
occur.

$ 398. To find the error and ascertain the rate of a chronometer,
the observer being in lat. 40° 42' N. and long. 74° 00' W.

§ a. Aug. 4th, A. M., 1834.

Chro. 12A 58m. App. alt. Sun's L. L. 30° 17' 20"
Sun's semi-diam. =* 15' 47"

App. alt. Sun's centr.=30° 33' 7"
Refraction - « l'37' i

Sun's Tr. alt.^30 31' 30"

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NAUTICAL ASTRONOMY. 149

lb.

Sun's tr. alt. = 30° 31' 30" (§ 387.)

Lat. = 40° 42' 00" sec. =0.120254

Sun'sP.dist.= 72° 40' 30" co-sec. =0.020 164

2)143° 54' 00'

i sum= 71° 57' 00" cos. =9*491147
(Alt w i sum) = 41° 25' 30" sin. =9.820621

2)19.452186

Sin. i Horary angle= 9.726093=2A 8m 37*+

12A00m00*(§383.)

Horary angle= 4A 17m 14* - =4A 17m 14*

7A 42m 46* App. time
Equation + 5m 47* (§388.)

Ih 48m 33* Mean time, A. M.
Chron. 12A 58m 00* (§ 398. § a.)

5A 9m 27* 1st comparison, (§ 394.)

$c. Aug. 15, A. M. Chron. 12A34m25*.
App. alt. Sun's L. L. 24° 2' 10"
Sun's semi-diam. 15' 49"

App. alt. Sun's centr. 24° 17' 59"

Refraction - 2' 9" (Table IX.)

%d. Sun's tr. alt. 24° W 50" %

Lat - 40° 42' $' sec. =0.120254

Sun's P. dist 75° 51' 10" cosec.=0.013376

2)140° 49 / 00"

i Sum - 70° 24' 30" cos. =9.525452
(isumcoAlt) 46° 8' 40" sin. =9.857989

2)19.517071

Sin. J Horary angle=9.758535-2A19m58j*

Horary angle=4A39m57*

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150 LONGITUDE BY CHRONOMETER.

12A 0m 0*
Horary angle 4A 39m 57*

7 A 20m 3* App. time
Equation . + 4m 15a

7A 24m 18a Mean time, A. R|.
Chro. 12A 34m 25a

5A 10m 7a 2d comparison.

S e. 2d Compr. 5A 10m 7a

1st do. 5A 9m 27a (§ 394. $ a.)

Chro. gains 40a in 11 days.

$/. 40a-$-ll a =a&a (§394. $ ftj the daily gain, or rate.
$ g. Long, of the observer 74° W. = 4A 56m
Time of (§ d.) last. obs. - = 7A 24m 18a

f$ 396. $ a.) Greenwich time do. =12A 20m 18a
Chro. (§c.) - - =12A34m25a

Error (§ 393.) of chro. Aug. 15th, »14m 7a (fast.)

§ A. To find long, by chronometer ;

Time per watch 3A P. M.=15A 0m 0a
do. chron. - «HAl5m22a

Diff.= 3A 44m 38a Chr. slow.

Time of obs. per W. 15A 2m 30a
Chro. slow of W. 3A 44m 38a

Chro. time of obs.=llA 17m 52a

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NAUTICAL ASTRONOMY. 151

Formula for calculation.
Sun's Alt. 31° 18' (corrected)
Lat. 36° 19' sec. =0.093796

P.D. 104° 9' co-eec.=0.013381

Sum«2)171° 46'

I sums 85° 53' cos. =8.856049
RemV.= 54° 35' sin. =9.911136 (Alt.w i Sum.)

2)18.874362

Sin. i App. time=9.437181=lA3m31£*

2

App. time of obs. =2 A 7m 3* P. M.
Equation - + 4m 16*

Mean time of obs. 2h 11m 19* P. M.

Time of obs. per chro. HA 17m 52*
Chro. slow of Gr. time 1A 3m 8*

Greenwich time 12A21m 0*

Time of obs. - 14A 1 lm 19*

Long, in time (§ 391. $ c.) 1A 50m 19*=27° 34' 45" E.

§ 399. By inspecting the triangle (P B A) of the pro-? p
blem for finding the apparent time of day, it becomes evi- 5
dent, that if, of the lat., time of day, dec, alt., and azimuth, of the
sun, or any other body, any three be known, the two others are de-
terminable.

$ a. The azimuth gives the angle P A B ; and the apparent time
of day gives the angle A P B ; the altitude (B /) determines the ze-
nith distance A B ; the latitude A Q, determines P A, the co-lat.,
and the declination k B, determines the polar distance P B.

LATITUDE BY MERIDIAN ALTITUDES.

§ 400. The most common method of determining latitude at sea,
is by means of an altitude of a celestial body, measured when the
body is on the meridian.

§ a. Then the circle of the body's declination, coincides with the
meridian (N AS) of the observer.

$401. Suppose the body be a star; the zenith distance (Ac')
($ 282. § b.) expresses the number of degrees, etc., from (A) the
zenith to the star.

§ a. And the declination Q e' (§ 255. § b.) gives in the same mea-
sure, the distance of the body from the equator.

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Plate 3.

152 LATITUDE BY MERIDIAN ALTTTUDE8.

C § b. Wherefore the difference between the declination
t (Q e')and the zenith distance (A e') gives A Q the latitude.
§ 402. The sun being south of the observer, and its declination
10° 20' south, its meridian altitude (corrected) was 59° 40'.
To find the latitude of the observer,

$ a. "The sun's altitude (§ 282. $ a.) is e S ; 90°— e S=A e (§ 282.
§ 6.) the zenith distance.

$6. The observer being at A, north of the sun, makes ($281.
$ 6.) the zenith dist., A e, north.

§ c. The dec. Q c, being taken from the zenith dist. A e, leaves
A Q« the required latitude.
$ d. Sun's Z. D. (§ a.)=*30° 20' N. (§ b.)
Sun's dec. =10° 20' S.

Lat. =20° 00' N.

§ 403. Suppose the object to bear south, but its dec. to be 0° 3' N.
and its meridian alt. 79° 3'.

To find the latitude,

$o. The stars alt. is e' S ; A S — d S=A e? the *'s zenith dist.

§ b. The dec. Q e' being added to e' A the zenith dist., makes
A Q, which is the latitude.

§c. *'s Z. D.=10° 57' N.
#'s dec. = 9° 3' N.

Lat. =20° 00' N.

$ 404. Suppose the star to be north of the observer, its declina-
tion 60° 5' N., and its meridian alt as 49° 55'.

To find the lat. of the observer,

§ a. N e" is the *'s alt. The zenith dist. (§ 281. § c.) is south ;
N A— N e"=A e", the *'s zenith dist.

§ b. The dec. Q e"— A c" (the zenith dist.), gives A Q, the lat.

|c. *'s dec. =60° 5' N.
#'s Z. D. =40° 5' S.

Lat. =20° 0' N.

§ 405. Wherefore, when the dec, and the meridian zenith dist.
are both north, or both south, their sum is the latitude.

§ 406. And, when one is north, and the other south, their differ-
ence is the latitude.

$ 407. When the body is in the zenith, there is no zenith dist.,
and the dec is the lat.

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Itote J.

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NAUTICAL ASTRONOMY. Jfij|

LATITUDE BY DOUBLE ALTITUDES.

$408. The sun, (and every other celestial object), is frequently
obscured by clouds, so that a meridian alt. cannot be measured.

§ a. Under such circumstances, data sufficient for determining the
lat may be obtained by taking at two different hows of the day the
sun's alt., and noting the time that elapses between the two obser-
vations.

$ 409. The time, expressed in degrees, or hours, etc. (§ 348. § o.)
constitutes, with the zenith and polar distances, the required data.

$410. This method of finding the lat. is called "by Double AU
titudes."

$ a. The process of calculation is tedious, but the result wilt
give the latitude as correctly as It can be obtained through any other
process of oosertatkjft and caWlatkm*

§ 412. For determining the lat- by " double altitudes," ?
B D represents the declination, and B / the alt. of the sun, 5
at the first observation; asd & D / r and B' /' the dec. and alt., at the
second observation.

$ a. A P & is the sun's horary angle at the first, and A P B', its
horary angle at the second, observation.

§ o. The difference (B'PB) between these two angles, is the
angular value of the time that elapses between the observations.

§ c. B B' is the arc of the great circle, which passes thfdugh the
points, in which the centre of the sun was at each observation.

$ d. The other circles, arcs and angles, of the Fig., afg explained
under § 317. ($«,).

$ 413. The problem of " double altitudes'' involves three triangles
(B' P B, A B B, & A P B') in the process of solution.

$ a. The side B' B is common to the first and second triangles ;
P B' to the first and third; and A B' to the second and third.

$ 414. The parts that are given in the first triangle (B' P B), are
the two sides, P B and P B'(§ 412.), which are the P.dist (§ 286.)
at the first, and the seeond observation ; and (§412. § b,) their con*
tained angle B' P B«Elapsed Time.

$ o. And the parts required are, the third side B' B, and the angle
PB'B.

$ 415. The value of P B' B and B' B being determined by calcu-
lation, the parts that are then known in the second triangle ABB',
are B A and B'A(§412.); which are the zenith dist. (§ 281.) of
the sun at the first and the second observation ; and the common
side B' B.

§ a. The part here required, is the angle A B' B.

§ 416. The difference between the* angles A B' B, and PB'B
($414. § a.) gives the value of the angle A B' P, in the third trian-
gle A P B'.

§417. Then in the third triangle, the two sides AB', P B' and
their contained angle A B' P, are the given parts; and PA is the
part required, and the co-lat.

X

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154 LATITUDE BY DOUBLE ALTITTJDEa

$ 418. In north latitude, at 7 A. M. by a watch*
Sun's true alt 24° 68' 12"
Sun's dec. 22° 25' 00" N.
C $ a. And at the 10A 15m by the same watch,
*** 4 ' I Sun's true alt. 62° 49' 20"

Son's dec. 22° 24' 00 N.
$ b. The elapsed time is Sh 15m.
5419. P B (1st P. D.)=67° 35'
PB / (2dP.D.)= s 67°36 /
AB(lstZ.D.)=65° 1'48"
A B' (2d Z. D.)=27° 10' 40"
B' P B (E. time) — Bh 15m. ($ 409.)
§420. To find the side B'B and angle P B'B ($414. $a.)in th©
first triangle B'PB.
$ a. 1st. To find the Talue of V B ($200. $«. & $ *.).

B' P B (E. T.)«3A 15m cos. =9.819113 (§348.§c.)
P B' (2d P. D.)«67° 86' tang. =0.384923

Tang. aux. a»0.204036s57° 50' 23"

P B=67° 35'

(dooPB)- 9° 36' 37'

P B> (2d P. D.)« 67° 36' cos.«9.581005

Auxl. a»57° 59' 23" sec.=0.275666

(an 1st P. D.)»9° 35' 37" cos. =9.993884

Cos. B'B»9.850555»44 o 51' 30"

§ 6, To find the value of P B' B (§ 144.).
B' B »44° 51' 30" co-sec.»0.151591

BTBfE. T.) =3^ 15m sin. =9.876125
P B (1st P. D.)=67 Q 35' sin. =9.965876

Sin P B' B=9.993592=80° 10' 53"

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NAUTICAL ASTRONOMY. 1(5

§ c. In the second triangle, to find the value of the an- ? t>, _ A
gle A B'B($ 201. $<*.). 5

AB(lstZ.D.) «65° 1'48"
AB' (2d Z. DO =27° 10' 40" co-sec. =0.340319
B' B (§41.) =44° 51' 30" co-sec.=0.151501

2)137° 3' 58"

i Sum=68° 31' 50" sin. =0.068776
(I S. « 1st Z. D.)= 3° 30' 11" sin. =8.786063

2)10.246730

Cob. dJL?=0.623360= 660 0'
2

2

AFB =130° 18' 58"

PB'B (§&.) = WltfW

AB'P = 60° 8' 5"

$ d. In the third triangle, to find the value of the third side A P;
or to find the latitude (§ 200. §*. & $*.)

AB'P=50°8'5" cos. =0.806848

P B (2d P. D.)=67° 36' tang. =0.884023

Tang. auxl. a=0.101771=57° 15' 20"

1^=27° l(f 40"

(o»AB')=30° 4' 40"

P B' (2d P. D.)=67° 36' cos.=0.581006
Auxl. a=57° 15" 20" sec.=0.266018

(a»2d Z. D.) =30° 4' 40" cos. =9.937 178

Cos. AP=0.785101 sin.=lat. 37°33'50"N

$ 421. In the example last quoted, the two observations are taken
at the same place ; but in practice, this cannot always be done, es-
pecially at sea, when the vessel is changing her place during the
time that elapses between the taking of the observations.

§o* Therefore the 1st altitude must be corrected, in order to
know what it would have been, had it been taken at the same time
per watch, but at the place where the 2d obs. was made.

% 422. When the first observation is being made, note the bear-
ing of the sun and the course of the ship ; and after the second ob-
servation has been taken, find in a traverse table (Table IV.), the

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1(6 LATITUDE BY DOUBLE ALTITUDES.

course end distance from the place where the first, to that where
the second, observation was taken.

§ a. The angle, which this distance makes with the bearing of
the sun, being used in the traverse table, as a eourse, the correction
which is to be applied to the first alt., is found under this coarse,
and opposite to said distance, and in the column marked " d. lot."

$ 423. If this angle be less than 90°, the correction thus found is
addative.

$ 424. If it be greater than 90°, its supplement is the course ;
and,

$ a. The correction is subtractive.

§ 425. If the ship sail on the line of the sun's bearing, the dis-
tance sailed in miles, is the correction in minutes (') of a degree.

$ a. It is addative when the ship sails towards the sun ; and,

$ 6. Subtractive when she sails from the sun.

§ 426. The corrections are found as geographical miles, but are
to be applied to the altitude as minutes (') of a degree.

$ 427. When the angle between the bearing of the sun, and the
comae which the ship makes, is 90°, there is no correction. •

§ 428* The corrections thus obtained and applied, do not show
exactly what the 1st altitude would have been at the place of the
second observation ; but the approximation is sufficiently clpse to
answer the purposes of common navigation.

§ 429. The time that elapses between the taking of die observa-
tions, should not be less than 30m, or more than 6A, if avoidable,
especially if the observer be at sea.

$ a. If the body under observation do not change much in decli-
nation during the " elapsed time," the mean of its declination at the
twp observations may be used as the body's declination at each ob-
servation.

$ 6, This " mean" declination is found by taking out the declina-
tion for half of the " elapsed time," plus the time of the first ob-
servation.

m Z 5 c * ^*en the mean declination is used, P B«PB';
Fhto 4# 5 and the first triangle B' P B (§ 140.) is isosceles.

$ 430. The latitude at the time and place of the first observation,
may be found, if required, by making A P B the third triangle, and
using the angles P B B', A B B' and A B P, instead of P B' B, A B'
B and A B' P (§ 420. %b. & § c).

$a. But generally, in navigation, the latitude of the time present,
is most desirable to be known ; and usually, when the problem is
•solved, the vessel is nearer to the place of the second, than she is
to the place of the first observation.

€ b. A formula for practice in finding the latitude at sea by " double
altitudes ;" the mean declination being used. Under the example
$418.

Sun's mean dec.«22° 24' 30", P. D. =67° 35' 30"
«un*a 1st alt. «24° 58' 12", 1st Z. D.=65° 1 ' 48"
Sim's 2d alt. «62° 49' 20", 2d Z. D.=27° 10' 40"
£. time - 3A 15m

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NAUTICAL ASTRONOMY. 157

ft. Thft lint triangle ($420. $ e.) is isosceles ; wherefore ($900.
$**, $ze. & §r/.)

P. D. «67° 85' «0" sra.»9.965903 see.^0.418842

(i E. time)«lA 8Tm 30* sin. =~ 9. 61 5642 cot. =0.3438 12

11' D . ' »' Tl »■ f ■■»■*'< 1 111 I

ABS-r-a^fi'W" Sin,-9.581545

2 1st Ang. 80° 12' 2" Tang.«0.762654

ArcB'B«44 51'32" coHsec.=0.151587 (§420. $c.)
34Z,P.=27°10'40" co-sec. =0.340319
1st •• =65° 1'48"

2)137° 4' 0"

i S.=68°32' 0*' sin. =9.968777
(J8.«Z.D.)3°30'12" sin. =8.786087

2)19.246770

Cos. h 2d Angle« 5 =9.623386= 65° 9' 25"

2

2d Angles 130° 18' 50"
1st " =* 80° 12' 2''

3d " = 60° 6' 48"

J (3d Angle) =25° 3' 24" sin. x2= 19.253736 (§ 200. $ */.)
P. D. - =67° 35' 30" sin. =9.965903
2d Z. D. =27° It/ 40" sin. = 9.659681

2)38.879320

( P.D. «2dZ.D.)=40° 24' 50" 19.439660

PP ' C0 y Z ' P '=20 o 12 25" co-*ec.=0.461662

Tang. a=9.901322=38°32'46"
a. co-sec. =0.205412

Sin. i co-]at.=9.645072=26° 12' 32''

2

Lat.=37° 34' 56" or co-lat.=52° 25' 4"

§ d. The result by this method differs 57" from the truth. Like
every other trigonometrical calculation, in which true data are not

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158 LATITUDE BY DOUBLE ALTITUDES.

operated upon, it brings an error into the result ; but the result is
an approximation, which oscillates about the truth, and within
limits that depend upon the correctness of the assumed data.

$ e. Some navigators solve the problem of " double altitudes" by
the method proposed by a Mr. Douwes ; but the correctness of the
result by his method, depends upon repeated calculations, or the
proximity of a supposed, to the correct, latitude.*

$ 431. Latitude by " double altitudes" may also be determined by
taking, at the same instant, the altitudes of any two bodies, whose
right ascension and declination are given.

§ 432. The method of solving this problem consists in a process
of operation, precisely similar to that (§ 420.) for finding the lati-
tude by means of two altitudes of the same body, taken at different
times.

Plate 4. 5 $ * 33, ^ ie diff erence m right ascension of the two bo-
£ dies gives an angle, which corresponds to the angular
value (R' P B) of the " elapsed time," between the two observations.
§ 434. When the difference in right ascension of the two bodies,
exceeds 180° or 12A, the greater right ascension, subtracted from the
less plus 24A, gives the angle corresponding to B' P B.

§ 435. In this method of finding latitude, the altitude of the moon,
unless the Greenwich time be known, cannot be used with cer-
tainty of success. For the moon's variation in declination and
right ascension, is so rapid, that unless the Greenwich time of day,
when the observations are made, be known, the proper value of the
angle corresponding to B' P B, cannot be obtained.

§ 436. The stars afford the greatest facilities for the application
of this problem to practice.

§ a. Their variation in declination and right ascension, being for
the most part, not very rapid ; consequently when the horizon is
well defined, the data requisite to the solution of the problem, can
be obtained by means of the stars, with sufficient accuracy.

$437. In north lat. June 2, 1834, Jupiter being east, and Mars
west, of the meridian; their altitudes were taken at the same, in-
stant;

49*
0' 40" N
27m 36*
9m 20*
54' 20" N
29' 12"

Plate 6 5 $ a ' Difference in right ascension is 2A 18m 16*.

' I § 438. B f and B D are Jupiter's altitude and declina-

• Mr. Douwes' method is both supported and questioned by high authority.
Bowditch has adopted it in his Pbactical Natioatob. And there is a paper
from M. Delambre, showing that the result by this method is not always an ap-
proximation, but that, in some cases, if repetitions be made, the result will re-
cede from the true latitude.

J.'s alt.

71°

" dec.

18°

" rt. asn.

dh

M.'s "

1A

" dec.

6°

" alt.

70°

To find the latitude.

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Hat* .J.

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NAUTICAL ASTRONOMY. 159

tion ; and B> t, and B' D' are Mars* alt. and dec; and B' P B ) piate 5
is the angular value of their difference in right ascension. 5

$ 440. B P B' is the first triangle in the problem, that is brought
under calculation ; B' B and B' B P, are the required parts.

§ a. B'AB is the second triangle, and A B B' is the part in it,
that is required;
§b. ABB'* B'BP«ABP.

$ c. A P B is the third triangle, and A P is the part required to
complete the problem.
§441. PB(J.'sP.D.) =71° 59' 20"
PB'(M.'sP.D.) =84° W 40"
B'PB (diff. R. A.) = 2h 18m 16*
AB(J.'sZ.D,) =18° 11' 0"
AB'CM.'sZ-D.) =19° 30' 48"
$a. To find the value of B'B(§ 200. §*.&§*.)
B' P B (diff. R. A.) 2h 18m 16a cos.»9.915646
P B (J.'s P. D.)c*71° 59' 20" tang.=0.487937

Tang. auxl. a=0.403683=68° 27' 15"

M.'s P. D.=84° 5> 40"

M.'s P. D.«a»15° 38' 26"

P B (J.'s P. D.)=71° 59 / 20" cos. =9.490243

Auxl. a=68° 27' 15" sec.»0.435044

(am PB (M.'s P. D.)=15° 38' 25" cos. =9.983614

Cos. B' B=9.908901=36° 49' 42"

%b. To find the value of the angle PBB'(§ 420. §6.)
B B' 30° 49' 42" co-sec. =0.232578

B P B' (diff. R. A.)=7A 18m 16a sin. =9.753862
P B' (M.'s P. D.)=84° 5' 40" sin.=9.997689

Sin. P B B'«9.984129= 105° 23' 52"

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160 LATITUDE BY DOUBLE ALTITUDES.

Plate 6. | § c . To find the value of AB B' (§201. €/)
A B' (M .*s Z.D.) =19° 3W 48"
A B (J.'s Z. D.) -18° 11' 00" cosec,*=0.606*e4
B'B - • - —35° 49' 42" " a0.232678

2)73° 31' 30"

i Sum —36° 45' 45" sin. -»9.77706S
(iSQDAB',M. , sZ.D.)=17° 14' 67" " =9.472066

* - ■- - -

2)19.987471
ABB' — — —

Cos. ^?- =9.993735«9° 42' 30"

2 -

2

ABB* ±=i 19* 25' 00"
PBB' *=105°S3'52"

ABP = 85° 58* 52'

$rf. To find the value of A P or co-lat. (§ 200. § ti.)
A B P=85° 58' 52" cos. =8.845627
P B=71° 59' 20" tang.=0.487937

Tang. Auxl. a= 9.333564^12° 9' 62"
A B (J.'s Z. D.>*18« 11' 00"

(a odJ.'sZ. D.)= 6° 01' 8"

P B (J.'s P. D.) =71° 50' 20" cos.=9.490243

Auxl. a= 12° 9' 52" sec. =0.009864

(arc A B, J.'s Z.D.^6 1" 8" cos.=9.997599

Cos. A P (co-lat.)=9.4977b6S.— Iat.l8°«/4' , N.

§ 442. The data for determining latitude by " double altitudes 9 *
may be obtained also by taking the altitude of two objects, each at
a different time of the day.

$a. This method may sometimes be found useful in cloudy
weather, when the two objects cannot be seen at the same time, and
when it is highly important to know the latitude.

$ 443. When the first object (say a star) is seen, let its altitude be
taken, and the time, per watch, when the observation is made, noted
down.

$ a. The bearing of the star at the same time, must also be ob-
served, in order that the angle may be known, which is contained
between this bearing, and the rhumb line, upon which the ship sails
until the second observation may be made.

§ b. This angle, and the distance sailed during the interval that
elapses between the taking of the two observations, are to be used

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ri,itt £

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NAUTICAL ASTRONOMY. 181

in finding the correction (§ 422. $ a.) which must be applied to tfre
first altitude, to make it what it would have been, had it been taken
at the place where the second observation was taken.

§ 444. The right ascension and declination of this star must be
taken out of the Nautical Almanac, also for the time at which its
altitude was measured.

§ 445. When the second object (say the sun) is seen, its altitude
also must be taken, and the time of its being done noted down, in
order to know the time which has elapsed since the first observation.

§ 446. The right ascension and declination of the sun, when its
altitude is measured, must also be found, in order to obtain the dif?
ference in right -ascension of the two bodies when their altitudes
were taken.

§ u. This difference is an angle formed at the pole, by the inter-
section of the circle of right ascension, in which the star was, when
its altitude was taken, with the circle of right ascension which
passed through the centre of the sun, when the alt. of it was mea-
sured.

J 447. This angle corresponds to B'PB; and with the? p lato fl
portion of apparent time that elapses from the taking of) '

the first, till the taking of the last, observation, it constitutes the
angle which is contained between the P. distances of the two bodies;
and with the polar and zenith distances of these bodies, it also con-
stitutes the required data of the problem.

§ 448. The sum or difference of the difference of the two bodies
in right ascension, and the portion of mean time from one observa-
tion to the other, give the required contained angle.

§ a. If the sun be the second object, the apparent instead of the
mean time, from one observation to the other, must be taken.

§ 449. The sum of said two quantities is the value of this angle,
when the first object is to the eastward of the second.

$ 450. But when the second is the more eastwardly object, the
difference of the two in right ascension, minus the elapsed time,
gives the value of the required polar angle.

§451. If, when the second is the eastern object, the elapsed
time exceed the difference in.right ascension of the two bodies, the
second will have crossed the hour circle (§ 263. $ c.) which coin-i
cides with the circle of declination, in which the first object was,
when its altitude was measured.

$452. The excess of the elapsed time above this differ-) p]at0 6
ence in ascension, is the value of the required polar angle )
(B'PB).

§ 453. B represents the place of Venus (the fivst object), when
its altitude was taken.

$ a. Venus was then the morning star, and consequently to the
westward of the sun, (which is the second object).

§ b. Venus' altitude was taken in the morning, when the sun
was below the horizon.

$ c. P b is an arc of the circle of right ascension in which the sun
was, when Venus' altitude was taken.

Y

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162 « LATITUDE BY DOUBLE ALTITUDES.

Plate 6. 5 § <*• B P 6, is the angular value of the difference in
( right ascension of the sun and Venus, when the altitude
(B I) of the latter was taken ; it is about 2h 36m 54*.

$ 454. The sun's altitude, B' ?, was taken 4A 29m 45s after Ve-
nus' was measured.

$ 455. The sun had then crossed over the hour circle P B ($ 263.
$ c.) in which Venus was, when its altitude was taken.

§ a. The sun was at B', to the westward of this circle, when its
altitude was taken.

§ 456. B' P b, is the horary angle, in apparent time, which the
sun described during the interval between the taking of the alti-
tudes.

$ a. Consequently the difference (B P b) of the sun and Venus
in right ascension, when their altitudes were measured, subtracted
from (B' P b) the angular value of the apparent time between the
observations, gives the value of the angle B'PB.

§ 457. If the portion of time that elapses, between the observa-
tions, be given in mean time, it may be converted into apparent
time, by applying to it the fraction which during the elapsed time,
the second object's equation of time gains or loses, on mean time.

§ 458. In surveying expeditions the elapsed time should be cor-
rected and changed into apparent time, in order to determine the
latitude with exactness.

§ 450. But upon the open sea, this nicety in operation may be
omitted, for it only advances, by a very small fraction, the accuracy
of the result.

§ 460. If the altitude of the more easterly object, be taken last,
and the elapsed time be equal to their difference in right ascension,
the same hour circle (§ 263. § c.) will pass through each object
when its altitude is taken ; and the process of deducing the latitude
will be confined to the simple operation of finding an angle with the
three sides of a triangle as data ; and thence in deducing, from the
two sides and their contained angle, as data in a second triangle, its
third side, which is the co-lat.

§ a. The three sides of the former of these two triangles, would
be the zenith dist. of each body, and # the difference between their
polar distances.

$461. Feb. 25, 1832. Venus being the morning star, and the
observer being in south latitude when he made his observations.

§ a. Venus' Alt. 24° 50' (True.)

" Dec. 20° & S.

" Rt. Ascen. 19A 59m
" bearing E.{S. ($443. § a.)
Time per watch Sh 16m 19*

$6. The ship then sailed N. E. i N. 18 miles, when the sun'
seen, and its altitude taken.

Sun's Alt. 43° 44' (Correct. $385. $ A.)

Sun's Dec. 9° 16' S.

Sun's Rt. A. 22A 30m 69*
Time per watch Ih 46m 8*

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NAUTICAL ASTRONOMY. 163

§ 462. The angle between the bearing of Venus (§ 461. $ a.) 9 and
the ship's course, is five points, which (§ 422. § a.) with the dis-
tance sailed, (18 miles), gives l& (§ 426.) in the column (d. lot.) of
the traverse tables ; which 10' ($423.) being added to Venus' altitude,
makes it 25°, (what it was at the time of its being taken, but at the
place where the sun's alt. was observed.)

$ 463. The watch kept mean time. The elapsed time ($ 457.)
must therefore be corrected, in order to obtain the portion of appa-
rent time, that elapsed from the first, until the second observation
was taken.

$ a. During this interval there were 2* less of apparent, than of
mean time, which being subtracted from the mean time elapsed,
gives the elapsed apparent time.

$ 464. This correction is obtained from the ephemeris. It is the
quantity which the equation of time gains or loses from the first to
die second observation ; and it must be applied accordingly.

§ 465. The triangulation, through which the process ) piato 6
of finding the latitude by means of the data in this pro- J
blem is conducted, is represented in the triangles B' P B, A B' B and
AB'P.

$ 466. A B (star's Z. D.)=65° 0'

P B (star's P. D.)«69 56'

A B' (sun's Z. D.)=46° 16'

PB'(sun , sP.D.)=80°44'

Elapsed mean time = Ah 29m 49*

Correction ($ 463. $ a.) 2*

Elapsed app. time - »4A 29m 47*
Diff. in R. A. of sun and star=2A 31m 59*

B' P B=1A 57m 48* (§ 456. $ a.)

To find the latitude.

$ a. In the triangle B' B P, to find the value of B' B. ($ 200.

B'PB«lA57m48* cos. «= 9.939911

PB'(Sun'sP.D.)=80°44' tang.= 0.787389

Tang. auxl. «= 0.727300=79° 23' 15"

PB=69°55' 0"

(aa>star'sP.D.)= 9° 28' 15"

P B' (sun's P. D.)= 80° 44' cos. =9.206906
Auzl. a»79° 23' 15" sec. =0.734791

(aooP B) - =9° 28' 15" cos. =9.994040

Cos. B' B=9.935737=30° 24' 24"

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-\

154 LATITUDE BT DOUBLE ALTITUDES.

« * a * $ *. To find the value of PB' B.
FWa I B' B =30° 84' 24" conse* =0.295734
B' P B - —I* 57m 48* sin. =9.691668

P B (star's P. D0=69° 56' 0" sin. =9.972755

Sine P B' B =9.9601 67 =65° 49' 50''

Cc. To find A B' fc (§ 201. §/.)
A B (star's Z.D.)=65°

A B' (sun's Z. D.)=46° 16' co-sec.=0.141 123
B' B - =30° 24' 24" co-sec* =0.295734

2)141° 40' 24"

i snm=70° 50' 12" sin. =9.975241
(JS.* star's Z.D.)= 5° 50' 12" sin. =9.007291

2)19.419389

Cos. ^5=9.709694=59° 10' 10"
2 ^

2

AB'B=118°2(K20''
PB'B= 65° 49' 50"

A B' P= 52° 30" 30*

§ i. to find A P, the co-lai.

AB'P=52°30'30" cos. = 9.784365
P B' (Sun's P.D.)=80° 44' tang.= 0.787389

Tang. auxL a= 0.571754=74° 59' 37"

AB'=46°16' C

(a»AB')=28"43'3r

P B' (sun's P. D.)= 80° 44' eos.=9.206906
Auxl. c=74° 59' 37" sec.=0.586823

(<•» sun's Z. D.)=28° 43' 37" cos.=9.942959

Sin. Complement A P, or laL=9.736688=33° 2' 59" South.*

• This difteta 1" frtoa the train.

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NAUTICAL ASTRONOMY.

LUNARS.

5 467. The problem of finding the true distance between th6 sun,
or a star, and the moon, for the purpose of thence determining the
longitude, is one of great importance in navigation.

$ 468. The necessity for the circuity of calculation in finding the
longitude by lunar observations, arises from the circumstance that
the two observed bodies are not seen ($ 289.) in their true places.

$ a. They always appear, each in its proper azimuth circle, but
higher up, or lower down in it (§283. § c), than its true place,

$ b. Aiid consequently (§ 283.) they appear either nearer to, or
farther from, each other, than they really are, which makes the dif-
ference between the true, and apparent lunar distance*

$ 469. Only two triangles are involved in the process of finding,
by trigonometrical calculation, the true from the observed lunar dis-
tance.

$ a. Of the first triangle, the three sides are given ; and the angle
required is that which is contained at the zenith, between the zenith
distance of the moon, and of the other body.

$ b. Of the second triangle, this angle and its adjacent sides, are
the given parts ; and the third side, or true lunar distance, is the
part required.

$ e. The three sides of the first triangle are the apparent lunar,
and the app. zenith distances of the two bodies.

$ d. And the three sides of the second triangle, are the true lu-
nar and zenith distances.

$ 479. The lunar, or the true lunar distance of the sun, or a
star, is an arc of a great circle, contained between the centres of
either of those objects, and the moon.

§471. The distance between the perfect limb ($283.), or the
round edge, of the moon and the near limb of the sun, is the dis-
tance usually measured with a sextant.

§ 472. The apparent is obtained from the observed distance, by
applying, with proper signs, the corrections for the error of the sex-
tant, for the semidiameters of the two bodies, and for the augmenta-
tion of the moon's semidiameter.

§ a. The sun and moon's semidiameter is given in the ephemeris ;
the sun's for every 24A, and the moon's for every noon and mid-
night.

$ b. The correction for augmentation is found in Table X*

§ 473. Augmentation of die moon's semidiameter, is the differ-
ence between the visual angle of the moon's semidiameter at the
centre, and its visual angle at the circumference, of the earth.

$ 474. The difference between these two angles, is proportional
to die ratio between the length of the semidiameter of the earth, and
its distance from the moon.

§ 475. On account of its great distance from the earth, the sun's
semidiameter has no augmentation.

$ a. As the earth's semidiameter measures more in proportion to

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166 LUNARS.

the moon's geocentric distance, than to that distance of the sun, or
of any other heavenly body, the difference of the moon's semi-
diameter, when seen from the circumference and from the centre of
the earth, is greater than that of any other body, when seen from
the same positions.

§ 476. The semidiameter of the moon when it is in the horizon,
appears under nearly the same visual angle (barring refraction) that
it does from the centre of the earth.

§ 477. The moon appears larger when it is in the horizon, than
it does when in the zenith. This though is an optical delusion, for
the moon actually subtends the largest visual angle when in the ze-
nith ; as it is then nearer to the observer than it is when in the ho-
rizon, by a little more than the semidiameter of the earth.
Plate 7 I § * 78 ' B' A B represents a diagram of the lunar problem.
' J b' b is the apparent, (§ 472.), and B' B the true ($ 470.)
lunar distance.

§ 479. The three apparent distances, (b' 6, b 1 A, A 6), are the given
parts ($469. §a.) of the first triangle b' A 6, and the angle V Ab is
the part required in it

$ 480. The angle V A b is common to the second triangle B' A B,
of which the true zenith distances A B, and A B' are the given, and
the true lunar distance B' B is the required part

§ 481. The apparent Z. D. of the sun, or of a star, ($ 284. $ a.)
being less, and the app. Z. D. of the moon (§ 286. § a) being greater,
than the true Z. D.; it appears from the diagram, that, when the
moon's is the greater of the two Z. D., the true is always less than
the apparent lunar distance.

$ a. And when the moon's is the less Z. D., the true is some-
times greater, and sometimes less, than the app. distance.

§ b. The true is the less, when the lunar distance exceeds 90°.

| c. And it is generally the greater, when the distance is less than
a quadrant, and the ratio of cos. moon's Z. D. to cos. sun or star's
Z. D., is greater than the ratio of rod. to cos. of the lunar distance.

$ 482. The reason why the difference between the app. and the
true L. dist, appears to be governed by the moon more than by
the other body, is, that the moon being nearer to the earth, has the

£ eater parallax, and is generally seen further from its true place in
e heavens, than any other body, from which lunar distances are
measured, appears from its true place.
$ 483. In N. lat, Aug. 27th, 1834.
Obs. Dist sun and moon —89° 51 ' 40"
Obs. Alt sun • =34° 4' 60"

Obs. Alt moon's upper limb«64° 27' 50"

$ a. Obs. dist sun & moon =89° 51 ' 40"
Sun's semidiameter - 15' 51"

Moon's " - - 15' 1"

Moon's augmentation - 13" ($ 472. § 6.)

Error of sextant - +6"

App. lunar dist =90° 22' 51"

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tlatf 7.

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NAUTICAL ASTRONOMY, tflT7

§k Ban's obs. nit. p 34° 4' 50"
Sun's seui .diameter * 15' 51"

♦Dip (§ 275. § a,) - — 3' 31"

Sun's app. alt - - 34° 17' 20"

§c. Refraction for sun's alt- — 1' 20" (Table IX.)
Sun's P U. in alt 8" (Table XL)

Sun*s true alt. - - 84° Iff 8"

$ d. Obs. alt moon'* XL L. 54* 27' 5<T
Moon's semidiameter - —15' 1"

Moon's augmentation - < — 13" (Table X.)

Dip (§275. §a.) - — V&l*

$ *, Moon's app, alt 54* 9 H 15" cos.=9,767805 (§288.)

Moon's hor. pix. (§ 287.) 55' 7" sin. =8.204991

Moon's plx, in alt. (§287. §<H 3 ^' 1G " ain - = 7 ^2596

§/. Refraction for moon's alt= — 13"
Correction for moon's app. ait^ 31' 33"f

$ g. To find the value of b f k b (§ 201. §/.) in the first > p ktB r
triangle. S

4' 6 (app. lunar di*t.) =00° 22' 51 "
6' A (moon's app. Z, D,)=35 Q 50' 45" co-see,=0.232394
Kb (sun's app. Z. D.) =55° 42' 40" co-sec. = 0*0829 11

2)181° 56' 16"

I mm 90* 58' 8" sin. ^0,999938

(i S.« app. lunar diet.) = 0° 35' 17" sin. ^8.011288

2)18.326531

Cob. 81° 37' 34" (=^-^) 9-168885

©'A 6=163° 15' 8''

• The dip , when the eye is 9 feet above the water, is about # ; and 4', when
the height of the eye is 1 7 feet, On hoard ship 4' is generally allowed for dip.

\ When the nicest accuracy ia required, corrections may be applied for the
difference in the effects of refraction ou the upper and tin* lower lkiib of the sun
and moon; for the moon 1 a parallax in different latitudes, supposing the equato-
rial to be ftreaier than the polar diameter ; and also a correction for the actual
barometrical, and therm oroelrical state of the atmosphere ; but at sea, greater er-
ror* are unavoidable* hence these corrections are of minor importance*

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168 LUNAR8.

§ h. To find B' B (§ 200. § v.) the tr. lunar dist. (§ 480.)
o' A 0=163* 15' 8" cos. = 9.981176
ABCSun'str.Z.D.)=55°43 , 52 ,/ tang.= 0.166625

Tang. auxl. a= 0.147801 = 125° 26'

Moon's tr. Z. D.= 35° 19' 12"

90° 6' 48"

A B (Sun's tr. Z. D.)=55° 43' 52" cos. =9.750567
Auxl. a=125° 26' - - sec. =0.236755
(aw Moon's tr.Z.D.)=90°6'48" cos.=7.296235

Tr. lunar dist.=89° 53' 24" Cos. B' B=7.283557

§ 484. By referring to the Nautical Almanac, page xvli., of the
ephemeris for Aug. 1834, it will be observed, that it was near noon
of the 27th at the Greenwich observatory, when the moon was 89°
53' 24" from the sun.

§ a. If the latitude of the place at which the observation (§ 488.)
was made, be known, the time of day at that place, and when the
observation was taken, may be found according to the rule § 387.

§ b. And the difference (§ 391. § a.) between the time of day thus
found, and the time in the ephemeris, which corresponds to the cal-
culated lunar distance, is the longitude in time.

§ 485. Distances for determining longitude are always measured
from the moon, because the change of the moon's position in the
heavens is more obvious than that of any other heavenly body that
is visible to the naked eye.

§ 486. The moon (§ 360.) has a daily motion of about 13° 10' in
the heavens ; owing to which the moon rises, culminates, and sets,
later and later every day, until, having passed through all its phases,
it crosses the circle of the sun's altitude, gets to the eastward of the
sun, and presents the appearance of a new moon.

§ 487. At a mean, an error of 2' 12" in the lunar distance, will
produce an error of 1° in the longitude.

§ 488. The moon changes its distance from those bodies, which
lie directly in its path, more rapidly than it does from those to-
wards or from which it moves more obliquely.

§ a. Therefore the stars from which the change of the lunar dis-
tance for the time being is the most obvious, should be preferred in
taking lunar observations ; for the more rapidly the distance changes,
the less will the longitude be affected by a small error in the ob-
servation.

§ 489. In the Nautical Almanac ; pages xiii. to xviii., of the ephe-
meris for each month, contain at intervals of three hours, the dis-
tance of the moon from the sun, from four of the planets, and eight
of the principal fixed stars.

§ a. The small column, marked P. L. contains the proportional

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NAUTICAL ASTRONOMY. 169

logarithm of the quantity, which the distance preceding it in the
left hand column, varies during the three hours between which the
P. log. stands.

§ 6. Table III. of this volume, contains the proportional log. for
every second from 1" to 3° ; or for every second of time from 1 * to 3 h.

§ c. Proportional logarithms are nothing more than the difference
between the log. of 3, and the log. of the given number less than 3.

Thus, the log. of 3 - - - =0.4771 +

Log. of 12' ; (12' reduced to the decimal of a de- ? Q Qm ft ■
greeis0.20.) Log. 0.20 - - J W, * U1U+

P. log. of 12', or 12m=1.1761

§ d. This Table (III.) is always used for finding the Green-
wich time when a given lunar distance occurred.
Thus, to find the time af

Greenwich when the sun

was distant from the moon

84° 17' 10"

L. Dist. at 0. 83° 56' 19"

" at 9. 85° 16' 28"

^Diff. - 20' 51" P.L.=9362
Diff. for 3A=1° 20' 9" P. L.=3514

46m49«=5848

And 46m 49* added to 6A, shows the time (67k 46m 49*) at Green-
wich, when the sun was distant from the moon 84° 17' 10".

§ e. This is a much shorter way than that of arriving at the same
result by common logs. ; thus,

As 1° 20' 9"=1.336(decm'lsof adeg.) Ar.co.=9.874193
Is to 3A - - Log. =0.477121

So is 20' 51"=0.3475 (do. of a deg.) Log. =9.540955 (§ 90.)

To46m49#=0.7803 " " Log. =9.892269 (§93. §£.)

§ 490. In selecting a star to measure a lunar distance from, that
one should be fixed upon, whose lunar distance at the time has the
least p. log. after it in the ephemeris ; for, as the greatest variations
in the dist. are represented by the smallest p. logs., the lunar dist.
of those bodies, which has the smallest p. log. after it, is increasing
or decreasing, in a greater ratio, than that is which has after it a
greater p. log. ; consequently, (§ 488. § a.) those bodies are the
most favourably situated for determining longitude.

§491. When practicable, the distance of the moon should be
measured from two stars ; one to the east, the other to the west of
it ; and the mean of the longitude resulting from the observations
should be taken as the long, of the place.

§ a. If the sextant have an unknown error, the error will partially

Z

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176 LUNAR8.

correct itself by its own counteracting effects in distances measured
to the east and west of the moon.

$ 402. Latitude by " double altitudes" may also be determined
by means of the data in the lunar problem.

pi * 5 493 ' By inspecting the lunar diagram, it is seen that
riate 7. ^ ti^ j^g ( P B> p B /) of tne circleg of declination of the

sun and moon, join the extremities of the true lunar distance (W B),
forming thereby the triangle B' P B, of which the three sides axe
known.

. § a. Then the value of the angles P B B', and A B B' being found,
^their difference gives the value of A B P, which, with B A, and
B P in the triangle P A B, constitutes data requisite for determining
A P, the co-latitude.

$ 494. The lunar problem is one of the most useful and most
beautiful problems in navigation ; for besides his longitude and lati-
tude, the navigator may deduce, from the principles involved in it,
data for the solution of almost any problem in nautical astronomy*
§ 495. The three parts A P, P A B, A P B, in the fourth triangle,
are determinable from the data obtained by a lunar observation.
§ a. A P is the co-lat. of the place of observation.
$ 6. A P B is the horary angle, which (§ 382. § a.) shows the
app. time of day at the place of observation when the latter was
made. And,

$ c. The difference between this time converted into mean time,
and the Greenwich time in the N. A. that corresponds to the lunar
disk, expresses, in time, the longitude of the place of observation.

$ cf. P A B is the sun's true azimuth at the time and place of ob-
servation ; the difference between it, and the sun's magnetic azi-
muth, (§ 308. § a.) is the variation of the compass.

$ 496. By increasing the triangulation, and extending the opera-
tions, other quaesita may be added to the problem ; and the length of
the day and night ; the time of the sun and moon's rising and set-
ting ; the hour of the moon's passing the meridian ; the hour when
each object bears east or west; the duration of twilight ; and the am-
plitudes of the two objects on the day, and at the place of observa-
tion, may all be determined by means of the parts involved in the
common lunar problem.

$ 497. Therefore when, on account of unknown drifts, and of
gales during several days in succession of thick or cloudy weather,
a ship has lost her reckoning, it may be successfully restored by a
single lunar observation.

$ a. If the variation of the compass be required also at the same
time in addition to the lat., time of day, and longitude, the calcula-
tion, after being conducted into the fourth triangle BAP, should be
conducted by a process different from that for evolving the value of
A P, the co-lat., alone.

$ 6. In the latter case, the value of A P would be determined by
the rule § *. and § /., under Case IV. (§ 200.)

$ c. And in the former case, the proportion between the sines of
opposite sides and angles, would evolve it ; for, when the three un-

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NAUTICAL ASTRONOMY. 171

p^ 7 C known parts of this triangle (BAP) are required, the
* { most direct method of arriving at the required results,
consists in applying the rules § z. and § z a., Case IV., (§ 200.) to
calculation, in order to determine first the value of the horary and
azimuth angles (A P B, & P A B.)

$<?. In trigonometrical operations, that method of calculation
should ever be adopted, which, being less elaborate, arrives more
directly at the required result.

$ 498. When the latitude, etc., is required by means of a lunar
observation, the operation of correcting altitudes and of finding the
true lunar distance, is conducted as it is under § 483. ; and then the
value of A B B' is found, before the operation passes from the second
triangle B' A B.

$ a. The true lunar distance B' B being determined, the Green-
wich time corresponding thereto, is found (§ 480. § d.) in the lunar
tables of the Ephemeris; and then the declination and the right as-
cension of the sun and moon, at this time, are found in the appro-
priate tables of the Nautical Almanac.

$ 400. The declination of the two bodies thus found, gives ($ 258.
$ b.) the sides B' P and B P.

$ a. And the difference of their right ascensions (§ 433.), expresses
in time the value of the angle B' PB.

$6. The angle P B B' is the only part in this triangle, which is
required, and it may be found without the trouble of taking from the
Ephemeris, the right ascension of the two bodies.

$500. The true lunar distance (§ 483.) 80° 53' 24" of the pro-
blem occurred (§ 484.) at Greenwich, August 27, 1834, near mean
noon, when the sun's declination (page ii. of the Monthly Epheme-
ris) was 10° 10' 2" N.; and the moon's (page v. to xii. do.) was
10* 12' 30" N. Their difference in right ascension then was
6A 13m 52s.

$4. And the sun's magnetic azimuth (§304. $a.) at the time of
observation, was N. 100° 4' 30" East.

$501. The apparent and true zenith distances are found, and the
calculation for determining the true lunar distance (B' B) is con-
ducted precisely in the same manner, as it is (§ 483.) when the
true distance is the only object of the problem.

§ 502. In the second triangle B' A B, to find the value of A B B'.
B'B (Tr. lunar D.) =89 53'24" cosec. =0.000001

B'AB c=163°15' 8" sin.=0.459632

B' A (Moon's Z. D.) =35°10'12" sin. =9.762035

Sin. A B B'=9.221068=9° 35'24"

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172

LUNAR8.

Plate 7 $ § «• In the Aird triangle B' P B, to find the value of

JPBB'.
B' B (Tr. L. D.) =89° 53' 24" cosec. =0.000001
B'PB (Diif. R. A.)=6A 13m 52* sin. =9.999205
B' P (Moon's P.D.) 70° 47' 21" sin. =9.9751 16

Sin. P B B'=9.974322=70° 29' 27"

A B B'= 9° 35' 24' A

ABP=60°54' 3"

$ b. To find, in the fourth triangle, the sun's azimuth P AB, and
the horary angle A P B (§ 200. § z b.)
AB+BP J

AB<»BP

2
ABP
2

*67° 46' 55"
=12° 3' 3"

=30° 27' 1"

sec. =0.422357
cos. =» 9.990322
cot. =0.230713

cosec. =0.033505
sin. «9.319687
cot. =0.230713

=9.583905

Tang- PAB ^ APB «77° 11' 38^=0.643392

Tang. PAB : APB -20° 50' IT' ~~ .

A P B (H. Ang.) =56° 12' 21" ($77. § h.) *~

P A B (Azm.) -N. 98° 10' 55" E.

§ c. APB (JL Ang.)=56° 12' 21"«3A 44m 40#»(§ 348. $*.)
8A 15m 11* A.M.

lm 25# —Equation of time, (Naut Al., p. ii. Mon. Eph.)

Bh 16m 36# A.M. —Mean time of day at the place of observation.

$ d . To find A P, the co-lat.

APB(Hry.Ang.) -56° 12' 21" cosec. =0.080377

A B (Sun's Z. D.) =55° 43' 52" sin.«= 9.917192

ABP -60° 54' 3" sin.=9.941402

Cos.-20° 40' 8"N. Lat=Sin. A P(co-lat.)=9.938971

$ e. Sun's Mag. Azm. (§ 500. $ a.) N. 109° 4' 30" E.
Sun's Azm. ($ 6.) N. 98° 10* 55" E.

Variation ($312.)= 10° 58' 35" Westerly.

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NAUTICAL ASTRONOMY. . 1ft

§/. Time at Greenwich ($ 484.) > PIato 7

when the ohe. was made, 12/k 00m 17# P.M. J

Time (§ c.) at the place of observation, Sh 16m 38* A.M.

Long, of the obsr. (§ 391. $ 6.) W. 3A 43m 41s— 55° 55' 15"

$603. If the latitude and time of day, without the variation, be
required from a lunar observation, the process of calculation may be
varied from that above for finding the azimuth, time, and latitude;
and may be made more direct

§ a. but in either case, the order and method of calculation are
the same, in the process of arriving at the value of A B P in the
fourth triangle.

$ 504. The true lunar distance (B' B) must always be determined,
before the correct P. distances, (P B, P B') and the difference (B' P B)
in right ascension (§ 408. § a.) of the two bodies, can be known.

§ a. These parts must always be taken from the Ephemeras, and
for the time in it, which corresponds to the true lunar distance.

$ 505. Having found, in the second triangle, the required angle
ABB'; then having taken (§ 498. § a.) from the Ephemeris, the
proper P. D., etc., and calculated ($ 502. § a.) the value of the angle
(P B B') required in the third triangle ; the value of A B P is ob-
tained , and the lat. and time of day may thence "be deduced by the
following formula.

$ 506. Formula of calculation for finding the latitude and time of
day by a lunar observation.

To find the lat. (§ 200. § n.)

Contd. Angle (A B P) cos. =*.******
Sun's Z.D. (A B) cos. ac*.****** tang. =*.******

Auxl. a seers*. ****** Tang. auxl. a— *.******
(a » Sun's P. D.) cos.— ••——

Cos. co-lat. (A P) „♦.♦*****

To find the time of day, (§ 144.)

Co-lat. (AP) co-sec.—*.******

Contd. Angle (A B P) sin.=*.****»*
Sun's Z. D. (A B) sin.-*.******

Sin. Horary Angle (A P B) =♦.♦••**»

$ 507. If the second and third triangles fall on opposite 7 piat0 8
sides of the lunar distance B' B, then the sum of the an- 5
gles (ABB' & PBB) of the second and third triangles, gives the
value of the required angle (P B A) of the fourth triangle.

$ 508. When the polar distance of each body is greater ?
than the co-latitude of the observer, then the second and 5 ^ ate T '
third triangles fall upon the same side of the lunar distance B' B,

Digitized by VjOOQIC

174 U7NA8&

C and the difference between the angW* (A B B', P B B') is
£ the value of the angle (A B P) sought in the fourth triangle.
§ 509. It may always be known, whether the sum or difference
of the angles of the second and third triangle, will give the angle
required to be known in the fourth triangle, by observing the incli-
nation of the plane in which the sextant is held to bring the sun and
moon in contact.

$ a. If the sextant be inclined to the plane of the, horizon, to-
wards the pole of the observer, then his co-lat is greater than the
P. dist of either body, and the second and third triangles fall upon
different sides of the lunar distance, and the sum of the two angles
is the required angle of the fourth triangle.
$ WO. May 16, (P. M.), 1834.

Obs. dist. sun & moon 07° 16' 28"
Obs. alt. sun's L. L. —41° 9'59"
Obs. alt. moon's V. L.=37 Q 68' 24"
Required the latitude and longitude of the place of obseiration f
§ a. To apply the preliminary corrections.
1st. Obs. lunar dist. 07° 16' 28"

Sun's semidiam. - - W 60" (N. A., p. ii., M. E.)

Moon's « - 1* 6" ( « iii. " )

Moon's augmentation - 10^

App. lunar dist -97° 48' 84"

24* Obs. alt sun's I*. I*
'Sun's semidiameter

Sun's app. alt
3d. Sun's refraction
Sun's plx. in alt -

41° 3' 58"
WW

41°19'49" App.Z.D. 48°40'11"
—I'll"

Sun's true alt 41° 18' 46"

4th. Obs. alt moon's U. L. 37° 68' 24"

Moon's semidiameter
Moon's augmentation

Moon's app. alt
Moon's hor. plx. -

Moon's plx. in alt

M oon's refraction

— le' 6"

—10"
37° 42' 8"

46' 44"
—I' 14"

Z.D. 48° 41' 16"

cos. -9.898286
69' 4" sin. -8.236047

Sine— 8.18883*

Moon's true alt 88° 27' 88" Z. D. 61° 32' 22"

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Plate A

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NAUTICAL A8T10N0MY. 175

$ ft. To find the tr. taiar dist

1st To find b A 6' in the first triangle, (§ 201. 5/.) > __ 4
b 6' (Ap. lunar dist) -97° 4g' 34" £ Plato *

A 6 (sun's ap. Z. D.) -=48° 40' 11" co-sec. «0. 1244 10

A V (moon's ap. Z. D.) -=52° 17' 52" co-sec. =0.10 17 14

2)198° 46' 37"

i Bum=99° 23' 18" sin. =9.994144
(J S^b b f ) =* 1° 34' 44" fiin. =8,440173

2)18.660441

Cos, *A£=77° 38' 55"= 9.330220

2
2d. To find BB'?

($200, so J bkb t =i^nrm it cob. =9.953319

AB- 48°41'15" cos— 9 819653 - - tang=0.056056

AuxL a - B ec.=0,157822Tang.al34 a 3' 7" =0.014375

51°32'38" (M,'b Z, D.)
(a«M/sZ.D.) cos.=0.1H977 - 82°30'45" (A B' «a)

Cos. B B =9.092452 = 97° 6'25" (Tr. ft dist.)

When this distance (97° 6' 25") occurred, it was 9 P. M. at
Greenwich of the given day (N. A. p. xv., Monthly Ephemeris.)

§ c. To take ($498. § a.) the other requisite data from the ephe-
mera.

Moon's dec. 14° 59 / 6" N. P. D. 75° 0" 54" (N. A. t p. viii. M, E.
Sun's dec. 19 Q 8' 28" N. P. D. 70* 31' 32" ( » ii.
Moon's Rt A. 10A25ml5s - ( viii, «

Sun's *• 3A52m25* - ( m H. "

Polar angle =6fc 52m 50* =B P 11 '.

$ d* To find the required angle B' B A of the second triangle.
B B' (lunar dist) =97° 025" coaec. =0.003350

BAB' - 155°L7'50 " sine =9-621084

A B' (moon's Z. D,) =51°32'22" sine =9,893782

Sin, B'BA=9.51821G=19*lft'18'

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176 LUNAB8.

« ». » S $ «• To find the required angle B' B P of the third
Plato 8. « tri^gfe

BF (lunar dist.) -= 97° 6' 25" co-sec. =0.003350

B'PB (pl'r. angl.) — 6A 52m 50" sin. =9.988356
P B' (moon's P. D.) = 75° 0' 54" sin. «9.984975

Sin. B' B P«71°23 / 30" =9.976681
5/ 4th triangle B A P ; > B' B A-19°16'18" ($ 506.)

nd the co-lat. $

P B A— 90°38'48" cos. =8.052549
AB«48°4i'15" cos.=9.819653 - tang. =0.056056

Y

to fin

Auxl. a - sec.=0.000036 Tang, a 179° 15'51" -=8.108605

70°51'32"(S.'sP.D.)

(a od sun's P.D.) cos. =9.499325 108°24'19" (A Boo a)

Cos. A P, or sin. lat.«9.319014— 12°1'55"N. see.»0-009647

To find the Hr. > P B A=90°38'49" sin.»9.999973

angle A P B. J A B (sun's Z. D.)=48°41'15" sin.»9.875710

Sin.APB 3&20m41*=9.885330

Equation= 3m 56*

Mean timenSA 16m 45* P. M.
Gr. time (§ 6.)=9A P. M.

W. Long. (S 391. $ 6.) 5A43ml6*=85°48'4G"

$511. Examples under $ 483. and § 510. are drawn oat in the
solution, in order to familiarize the problem to the student. But
such a process of calculation as is there shown, is not desirable in
practice ; especially that part of it which has been made to come
under Case IV., (§ 200. § n.) on account of the perplexity in deter-
mining whether the value of the auxl. a be greater or less than 90°.

§ a. Therefore, in practice, those methods of calculation are ge-
nerally preferred, which give the trigonometric function of half the
value of the required pait; such as § a. (§ 386.), for half the value
of a required arc or angle should never exceed 90°.

$ 512. The learner has now become familiar with the triangula-
tion, which is brought under calculation in the process of finding
longitude, etc., from a lunar observation ; he will therefore readily
comprehend the manner by which such an operation is simplified,
and reduced to the following methods and formulae for practice.

§ 513. The process of correcting the obs. distance and altitudes
is fully shown under § 483. and $ 610. This part of the operation

Digitized by VjOOQIC

NAUTICAL ASTRONOMY. 177

is the same in every method for calculating the true from aa obs.
lunar distance. .

$ 514. An improved method for finding the trod from an-observed
lunar dist«*

$ a. Aug. 27th, 1834.
App, dist. sun & moon— 88° 26' 52"

Sun's app. alt.
Sun's refraction -
Sun's pl'z. in alt.

«47° 52' 12'
— 53"

7"

Sun's tr. alt.

47°51 v 26"

Moon's app. alt. -
Moon's nor. pl'x.

40° 52' 58" cos. s *9.8?855a
55' 0" sin.«8:2(Ht070

Moon's pl'z. in alt.

41' 35" sine=8:082620

Moon's refraction

- —1' 7"

Moon's true aft.

=4t»3&'3t6*'

$b. App. lunar D.
S«n's app. sit.
Moon's "

88° 26' 52"

47° 52' 12" sec.tttO.178897

40° 52' 58" see»~(U21449

2)177° 12' 2"

isnm=88°36' 1" cos. =8.387876
(iS. qo app. lunar D.) « 0° 9' 9" co8.=9.999998

Sun's tr. alt.
Moon's "

47° 61' 26" cos.«9.826710
41° 33' 26" cos.-s9.874073

Sun's + Mon's alt.

Sun's + Moon's Alt.
2

2)38.383503

19.191751
-44° 42' 26" cos.-9.851693t

Sin. auxl. a»9.340058«12 o 38 / 20"

Auxl. a cos.=9.989847f

ftin. Tt - lunaT dia1; --9.841040-43 o 54'25 T '

9. —

2

True lunar dist.»87°48 / 50"

• Vide KjaLi'i Sphkbics, p. 105.
Aa

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178 LATITUDE BY THE NORTH 8TA&.

$ & (N. A. f p. xvu., M. E;) 8A P. M.

Sun's lunar dist. 98° 2tf 8" P. L.»3277
True lunar diat. =87° 48' 50"

40° 18" P. L.=6500 Bh P. M.

3223=lA25m42*

Greenwich time =*4A 25m 42*P.M.

LATITUDE BY THE NORTH STAR. *

$515. In northern latitudes, the latitude may be found at any
time of the night, by an altitude of the north star.

§ 516. If the north star were situated in the pole of the world, its
altitude would always show the elevation of the north pole ; and
($251. $ a.) consequently to the observer, his latitude.

$ 517. But as this star revolves in a very small circle around the
pole, the elevation of the pole may always be determined by apply-
ing to the star's altitude at any time, the corrections opposite to
that time in the subjoined tables (A & B).

$ 518. The corrections in Table A, and those after ( *'s alt. +)
in the hour columns of Table B, are always to be added to the
star's alt. They are expressed in seconds (").

$ 519. And the other corrections in Table B, must be applied to
the star's altitude, according to the precept, which is at the head of
the hour column in which the sidereal time is found.

$ 520. To find the lat. by an alt. of the north star; the altitude
being corrected for dip, refraction, etc.

$ a Subtract 1' from the star's alt.

§ o. For the supposed time of night when the observation is
made, take the corresponding sidereal time ; the correction found
in Table B and opposite to this time, being applied, according to
the precept (§ 519.) at the head of its column, to the star's alt.,
gives the approximate latitude.

$ c. - And the corrections under the hour in Table B, and opposite
the hour, but under the " approximate latitude," in Table A, being
added to the approximate latitude, give the lat. of the observer.

$ 521. May 4, 1834, long. 85° W. the altitude of the north star,
at 7h P. M., was 35° 29' 43". To find the lat.

$ a. Supposed time - - 7h Qm Of P. M.
Longitude in time 5A 40m Os

Time at Greenwich - - 12A40m 0* (§ 396. $ a.)

Sidereal time, May 4 - - 2h 47m 34s (N.A.,p.ii.,M.E.)

Supposed time 7h

Acceleration of S. T. for 12A 40m 2m 59 ($ 213. $ b.)

Sidereal time of obs. - - =»9/i49m39i

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NAUTICAL ASTRONOMY. 179

ib. Star's alt. 35° 29' 43"

1' (§530. § a.)

35° 28' 43"
Oppos. 9A 49m 39* (Table B) correction + 1° 3' 38"

Apprx.lat. 36° 32' 21"
Under 9A, *'s alt.+(Table B) - 66"

Oppos. 9A 49m 39* and under 40° (Table A) 32"

Latitude 36° 33' 59" N.

§ 522. Jan. 14, 1834. Long. 45° East 9A P. M. ; the altitude

of the north star was 41° 11'. To find the lat.

$ a. Supposed time 9A

Longitude in time Sh

Time at Greenwich - - 6A ($396. §6.)

Sidereal time, Jan. 14 - I9h 33m 53* (N. A., p. ii., M.E.)

Supposed time of obs. - 9h

Acceleration of S. T. for 6A 59* (§ 213. 5 b.)

Sidereal time of obs. - - 28A 34m 52*— 24A=4A 34m 52*

§ b. Star's altitude - - - - 41° 11'

1'

41° 10'
Opposit. 4h 34m 52* (Table B) Cor. —56' 15"

Appro*, lat. - - ... - 40° 13' 45"
Under 4*, *'s alt.+(Table B) Cor. 56"

Oppos. 4h 34m 52*, and under 41° (Table A) 41"

Latitude 40° 15' 22" N.

5 523. Aug. 8, 1834, longitude 150° East. The alt of the north
star at 4 A. M. was 49° 33'. To find the latitude.

§ a. Supposed time 4A A. M. -16A* (§ 221. § b.)

Longitude in time - - =10A

Time at Greenwich - - « 6A

Sidereal time, Aug. 8 - 9A 6m 3*

Supposed time of obs. - - 16/k
Acceleration of S. time for 6A 59*

Sidereal time of observation - 2bh7m 2*— 1 A 7m 2*.

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180

LATITUDE »Y TO£ NORTH 8TAR.

$ b. Star's altitude

Oppos. Ih7m2e (Tabfe B) Cor.

51°

84/
1'

51°

— 1°

33'

41'**"

50°

r

6"

58"

0"

\ 60°

2'

4"

Approx. latitude «...
Under lh, *'s alt. + (Table. B) Cor.
Oppos. lh and under 50° (T. A) "

$ 624. An error of several degrees in the longitude, and of several
minutes in the time of taking the star's altitude, brings only a small
error into the latitude, therefore die ship's time, and longitude by
dead reckoning, may always be used for finding the latitude at sea
by the polar star.

Tables fbr finding the Latitude by the alt. of the North Star*
Table A.

Sidereal
Time*

APPROXIMATE LATlTCDC

*

BMenei

Time.

15°

30°

40°

50°

60°

65°

70°

A m

it

It

// .

it

n

ti

it

A m

0.

I

3

4

9

9

12

*?

12

30

1

1

2

2

3

4

30

1.

13.

30

p

1

1

2

2

2

3

30

30

\

3

4

6

9

11

14

14.

7

9

13 1

13

24

31

30

3.

5

11

16

23

34

42

53

15.

30

7

17

24

34

60

62

79

30

4.

10

22

33

46

07

83

106

16.

30

13

28

41

58

85

108

134

30

5.

16

34

49

70

101

128

160

17.

30

18

38

56

79

115

145

182

30

6.

19

42

61

87

126

159

199

18.

30

30

44

65

91

133

168

210

30

7.

21

45

64

93

135

170

214

19.

30

61

44

64

91

133

168

211

30

a.

20

42

60

87

127

159

200

90.

30

18

38

50

79

116

146

183

30

9.

15

44

43

70

102

129

161

91.

30

13

28

32

59

86

109

136

30

10.

10

23

29

47

69

84

108

99.

30

8

17

27

35

50

62

80

30

11.
30

5
3

11

7

17
10

23
13

34
20

42
35

54

32

23.
30

19.

1"

3"

4"

6"

9"

12"

15"

94.

SUetMl
Time.

15°

30°

40°

50°

60°

65°

70°

SUerttl
Time.

A

pproxii

CAT! hi

ITITUDI

• Vide N. A. for 1334, p. 483.

Digitized by VjOOQIC

NAUTICAL ASTRONOMY.
Table B.

181

SUmalTims.

Correction*.

Sidereal Time.

Correction!.

ri

M

H

H

M

H

_

+

_

+

12

1° 31' 12"

6

18

0° 24' 43"

•

10

•

32 12

•

10

•

20 44

£

r

20
30

t

33 1
33 39

•

is

r

20
30

r

16 41
12 36

+

40

\

34 7

+

40

+

8 31

*

50

34 24

%

60

1

4 24

1

13

1° 34' 30"

7

19

0° • 01' 6"

3

10

J

34 25

19

10

7

3 61

t
r

20
30

fc
r

34 10
33 44

•
m

3

20
30

•

Oft*

+

7 58
12 4

+

40

+

33 7

40

16 8

1

50

%

32 19

50

20 11

t

14

1° 31 / 21 //

20

8

0° 24' 12"

,?

10

•

SO 13

•

10

•

28 9

•
r

90
30

r

28 54
27 25

20
30

fc
r

32 4

35 55

+

40

+

25 46

+

40

+

39 41

1

50

%

23 57

1

50

%

43 23

S

15

1° 21' 59"

91

9

0° 4r 1"

5

r

1*

V

19 51

♦

10

•

50 33

9ft

30

I

17 34

15 8

OB

20
30

r

53 59
57 19

+

40

+

12 34

+

40

+

1° 32

5

50

o>

9 52

S

50

S

3 38

4

16

i° r i"

33

10

1° 6' 38"

•

10

•

4 3

J

10

•

9 29

•

20

•

it

r

57

OB

t
r

20

OB

it
r

12 13

30

0° 57 45

30

14 48

+

40

+

54 26

+

40

+

17 15

*

50

1

51

50

19 33

<

5

17

0° 47'29"|

23

11

1° 21' 42"

•

10

•

43 53

•

10

#

23 42

so

20

9

40 11

09

£

20

SB

25 32

30

36 25

30

27 12

+

40

+

32 35

+

40

+

28 42

1

50

%

28 41

'*

50

3

30 3

6

18

1° 24' 43"

24

12

1° 31' 12"

H

M

H

Camctfanf .

H

M

H

Corrections.

Digitized by VjOOQIC

182 TIDES.

TIDES.

$ 525. The flux and reflux of the waters, known under the name
of Tides, are caused by the attractive forces of the sun and moon
exerted upon the ocean.

§£526. The moon being nearer to the earth than the sun is, has
a greater effect than the sun upon the tides.

§ a. The action of the moon upon the tides, is about three times
greater than that of the sun.

§ 527. When the attractive forces of the sun and moon act in con-
junction, they produce the highest tides.

§ a. When this is the case, the moon (§ 367.) is in syzygy.

§ b. And the tides caused about this time are called spring tides.

§ 528. The tides have the least rise and fall, when the attractive
forces act perpendicularly to each other.

§ a. And this is the case, when the moon (§ 369. § 6.) is in
quadrature. Then the tides are called neap tides.

§ 529. That portion of the ocean which is immediately under,
and nearest to, the sun or moon, is more attracted by either than
the centre of the earth is. This portion then, has a tendency to
approach the attracting body, and rises up, until its tendency is
counteracted by the attraction of gravitation towards the centre of
• the earth.

§,a. About twelve hours afterwards, this portion of the ocean,
owing to the earth's diurnal motion (§ 206. $ a.), is at the furthest
point from the sun or moon ; and the waters about it, owing to their
tendency to restore the equilibrium, which is disturbed by the effects
of the attraction of the sun or moon on the opposite side of the
earth, rise up and make high tide again.

§ b. Hence, during the time in which the moon is performing
one revolution around the earth (§ 362. § c/.),the tide rises twice, and
falls twice, at the same place.

§ 530. The attractive forces of the sun and moon upon equal por-
tions of the surface of the sea, being (§ 526. § a.) about 3 to 1 in
favour of the latter ; if the solar tide at any place be 2 feet and the
lunar 6, and the moon be in syzygy, the two tides will happen at
the same time, and (§ 527. § 6.) make a spring tide of 8 feet.

§ a. .Then, as the lunar (§ 360. § a.) is longer than a solar day,
the following solar will happen earlier than the succeeding lunar
tide, by the difference between half a solar and half a lunar day.

$ b. Thus, the lunar tide continues to retard upon the solar tide,
until the moon quadrates; when the high lunar and low solar tides
coincide in time, and ($ 528. § a.) we have a neap tide of four feet
rise.

§ c. After this, the lunar still continues to retard upon the solar
tide, until the moon arrives in the other syzygy ; when the two high
tides again happen at the same time, and bring about other spring
tides.

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NAUTICAL ASTRONOMY. 183

§ d. At this second spring tide, the moon has completed half of a
revolution in its orbit, and has lost one tide upon the sun ; there-
fore, while the moon is completing one entire revolution through
its phases, there are two more solar, than lunar, tides.

$ 531. The attraction of the moon being more partial, and its ef-
fect upon small portions of the earth's surface being more obvious
than that of the sun, the combined tides are governed more by the
moon than by the sun*

§ 532. The effects of the moon's attraction upon the earth, tend
(§ 589.) to create high tide, both in that part of the ocean immedi-
ately under, and nearest to, the moon, and in that part diametrically
opposite ; and were not the motion of the waters resisted and ob-
structed, there would be high tide at the moment when the moon
crosses either the superior or the inferior meridian.

§ a. But, at some places, the passage of the moon across the me-
ridian, precedes, by several hours, the time of high water.

§ b. This retardation of the tides might be explained, by attri-
buting the retarding power to the effects of the resistance, which
the shores, unequal depths of ocean, etc., offer to the mass of mov-
ing waters, were it not, that the waters do not rise up into spring
tides, when the moon is in syzygy, and when the attractive forces
of the sun and moon are combined in their action upon the waters ;
but the highest tides in every synodical month ($ 362. § c), is gene-
rally about the third tide after the passage of the moon through
either syzygy.

§ c. And the least neap tide, is generally the third tide after the
moon quadrates.

_ § 533. Observations show, that the tides require a little longer
time to ebb, than they do to flow.

§ a. This difference observable at sea, becomes more obvious in
rivers of strong currents.

§ 534. Upon the smaller seas v and sheets of water, such as the
Lakes, the Mediterranean, etc., the effect of the lunar and solar attrac-
tions, are partially counteracted by the circumscribed limits to their
action ; and hence but little, or no tide is produced there. .

§ 335. There are a variety of causes, (and some of a local nature),
which, acting together, tend, some to retard at one place, and others .
to hasten at another place, the hour of high tide on the full and
change days of the moon ; so that, there are no general rules for de-
termining beyond certain limits of approximation, the time of high
or low water at any place.

$ 536. The greaiest interval between two consecutive tides, gene-
rally happens about a day and a half after the moon has quadrated ;
when the tides, (§ 532. § c.) being at the minimum of. their rise and
fall, are the weakest.

§ a. And the least retardation of one tide after another, is the
2d upon the 3d, or the 3d upon the 4th tide, after the moon's latest
passage through a syzygy ; the tides, having then (§ 532. $ b,) the
greatest rise and fall, are strongest.

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184 TIDES.

(537. On an average, the tides rise and fall once, in about 12A
48m.

♦ §a. Consequently every tide retards, at a mean, about 24m,
upon the one which it succeeds ; and each tide happens about 48m
later than it did the day before.

§ 588. Were there not so wide a difference in the tides, between
the minimum and maximum of their retardation, the* hour of high
or low water, at any place and on any day, might be found by add-
ing to the time of high water on full or change days, the product
of 48m, and the number of days from the latest full or new moon,
to the day proposed.

§ a. About the time of neap tides, there is sometimes a difference
of more than an hour and three quarters in the time of high water,
on two successive days.

§ b. And a similar difference about the time of spring tides, is
frequently less than half an hour.

§ 589. The annexed table (G) shows the retardation of the mean
tides, upon the hour of high tide on full and change days, for every
day between the full and change of the moon.

§ a. Therefore, to find the time of mean high water/ on any day,
we have only to look in the Nautical Almanac (page xii* M. Ephe*
meris), for the day of the latest new, or foil moos, in order to find
its age (in days) since the last change in syzygy ; the time, jn the
annexed table (C), found opposite to this age, and added to the tone
of high water at the place proposed, on full and change days, gives
the time of mean high tide for the day when it is so required*

§ b. The .hour of high tide at any port, on full and change days,
is called the establishment of the port*

§ c. And the establishment of the port for any one place is at
ways the same; for on every full ana change day throughout the
year, high tides are considered to take place there at the same hoar
' of the day.

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NAUTICAL ASTRONOMY.

185

Table C.

$ 540. A table for finding the time of high or low water, at any
place, the establishment of the port (§ 539. §6.) being known.

$ a. To find the time of high wa-
ter at Charleston, Aug. 29, 1S34.
The establishment of the port being ,
7A5m.

$ b. The last change in syzygy
(§ 367.) was full moon, Aug. 18th, .
20A (N. A., p. 12, M. E.); conse-
quently the moon's age is 10 days.
$ c. Opposite to lOrf, and in the
column, from O to •, is Bh 10m 9
which, added to the establishment
of the port, gives ISA 25m, the hour
of high water on the day proposed.
§ a. To find the time of high wa-
ter at Portland, Feb. 14, 1834; the
establishment of the port being 10A
45m

§ e. The last change in syzygy
was new moon February 3 ; conse-
quently the age of the moon is six
days.

$/. Opposite to (W, and in the
column, from #> to 6» is 4A 3m,
which, added to 10A 45m, gives
14A 48m, the time of high water.

§ 541. This method does not
show the precise hour of high wa-
ter; but it approximates the true
time of high water, within limits
which will serve on all ordinary oc-
casions.

§ a. The time of low water may
be found by adding 6A 15m to the
time of high water on ihe day pro-
posed.

* The arguments # to O* metn from .Yew to Full moon ; and O to •> fr°n
Full to JSTew moon.

From to O

FromOtof

*

Daya.

Correction*.

Correction*.

H

M

H

M

21

i

20

41

1

41

1

1*

1

5

1

19

2

1

28

1

as

**

1

47

1

58

3

2

e

2

18

3*

2

26

2

88

4

2

46

2

58

4J

3

5

8

19

5

3

24

3

40

5*

3

45

4

3

6

4

6

4

28

*i

4

28

4

51

7

4

50

5

15

n

5

15

6

41

8

5

44

e

14

«*

6

14

6

53

9

6

48

7

80

»*

7

28

8

5

10

8

10

8

38

10*

8

47

9

7

11

9

20

9

29

11*

9

48

9

61

12

10

14

10

18

l*i

10

36

10

48

13

10

85

11

10

13*

11

16

11

30

14

11

38

11

51

14*

11

69

12

1 12

15

12 | 20

Bb

Digitized by VjOOQIC

Digitized by VjOOQIC

NAVIGATION.

Digitized by VjOOQIC

Digitized by VjOOQIC

NAVIGATION.

$542. The geographical situation of places, is designated by
their distance, norm or south, from the equator ; and by their dis-
tance, east or west, from a prime meridian ($ 252. $ c).

§ a. These distances ($ 248. § b. & § 252.) are known by the
name of Latitude and Longitude of the places.

§ 543. It has been shown in Nautical Astronomy, how the lati-
tude and longitude of places may be determined by means of obser-
vations made upon the sun, moon, or stars ; but in practice these
means are not always at hand.

§ 0. It remains then to be shown, how the geographical .position
of places may be determined, by knowing in certain respects, their
relative situation with regard to each other.

$ 544. Washington city is to the southward and westward of Bal-
timore. The difference of latitude between them (§ 254. $ c.) is
the meridianal arc, which is contained between Baltimore and the
parallel of the latitude of Washington.

€ a. And the arc of this parallel which is between Washington
and the meridian of Baltimore, ($ 254. $ &.), is the difference in
longitude between the two places.

€ b. These two arcs are perpendicular to each other ; for the me-
ridian of Baltimore (§ 248. $ a.) cuts the equator at right angles,
and the parallel of Washington (§ 249.) is parallel to the equator.
v § c. Now, if a line (W B) be drawn

direct from Washington to Baltimore,
it will connect their difference of longi-
tude (W L) and of latitude (B L), and
form a right angled triangle (W L B),
of which the hypothenuse ( W B) is the
distance, and the legs, the difference of
latitude and of longitude, between
Washington and Baltimore.

§ d. The two acute angles (W & B)
show the bearings of each place from
the other. The angle W is the course
from Washington to Baltimore, and the
angle B is the course from Baltimore
to Washington.

$ t. Hence if, of the Disk, Diff. Lat.,
Diff. Long., and Course, between two
places, any two be given, the others
are determinable.

Digitized by VjOOQIC

190 NAVIGATION.

§ 545. Now, the section of a loxodromic curve, which a ship
traces while she is sailing upon a given course, is the hypothenuse
of a right angled triangle, and corresponds toWB.

§ 0. Then if the course and distance, which a ship sails in any
given time, he known, we have the hypothenuse and an acute an-
gle of a right angled triangle t given ; and the other parts are deter-
minable.

§ b. Therefore, knowing the latitude and longitude of the place
from which a ship takes her departure, and knowing the course and
distance which she sails from that place, in a specified time, we
may determine by right angled trigonometrical calculations, the lati-
tude and longitude of the ship, at the end of that time.

§ 546. The sides of all such triangles are curved lines ; for the
earth (§ 204.) is a spheroid, therefore all triangles upon its surface
are spherical triangles, and are properly subject to the rules of
spherical trigonometry, for investigation.

§ a. But the laborious operations of obtaining results from these
triangles, when brought under calculation, as spherical triangles,
would be inconvenient in practice ; and they are not necessary in
the ordinary purposes of navigation.

§ b. Therefore all such triangles are considered in navigation. as
right angled plane triangles, and are brought under calculation ac-
cording to the rules of plane trigonometry.

§ 547. A degree (°) of all great circles, such as the equator, me-
ridians of longitude, etc., whose radii and the earth's semidiameter
are the same, consists of 60 nautical miles, or of about 60£ statute
miles ; so that, calling the number of minutes ('), miles* which is
contained in an arc of one of these great circles, we have the value
of this arc considered as a straight line.

§ a. Wherefore, knowing the distance and the difference, of lati-
tude between any two places, the difference of longitude between
them, may be determined also in nautical miles, by right angled,
plane trigonometrical calculations founded upon § 75.

$ b. The difference of longitude thus found, is called departure.

5 548. Departure is the difference in longitude between any two
places, expressed in miles.

§ a. And the departure in any latitude, may be converted into its
corresponding minutes (') of longitude by means of the principles
established under § 76.

§ 549. It is evident, that, if in mathematical calculations, curved
lines be treated as right lines, there must be an error in the result ;
and this error increases with the number of degrees contained in
such curved lines.

§ 550. The result then, obtained by calculating the sides of a
spherical triangle, as though they were the straight lines of a plane
triangle, is only an approximation ; and the smaller the sides, the
closer is the approximation.

$a. This method therefore is particularly applicable in small
distances, and short runs ; it is used for working up " dead reckon-
ing", and for calculating the distance, the difference of latitude and

Digitized by VjOOQIC

NAVIGATION. 191

of longitude, and the course, from one place to another ; any two of
which four quantities, being known, (§ 544. § e.), the others are de-
terminable.

§551. The diagram shows by a mere inspection, the> piate x
solution of every problem which can occur in right angled 3
plane trigonometry, provided the hypothenuse of the triangle in-
volved, do not exceed 300 miles, or 300 units of any measure.

§ a. This diagram may answer the purpose of Tables IV. & V.;
and it also shows the principles upon which they are constructed ;
for the number in the Dist. column of these tables, is the length
of the hypothenuse of a right angled triangle, whose acute angles
contain, one the number of degrees at the top, and the other those
at the bottom of the page, and the value of whose legs is set down
opposite to that of the hypothenuse, and in the columns marked
Dep. and D. Lat.

§ 552. This diagram, then, shows the solution of every case, or
problem in loxodromic sailing ; for it shows the dimensions of the
triangle involved in every problem, provided the distance or hypoth-
enuse do not exceed 300 miles.

§ a. If it exceed 300 miles, the solution may be obtained by
dividing the given side or sides, by 2 or 3, or by any other divi-
sor, which will reduce the triangle of the problem within the limits
of the diagram ; then the value of the side or sides, obtained, by
using said quotient in the diagram, being multiplied by the same
divisor (§ 70.), gives the value of the required side or sides.
$ 553. The angle B A D is 45°.

§ a. The acute angles of a right angled triangle (§ 32. § d.) being
complementary to each other, the less must always be located'at A.
$ b. And the leg adjacent to it, must be called the base, and be a
part of A D.

§ c. And the leg opposite to it, must be called the perpendicular.
§ d. All the straight lines standing upon A D, are alluded to as
perpendiculars.

§e. B C is the greatest perpendicular in the diagram. Equal
portions of it are transferred by means of the parallels b d, to the
graduated arc B D, for admeasurement.

§/l The graduated arc B D is thus made to answer the purpose
of a graduated perpendicular, for the arc contains the same divi-
sions which the perpendicular B C would have shown, had it been
graduated to the scale of A B and A D.

§ g. The height of every perpendicular is therefore shown on
that portion of the graduated arc, which is between the base (§ b.)
and that parallel, under which the perpendicular stands. Thus the
height of the perpendicular b C, transferred by the parallel b d, to
the graduated arc, is D c?=11.3.

§ 554. The arcs in the diagram show the angular value of any
course, provided it be not more than 4 Points, or 45°.

§ a. If the course be greater than this, they show the angular
value of its complement.
§6. The arcs, 3, 4, 5, etc., also show the length of the base and

Digitized by VjOOQIC

192 NAVIGATION.

Plato 1 \ -hypotiieiMwe of the triangle- proposed; while the scale on
' 5 the graduated arc (§ 553. $ g.) shows the value of the per-
pendicular (6 c) in the several triangles.

§ c. The section of any arc? which is contained between the base,
and the lines A e, shows the angular value of the courses which are
marked upon these lines ; and the angular value of the complements
(§ 553. § a.) of the courses, marked under these lines.

§ d. The marks on the arcs show the value of these arcs in de-
grees.

§ e. The eyen numbers (2°, 4°, 6°, etc.) of degrees are marked
on the concave side of the arcs ; and the odd numbers (1°, 3° 5°,
etc.) are marked on the convex side.

$/. The degrees (46°, 47°, 48°, etc.) that stand on the outside
of the graduated arc, ere the complements of those (42°, 43°,- 44°,
etc.) on the inside of the arc.

§ g. The entire length of every arc (§ 553.) is 45°.

§ 555. The dotted lines, A e', represent quarters of Points ; the
broken lines A e, represent halves of Points ; and the continuous
lines A e whole Points.

§ a. The distance on a given course is shown on the lines A e, at
the end of which lines that course is found ; then the portion of that
line (A e) which is contained between A, and the arc 3, 4, etc., which
is marked with the distance proposed, is hypothenuse to the triangle
required.

§ b. Thus 24 miles on a 2i Point course, is the distance A b on
the broken line A e, at the end of which 2i is marked ; and A C b
is the right angled triangle, of which the course and the distance
proposed constitute an angle and the hypothenuse.

§ c. The two legs of this triangle are the boat A 0, and the Jier-
pendicular b C which stands under the parallel b d.

§ d. The graduated scale On A D shows the value of A C**21.1,
the D. lat.; and the portion (D d) of the scale on the graduated arc,
between the base and the parallel b d (§ 553. § g.) 9 shows the height
of the perpendicular b C =1 1.3, the Dep.

§ 556. Now as the distance which a ship may sail upon any loz-
odromic course, is considered (§ 546. § 6.) as the hypothenuse of a
right angled plane triangle, if the distance sailed upon such course
can be found in the diagram, the departure and difference of lati-
tude, which correspond to that course and distance, may be found
by means of the graduated base and arc, A D and B D.

§ a. Or* if any two of these quantities; viz., Course, Dist., Dep.,
and Diff. Lat., be known, the other two are determinable by means
of the diagram.

§ 557. A ship sails N.W.|N. 19 miles, What departure and dif-
ference of latitude does she make ?

$ a. The given course is found upon the dotted line A *'* *&d that
portion (A b) of this line, which is between A and the 19th arc, is
the hypothenuse of the triangle (A c b) involved.

$ b. The perpendiculcr (b c) intersects the bate at 14.1, which is
the difference of latitude.

Digitized by G00gle

NAVIGATION. 193

$ t. And the parallel, from the top of the perpendicular, £
intersects the graduated arc at 12.7, which is the departure. J '

$ 558. A ship sails N.W. by N. and makes 12.7 miles of depar-
ture, What distance and diff. lat. does she make ?

$a. The parallel from 12.7 on the graduated arc, intersects the
N. W. by N. line, in the point, (nearly), where the 23d arc crosses
it ; then nearly 23 (22.9) miles in the distance sailed.

$ 6. And the perpendicular from this point of intersection, falls
upon the base, and shows the difference of latitude required, to be

19.1 miles.

§ 559. A ship sails N. by W.|W. and makes 19.8 miles diff.
lat., What departure and distance does she make ?

$ a. The perpendicular at 19.8 on the base, being traced up, is
found to intersect the N. by W.fW. line on the 21st arc; then 21
miles is the distance sailed.

$ 6. And the parallel from this point of intersection, being traced
to the graduated arc, shows the height of the perpendicular 7 miles,
which is the departure required.

$ 560. A ship, after sailing 18 miles, finds that she has made

16.2 miles diff. lat, What course did she sail, and what departure
has she made I

§ a. The perpendicular from 16.2 on the base, intersects the
18th arc on the dotted line marked 2 J Points, which is the course
^required.

$ 6. And the parallel from this point of intersection, shows, on
the graduated arc, the height of the perpendicular to be 7.8 miles,
which is the departure required.

$ 561. A ship sails 28 miles, and finds that she has made 15.6
miles departure, What course and diff. lat. has she made ?

§ a. The parallel from 15j6 on the graduated arc, cuts the 28th
arc, on the line marked 3 Points, which is the course sailed.

$ 6. And the perpendicular from this point of intersection, falls
upon the base at 23.3 miles, which is the diff. lat. required.

§ 562. The departure between two places is 8.7 miles, and the
diff. lat between them is 28.6 miles, What is the course, and the
distance from one place to the other?

$ a. The perpendicular from 28.6 on the base, intersects the
30th arc on the broken line, l£, which shows the required course
and distance to be 1$ Points, and 30 miles.

$ 563. If the distance sailed be not more than a mile, or if the
hypothenuse of the triangle proposed, be not greater than 3, the
divisions on the scales must be decimated; then the figures 1, 2, 3,
4, 5, etc., stand for T ^, *fc, y' 5 , -^, -fa, etc., and the subdivisions
stand for lOOths.

§a. Wherefore, Nvhen the distance is not more than 3 miles,
the distance from A, to the 10th arc, is 1 mile ; to the 20th, 2 miles ;
and to the 30th, 3 miles ; and all the numbers in the diagram stand
for lOths.

$ 564. What departure and difference of the lat. would a ship
make by sailing 2 miles N.W. IN. ?

C c

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194 NAVIGATION.

C $ a. The parallel and perpendicular which would past
Plate l. ^ trough tne point in which the 20th arc intersects the N.
W.iN. line, measured by the eye, would cut the graduated arc and
base in 12.7 and 15.4.

§ b. And using only one decimal place ($ 563.), the nearest would
be 1.5 diff. lat. and 1.3 departure; but the exact is 1.54 and 1.27
diff. lat. and dep.

$ 565. When the lines in the diagram do not pass through the
point required by the conditions of the triangle involved, the ima-
ginary lines required, may be traced by the eye with all the preci-
sion which is necessary ; and after a little practice, with the utmost
accuracy and facility.

§ a. Thus, the distance of one place from another, is N. by W.
27 i miles ; the triangle involved in this problem, being filled up by
the eye, the imaginary parallel and perpendicular, cut the graduated
arc and base in 5.3 and 27 ; which shows the departure between the
two places to be 5.3 miles, and the diff. lat 27 miles.

§ 566. If the distance sailed on a single course exceed 30 miles,
or the hypothenuse of the triangle involved, be greater than 30, all
the numbers in the diagram must be increased tenfold.

§ a. Then the arcs must be reckoned as being 10 miles a part;
the numbers 1, 2, 3, 4, 5, etc., will stand for 10, 20, 30, 40, 50,
etc. ; and the distance between these numbers being divided, every
one in 10 equal parts, every one of these subdivisions must be
counted as 1 mile.

§ 567. A ship sails N* W. 240 miles ; what departure and diff
lat does she make ?

§ a* The parallel and perpendicular from the intersection of the
24th arc (now counted ($ 566 ) 240), with the N. W.line, cuts the
graduated arc and base ($41.) at equal distances, and the departure
and diff. lat. (§ 14*) are the same.

%b. Departure =169. 5. diff. lat =169. 5. This is -& of a mile
less than the result by logarithmic calculation. This difference is
owing to the smallness of the scale upon which the diagram is pro-
jected, and not to any defect in the principles involved in the pro-
jection of the diagram.

$ 568. In the examples quoted above, the courses are on the
lines, and the diff. lat., except (§ 567.) when the course consists of
4 Points, is always greater than the departure.

§ 569. The courses and figures under the lines, (§ 554. § e.) are
the complements of those on the line above them ; and when the
course is found under the line, the departure is the longest leg of
the triangle involved, and must be read off upon the base A D ; and
the diff. lat. is taken from the graduated arc B D.

§ a. A ship sails £. i N. 28 miles. What departure and diff. lat.
does she make ?

$ b. The parallel and perpendicular which pass through the point
where the 28th arc cuts the dotted line, under which the given
course is marked, meet the graduated arc and base in 1 .3 and 27.9+.

§ c. Then ($ 569.) the departure =27.9. and the diff. lat. =1.3.

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NAVIGATION.

195

§570. The four cardinal points of the horizon ($ 272.) £
divide it into four quadrants, each containing 8 Points. £
$ a. Wherefore a Point=ll° 15', the quotient of 90° by 8.
$ 571. The Points, when expressed by numbers, begin at the
north and south, and are reckoned in numerical order, to the east
and west points.

$ a. Hence, it is easy to conceive that a vessel that sails N. 3
Points W., or N. W. by N., would make more northing than west-
ing, and that her diff. lat. would be greater than her departure.

§ b. And conversely, that if a vessel sail on any course between
4 and 8 Points, e. g., N. 5 Points W., or N. W. by W., she would
make more westing than northing, and that consequently her de-
parture would be greater than her cliff, lat.

$ 572. From what has already been advanced in explanation of
the diagram, it appears that the triangle involved in the trigonome-
trical solution of any problem in loxodromic sailing, is a right an-
gled plane triangle.

$ a. Loxodromic sailing,* is so called from the sort of curve
which the track of a ship forms, when she is sailing upon any
course not exactly due east or west, north or south.

f b. Loxodromic curves are spirals, which continually approach
the poles, but can never reach them.

§ e. And any course, not due east or west, north or south, is
called a loxodromic course.f

$ 573. A number of formula;, varied in the trigonometric func-
tions to be used, may be drawn up for the solutions of the several
cases in loxodromic sailing.

§ a. The terms in these formulae
depend upon the side of the triangle,
whether it be the distance, the de-
parture, or difF. lat., which is made
radius.

§ 574. Baltimore is 35 miles N. N.
E. i E. from Washington. The for-
mula for finding by calculation the de-
parture and diff. lat. between the two
places, depends upon the trigonometri-
cal construction which is given to the
triangle involved.

§ a. If the hypothenuse be made ra-
dius, to an arc B C, the dep. becomes
(§ 54.) sine to the course, and the diff.
lat. its co-sine.

§ 6. Then, (§ 75.), rad. : dist. : : sin.
of course : dep. : : cos. course : diff.
lat.

• The subdivisions of the »ailing», m^ Plane, Traverse, Parallel, and Middle
Latitude Sailing, are not here preserved. Theae distinctions are by no means
necessary for the purpose of facilitating the navigator's calculations; they have
therefore been generalized under the term of loxodromic tailing,

f E. by N. is a loxodromic course. 8uppose a vessel could sail E. by N.

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19B NAVIGATION.

§ e. Hence (§ 74.) are deduced the following formulae ; the
three first being the given terms, and the fourth, the unknown and
required term* of the proportion;

§ d. Sin; course : dep. : : rad. : dist. (§ 75. $ b.).

§ e. Dist. : rad; : : dep. : sin. course;

§/. Cos. course : diff. lat. : : rad. : dist.

§ £% Dist. : rad. : : diff. lat. : : cos. course;

§ X. Cos. course : diff. lat. : : sin. course : dep.

§t. Dist.=35. log.= - - - 1.544068
Course=2f pte. (§ 571.) sin. (Table D)=9.711049

Log. dep.=1.255117=18 miles.

§7. Dist. 35. log.ce - - 1.544063

Course, 2| pts. cos.= - - 9.933350

Log. diff. lat.=l. 477418=30 miles.

§ 575. If the triangle be constructed upon the diff. lat. as radius
of the arc A E, then the dfet. becoihes (§ 57.) secant, and the dep.
(§ 56.) tangent, to the course.

§ a. And (§ 75.) sec. course : dist. : : rad. : diff. lat. : : tang, course
: dep.

$ b. Wherefore, the diff. lat. being taken as radius, the following
formulae are derived either (§ 75. § c.) immediately from the con-
struction of the triangle, or (§ 67. § e.) from the relation of the
terms, as they are expressed in the proportion § a.

$ c. Diff. lat. : rad. : : dist. : sec. course.

§ d. Rad. : diff. lat. : : sec. course : dist.

§ e. Diff. lat. : rad. : : dep. : tang, course.

§/. Tang, course : dep. : : rad. : diff. lat.

$ g. Tang, course : dep. : : sec. course : dist.

§A. Course 2| pts. cos. =9.933350 (§101. $6.)
Dist. 35mfles, log.sss 1.544068

Log. diff. lat.fel. 477418=30 miles.

$*. Course 2} pts. cos.«9.933350
Dist. 35 miles, log. =* 1.54 40 68
Course 2j pts. tang. =-9.777700

Log. dep.=» 1.2551 18=18 toilet.

without any obstruction, or without ever deviating from that course ; the N. pole
would then be always one Print forward of her beam. It is difficult to imagine
how a vessel could ever arrive at rach a point

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NAVIGATION.

197

»«*■

$ 576. Or if the triangle be construct-
ed upon the dep., as radius of the arc
A F, the dist. becomes secant, and the
diff. lat. the tangent, of B, the comple-
ment (§ 32. § d.) of the course.

§ a. Wherefore (§ 75. § c.) co-sec.
course : dist : : rad. : dep. : : co-tang,
course : diff. lat.

$ b. Other formulae of proportion,
similar to those under § 575., might be
drawn out here, but these for the most
part would be a repetition of those, as
the expression 4x6 is a repetition of
the form 6x4 of multiplication. Indeed
several sets of the proportions under
$ 574. & § 575., are similar repetitions
of each other.

$ c. Course 2| pts. sin.=9.711049 (§ 101. $ b.)
Dist 35 miles, log. = 1.544068

Log. dep.= 1.2551 17=18 miles.

id.

Course 2| pts.
Dist. 35 miles,
Course 2£ pts.

sin.:

l0g.=

co-tang. =

=9.711049
=1.544068
=0.222300

Log. diff. lat. =1.4774 17 =30 miles.

§ 577. The principles of the above trigonometrical con-> p lfttc x
struction are also developed in the solution of problems 5
by the diagram.

$ a. Thus, to find the course and diBt. which correspond to 12
miles diff. lat., and 5 miles dep.

§ b. The perpendicular which cuts the base in 12, is tangent to
the 12th arc, and A 12 is its radius.

$ c. The parallel from 5, on the graduated arc, cuts this perpen-
dicular on the 13th arc, at its intersection with the 2 Point line, and
A 13 is secant to the course (2 Points) and the dist. required.

$ d. If the dist. A 13, on the 2 Point line be called radius of the
13th arc, then the portion of the perpendicular, between the point
of this intersection, is sine of the course, and A 12 on the base, is
its co-sine.

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198 NAVIGATION.

§ 578. Table of log. sines, etc., of the Points of the Compass.
TableD.

Points.

Sine.

CO-MC.

Co*ln.

Sec.

Tang.

Co-Tang.

i

8.690795

1.309205

9.999477

0.000523

8.691319

1.308681

7*

4

.991302

.006698

.997904

.002096

.993398

.006602

74

t

9.166520

0.833480

.995274

.004726

9.171246

0.828754

7*

1

.290236

.709764

.991574

.008426

.298662

.701338

7

1*

.385572

.614428

.986787

.013213

.398784

.601216

H

H

.462824

.537176

.980885

.019115

.481939

.518061

64

H

.527488

.472512

.973840

.026160

.553647

.446353

H

2

.582840

.417160

.965615

.034385

.617224

.382776

6

2*

.630992

.369006

.956163

.043637

.674829

.325171

*i

24

.673387

.326613

.945430

.054570

.727957

.272043

54

2}

.711049

.288951

.933350

.066650

.777700

.222300

5*

3

.744739

.255261

.919846

.080154

.824893

.175107

5

3*

.775037

.224963

.904828

.095172

.870199

.129801

4}

34

.802359

.197641

.888185

.111815

.914173

.085827

44

3}

.827083

.172917

.869790

.130210

.957294

.042706

4*

4

.849485

.150515

.849485

.150515

o.oooood 0.000000

4

Co-sine.

Sec.

Sine.

Co-sec.

Co-tang.

Tang.

Points.

§ 570. In every case, except when the dep. and din*, lat. consti-
tute the two given parts, the so-
lution of the problem proposed,
may be obtained by means of
the proportion (§ 74.) between
the sides of a triangle and sines
of their opposite angles.

§ a. In the triangle ABC
f § 74.) right angled at B, ($ 74.)
dist. : sin. B=90° : : dep. : sin.
course; but the sine of 90
(§ 62.) is radius, wherefore we
have again the formula $ e>,
(§ 574. \ viz., dist. : rad. : : dep.
t sin. course, and a repetition of others under § 574.

$ 580. Some one of the following formulae (rejecting or borrow-
ing 10 for rad., (§ 197. $/.), in the index of the second member of
the equation), may be used in every case where two of the four
quantities, dist., course, dep. and diff. lat., are given.
§ a. Dep. (S 574. S &.Wdist.xsin. course.
§ 6. Dep. (S 575. § e.)~diff. lat x tang, course.
§ c. Diff. lat. ($ 574. $j.)»dist.xcoe. course.

§<f. Diff. lat. ($ 675. 5/.)-^^
course*

course

(§ 101. S 6.)«dep. xcot

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NAVIGATION. 190

$ «. Dist. ($ 675. §<i.)»diff. lat. x sec. course.
§/. Disk (§ 576. S a.)»dep.xco-sec. course.

$ g. Sin. course (S 574. S *)— -jgp

$ A. Cos. course (§ 574. § £•)==■— -jr — "

dist.

§ t. Tang, course (§ 575. $ «Q— ffif' ■

din. lat.
$y. These formulae will always be found convenient in practice.

OP TURNING DEPARTURE INTO DIFF. LONG.

$ 581. If the difference of longitude between two places ($ 548.)
be expressed in miles, it is called departure.

$ a. A minute of longitude is no where, except at the equator,
equal to a mile of departure ; for the distance between any two me-
ridians of longitude (§ 248. § a.) is greatest at the equator, and the
least about the poles.

§ 6. Therefore the dep. between any two meridians iB greater at
the equator than in any latitude.

§ c. Furthermore, as all meridians ($ 248. § a.) intersect at the
poles, the dep. between any two becomes less ana less, as you ap-
proach the poles, where there can be no dep. ; whereas the differ-
ence of longitude, between these meridians in any latitude, ($ 254. § a.)
is the same, being expressed by the same number of degrees (°),
minutes ('), etc.

§ 582. Suppose two places to be situated in the same latitude,
the arc of this parallel of latitude, contained between them, ($ 254.
$ o.) is their difference in longitude.

$ a. And parallels of latitude ($ 249.) are small circles.

$ o. Suppose also an arc of a great circle ($ 122.) io be drawn
from one of these places to the other, this arc (§ 122. $ a.) would
be the shortest distance between the two places, and the minutes (')
contained in it, would show ($ 547.) the departure between them.*

$ 583. These two arcs are contained between the same two
points, and ($ 76.) the arcs are to each other, as the radius of the
great circle is to the radiuB of the small circle.

$ a. The rad. of such a great circle, as well as of the equator,
etc., ($ 122.) is semidiameter to the earth.

i b. And the radius of the small circle, or the parallel, (§ 130. § a.)
is the cos. of its latitude.

§ 584. Then, calling the semidiameter of the earth, unity, or
rad., we have (§ 76.) for any latitude, diff. long. : dep. : : rad. :
cos. lat.

$ a. Also cos. lat. : dep. : : rad. : diff. long.

b. And rad. : diff. long. : : cos. lat. : dep.

* Hence it appean that the course between two places, is never the shortest
distance between them, unless they be either upon the equator, or upon the same,
meridian. * *

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200

NAVIGATION.

§ c. Wherefore, if any two of these three quantities, (rad. is al-
ways known), be given, the other and remaining one is determinable.
§ 585. The terms of these proportions represent quantities, com-
prised in the elements of a right angled plane triangle.

$ a. Wherefore, if the figure which furnishes such proportions,
be considered a plane triangle ; it will be right angled, the latitude
will be one of its acute angles ; the diff. long., the hypothenuse ;
and the dep. will be one of the legs of such triangle.

$ b. The result obtained from calculations conducted upon such
principles, (§ 549.) must be only an approximation.

§ 586. Suppose a ship
sailed E. 15J miles on
the parallel of 56° 15'
south latitude, her de-
parture would be 15i
miles, the distance
sailed.

§ a. And in the right
angled triangle ABC,
the angle A=56° 15',
the lat. ; the leg A B—
15 J, the dep.; and the
hypothenuse ($ 585.
§ a.) is the diff. long.
$6. Now, if the diff.
long, be made radius, of the arc C E ; C B becomes its sine, and
A B its co-sine, whence (§75.) arise the proportions, under
$584.

$ c. Rad. : diff. long. : : cos. lat. : dep.
§ d. Diff. long. : rad. : : dep. : cos. lat.
§ e, Cos. lat. : dep. : : rad. : diff. long.

§/. To find the corresponding diff. long, by calculation (§ e.).
Lat. 56° 15' sec. =0.255261 ($ 101. $i.)
Dep. 15.5 log. =1.180332

Log. diff. long.=l. 445593=27.9 minutes (')

§ 587. If the dep. (A B) be made radius, to the arc B H, then
the diff. long, becomes secant to the lat., and we have (§ 75. $ c.)

$ a. Rad. : dep. : : sec. lat. : diff. long.

§ b. Sec. lat. : diff. long. : : rad. : dep.

§ c. Dep. : rad. : : diff. long. : sec. lat.

§ 588. Now, if the other leg (C B) be made radius, the dep. be-
comes co-tang. (§ 57. $ &.), and the diff. long, the co-sec., of the la-
titude, whence ($ 75.) arise the following sets of proportions.

$ a. Co-tang. lat. : dep. : : co-sec. lat. : diff. long.

$ 6. Co-sec. lat. : diff. long. : co-tang, lat : dep.

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NAVIGATION.

201

§ 589. To find the number of miles, or the dep., which corres-
pond to 1° or 60' on the parallel of 67° 30'.

§ a. The cliff, long, being made radius, we have, (§ 586. § c),
Rad. : diff. long. : : cos. lat : dep.

Diff. long. 60 log.=1.778151

Lat. 67° 30' cos. =9.582840

Log. dep.= 1.360991 =22.9

§ 590. G (§ 32. § d.) is the complement of A, and is equal to
22° 30'. C B is the cos. of C, and shows the number of miles con-
tained in 1° of long, on the parallel of lat. 22° 30'.

$fl. Diff. long. 60 log.=1.778151 (§589. § a.)
Lat. 22° 30' cos. =9.96561 5

Log. dep. =1.743766=55.4 miles,

which is equal to 1° of long, in lat. 22° 30'.

%b. Now (§75. § b.) C B : cos. C : : AB : cos. A. A B shows
the number of miles equal to 1° of long, in lat. 67° 30', and C B
shows the number of miles, that make 1° of long, in lat. 22° 30',
the complement of 67° 30'.

§ c. Wherefore, in any lat., the miles in a degree of long, are to
the cos. of that lat., as the miles contained by a deg. of long, in any
other lat., are to the cos. of this lat. And alternately.

§ 591. As a right angled plane triangle (§ 585,) is in- C p Jate l
volved in the solution of the several problems under § 586. \
§ 587. § 588., etc. ; the solution of every one of such problems
may be obtained by means of the diagram.

§ a. The diff. long. (§ 585. § a.) is the hypothenuse ; and the leg
adjacent to the angle which in degrees (°), is equal to the latitude
in which the dep. or diff. long, is made, is the dep.

§592. Of dep., diff. long, and lat., either is determined on the
diagram, by means of the two others, in the same manner, that any \

D d - :

Digitized by VjOOQIC

tot NAvmAtkm.

Plata l £ one of the three <l uantitieS » ▼!*•» ^ iff - la4 -» diMt. Mid eottrse,
' 3 is determined (§ 556. $ a.) by means of any two of these*

§ 593. When the latitude in which the diff. long, or dep~ is
made, is less than 45°, the degrees of lat. are to be found on the
inside of the graduated arc ; and the dep. on the base.

§ a. But when the latitude is greater than 45°, the degrees of it
are to be found on the outside, and the dep. on, the graduated arc.

§ b. Thus, 26' of long, is equal to 24.5 miles, in lat. 19° ; and in
lat. 71°, it is equal to 8.4 miles.

§ c. The perpendicular from the mark for the 19°th on the 26th
arc, cuts the base at 24.5, which shows the dep. for 26' of long, in
lat. 19°.

§ d. The parallel from 71° on the 26th arc, cuts the graduated
arc at 8.4, which is the dep. ($ 593. § a.) for 26' of long, in lat
71°.

§ 594. A ship in lat. 27°, makes 20.5 miles dep. To find the
diff. long.

§ a. The diff. lat. is found on the inside (§ 593.) of the graduated
arc, and the dep* on the base.

§ b. The perpendicular from 20.5 on the base 9 intersects 27° on
the 23d arc, then 23' is the required diff. long.

§ 595. A ship in lat. 60°, makes 12 miles dep. To find the diff.
long.

§ a. The degrees of the lat (§ 593. § a ) are found on the out-
side, and the dep. on the graduated arc.

$ b. The parallel from 12, intersects 60° on the 24th arc ; 24' is
the diff. long.

§ 596. The diff. long, between two places in lat 33° is 27'. To
find the dist between them.

§ a. The lat is found on the insifie (§ 593.) of the graduated arc,
and the dep. on the base.

$ b. The perpendicular from 33° on the 27th arc, falls upon the
base at 22.6, the dist. in miles between the two places.

$ 597. The diff. long, between two places in lat 54% is 24'. To
find the dist between them.

$ a. The degrees of lat are on the outside^ and the dist. (i 593.
$ a.) is to be found on the graduated arc.

§ 6. The parallel from 54° on the 24th arc, cuts the graduated
arc at 14.1, the dist. required.

$ 598. The dep. between two places on the same parallel, is 18.4
miles, and the diff. long, is 20\ To find their lat

§ a. The perpendicular from ia4 cuts the 20th arc at 23°, the lat
required.

§ b. The dep. being found on the base, the degrees ($ 593. % a.)
are token from the inside of the arc.

$ c. But if the dep. were found on the graduated arc, then the
degrees on the outside of the arc, would show the lat.

$599. The diff. long, being 20', and the dep. between two
places in the same latitude being 6.8 miles ; to find their lat

Digitized by VjOOQIC

$ «. The parallel from 7.9, cuts the 20th arc at 07°, ( OM ,
which ($ 598. $ c.) is the la*. J p,tla l -

4. When the minutes (') of lonf. exceed 80', every division on
the several graduated scales, ($ 566.) mast be counted as a mile, or
as a minute Q.

i e. And wnen the minutes (') of long, do not exceed 3', the di-
visions on the graduated scales (§ 563.) must be decimated.

$ 600. Of the diff. long., dep. and lat., either one may be found,
the others being given, by means of Table V.

^ $ a. The degrees at the top of the table, are those which are in-
ride of the arc ; and those at the bottom of the table, and on the
outside of the arc, are the same.

§ o. The numbers in the column marked (dist.) stand for minutes
(') of longitude ; they show the value of the hypothenuse of a trian-
gle, whose legs are equal to the quantities in the next two columns ;
and whose acute angles contain, one the degrees at the top, and the
other those at the bottom of the page.

$ c. When the lat. is found at the top of the page, the miles
which are equal to any number of minutes of longitude, are oppo-
site to them in the column marked, at the top, (d. lat)

$£ Thus, in lat. 41°, 60' of long.»45.3 miles.

$ «. If the degrees of lat be found at the bottom of the page, then
the miles which are equal to any number of minutes (') of long.,
are found opposite to them in the column marked (d. lat), at the
bottom.

$/. Thus, in lat 71°, 60' *f long.«19.5 miles.

$ 601. When a ship sails due east, or west, she sails either upon
a parallel of lat, or upon *he equator, and the whole distance sailed
is dep.; which aaaj be converted into diff. long, by the formula §/.
($ 586. )• or by means of the diagram (Plate I.), or Table V. ; either
of the two latter means is most convenient in practice.

$ 602. When a ship sails due North or South, she sails upon the
are of a meridian, and the whole distance sailed is diff. lat.

$ 603. It is frequently the case, that a ship sails during the day,
on more courses than one, making a zig-zag track.

$ a. In such cases, it would be a very tedious operation, after find-
ing the diff. lat. and dep. for every course and distance, to convert
the dep. for every such course and dist. into diff. long, by a sepa-
rate operation.

§ 6. Sueh circuity is avoided by taking the whole amount of dep.,
•ad converting it into diff. long, by a single inspection of the dia-
gram, or of Table V,

§ 604. Suppose a ship sails from lat. 64° N., 216 miles N.E. by
N.; she makes 120 miles of dep. and 179.6, say 180 or 3° diff. lat.

$ st Now this dep., or the diff. long, is not all made upon the
parallel of 64°, '«V° f '6°, or '7° ; or upon a.ny one of the intermediate
parallels, but upon ail of them together.

$ 6. And m order to convert, by a single operation, this dep. into
long., and to find the diff. long, which the ship has made by sail-
ing tiiis course and distance, the whole dep. is supposed to be

Digitized by VjOOQIC

204

NAVIGATION.

made upon that parallel (65° 30') which is midway between the
lat. left (64°), and that (67°) arrived at.

§ c. This lat. (65° 30') is called the middle latitude; it is half
the diff. lat. added to lat. left ; or to that arrived at, if the latter be
the less lat.

$ 605. If the number of miles in a degree of long, on the different
parallels, as you approach the poles, decreased, as the miles from
the equator to those parallels increase, this method would show the
true diff. long.

$ a. But the number of miles in a degree of long, at different
parallels, is (§ 590. $ c.) as the cosines of the lat. of such parallels.

5 b. Wherefore, the result given by this method is only an ap-
proximate one.

§ c. But in working up "dead reckoning," "days' works," etc.,
this method is generally used ; it is used in all cases, except in the
computation of great distances.

§ 606. On ship-board at the expiration of every hour, sometimes
of every two hours, the course and distance sailed during that time
are marked opposite each other, on the log slate.

$ a. And at the end of every sea day, or at every noon, the courses
and distances sailed in the last 24 hours, are " taken off," for the pur-
pose of " working up ;" and the whole distance Bailed on every sepa-
rate course is set down, opposite to that course, in a traverse table.

Plate \.\ $ ** ^^ e ^' * at ' 2ja ^ ^ e ^ e P' *° r everv course and
* 5 dist., are taken, either from the diagram, or from Table
IV., and set down opposite to their proper course and dist in the
" traverse table."

§ c. The dep. and diff. lat. for every course and dist, are then
added up ; the sums show the whole dep. and diff. lat.

§ d. And by means of the whole dep. and diff. lat., the course
and dist. "made good," are found on the diagram ($ 577. $ a.), or
are taken from Table IV. or V.

$«. The "middle lat." (604. § c.) is then found; and by means
of it and the dep., the diff. long, is found (§ 594.) on the diagram, or
in. Table V.

Fig. A.

Course.

Dist.

DuXLat

Depart

N.

8.

•e.

W.

N.byE.
S.W.

30
12

19.6

8.5

3.9

8.5

<

5 607. Fig. A is the for-
mula of a "traverse table."

§ a. The diff. lat is writ-
ten in the column marked
N. or S. according as the
course has northing, or
southing in it. Thus, the
diff. lat for 20 miles N. by
E. is put in the N. column,
and the diff. lat. for 12 miles
S. W. is written in the S.
column.

$6. The dep. must be
placed in the column E.-or
W., according as the course
has easting, or westing in

Digitized by VjOOQIC

NAVIGATION. 205

it. Thus the dep. for the dist. 20 N. by E., is written in the E.
column, and the dep. for the dist. 12, S.W. is written in the W. co-
lumn.

§ 608. The dep. and diff. lat. for any dist. less than 300 miles,
and for any course, are found in Table IV., in a manner similar to
that by which they are found on the diagram.

§ a. If the course be less than 4 Points, it is to be found at the
top of the page, and the diff. lat. must be taken from the column
marked at the top (d. lat.), and over which the course is found.

§ b. Thus, to take from Table IV. the dep. and diff. lat. for 120
miles S.£E.; find S.JE. at the top of the page, and 120 in the dist,
column ; then in the two columns, at the head of which S. £ E. is
found, and opposite to 120, are the diff. lat. 119.4, and the dep.
11.8.

§ c. But if the course be more than 4 Points, it is to be found at
the bottom of the page, and the precept at the foot of the column
must be the guide for taking out dep. and diff. lat.

§ d. Thus to take from Table IV. the dep. and diff. lat. for 30
miles E.N.E.; find E.N.E. at the bottom of the page, and 30 in the
dist. column, then in the two. columns, at the foot of which E.N.E.
is found, are 27*7 (dep.) and 11.5 (diff. lat) opposite to 30.

§ e. If the dist exceed 300 miles, some aliquot part of it is taken
as the 2d, 3d, etc., and the dep. and diff. lat opposite to this quotient
being increased by 2, 3, etc., as much as the dist was reduced, give
the required dep. and diff. lat Thus to find the dep. and diff lat for
400 miles N.N.W.; take 200, the half of it; then, under N.N. W. and
opposite 200, stands 184.8 in the d. lat. column, and 76-5 in the dep.
do.; each of which being multiplied by 2, gives the required diff.
lat 369.6 or 6° 9' 36", and the dep. 153 miles.

5/. Table V. is used in the same way for finding dep., diff. lat,
etc., when the course is given in degrees. Tlius the dep. for 18
miles N. 30° E. is 9 miles ; and the diff. lat for 40 miles S. 74° E.
is 11 miles.

Digitized by VjOOQIC

200

NAVIGATION.

TftATBBfK TABL*.

DUE Lat.

Dep.

Compaai Cooraes.

Coonoi Corrected,

Dtat.

N.

8.

E.

W.

N.NJ5.

NJB. by N.

14

11.6

—

T.8

—

8.8.E.

8. by E.

18

_

17.7

9.6

—

Sooth

8. by vi •

40

«~-

39.9

^-

74

8.4E.

N.W.JN,

8.JW.
N.N.W.JW.

19

p-»

18.9

•p.

14

9

7.9

— .

•^

44

N.byW.
W.by8.

North

6

6

—

—

—

West

9

—

—

—

8

Eaat

JE. by 8.

2

—

.4

2

—

N.E.

N.E. by E.

60

33.3

—

49.9

.— .

8.W. by 8.

S.W.

16

—

114

—

114

W.}8.
W.JN.

W.JN.
W.byN.JN.

7

4

—

—

7.

4

1

—

—

3.9

Lat. tailed from 41* 19" N.
Diff Lat - - 27*24"

a

60.1

874
60.1

68.2
89 1

89.1

8.

Latin - - 40° 48' 86"

N.

Mid.Lat(t604.*c.)4i*
Long. aailed from 70* 15' W.

27.4

24.1

Diff Long. 32' E.

Long, n 69° 43> W.

Variation l Point E.

§ 000. The courses in the 1st column of the "traverse table" are
the courses sailed per compass.

§ a. Those in the 2d column are those corrected (§ 398* ( b.) for
variation, which is 1 Point E*

§ b. The diff. lat and dep. in their columns, answer to the diet
and corrected courses to which they are opposite.

§ df. The diff. lat. 27'4 S. and the dep. 24.1 £. show that the ship,
by sailing the several courses and distances in the table, went to the
Southward and Eastward.

§ A The dep. 24.1 and diff. lat 274 are found opposite 80 (Table
V.) and wider 42°, which shows the course and dist " made good,"
(§ 008. § A) to be S. 42° E., 80 miles.

§ e- Under 41° (Table V.) 24.1 in the d. lat. column, stands oppo-
site to 32 in the dist. column; the diff. long, then is 32' East.

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NAVIGATION. 207

$ 610* The manner in which "dead reckoning," " days' ? PIate x
works/' etc., are worked up, and kept at sea, whether by )
means of the Tables IV. and V., or the diagram, may be learned
from the subjoined formulae.

$ 6. Aug. 28, lat. and long., 62° N. and 47° W.

$ b. Variation W. 14 Pt., which ($ 298. § b.) is to be allowed
towards the left.

Atro. 24.

DiffLeL

Sep.

Compass Courses*

Courses Corrected.

Dirt.

N.

S.

E.

W.

S.E.

8. E. by E. i E.

16

7.5

14.1

E. IN.

E. N. E.

13

4.6

11.1

E.byN.

N.E.byE.iE.

14

6.6

12.3

N.E.

N.N.E.iE.

30

17.6

9.4

N. N. E.

N.JE.

18

17.9

1.8

N. by E. i E.

North

66

66

North.

N.byW.JW.

40

38.3

1U

S.W.

8. 8. W. i W.

36

23.9

18.8

Yesterday's lat ft*) 63° N.
DhXlat - - 2° (X3e"N.

151.

30.4

48.7

23.9

Latin - - 64° O'Se"!*.
Mid. lat - - 63° /

30.4

33.9

Yesterday^ long. 47° (K W.
DuTlong. - 66' E.

120.6

24.8

Long, in - - 46° 6* W.

Course and dirt. « made good" N. 13°

E.133

miles.

§ c. Above 63°, the mid. lat. (Table V.) 24.8* in the d. lat. column,
stands opposite to 55 in the dist. column ; the diff. long, then is
55' E.

$ d. And the dep. 24.8 and diff. lat. 120.6 are found opposite to
each other under 12°, (Table V.), and opposite to 123, which
shows the course and distance made good.

• In this and all similar cases, the tabular number which is nearest to the
given number is always used, when the given number cannot be found. Thus,
in the present example, the given number 24.8 cannot be found in the d. lat
column over 63° ; 25 is the tabular number nearest to 24£ ; and W is opposite
26.

Digitized by VjOOQIC

208

NAVIGATION.
Aug. 25.

Compass Courses.

Courses Corrected.

Dist.

DUE Lat.

Dep.

N.

6.

E.

W.

East

E.S.E.

8.JE.

S.S.W.

S.W. by 8.

W.8.W.

W.JS.

West

W.N.W.

E.jS.

8.E. by E.*E.

South

S.8.W.JW.

S.W.*8.

W.byS.JS.

West

W.JN.

N.W.by W.JW.

30
18
16
19
25
22
17
30
29

4.4

14.9

4.4

9.3

18

16.3

18.5

5.3

29.7

15.4

9.8
16.8
21.3
17
29.7
24.9

Yesterday's lat 64° OWN.

> miles.

19.3

69.8
19.3

45.1

119 Ji
45.1

Diftlat. -

6C 30" 8.

Latin

63° 1C 6"

50.5

1 744

Mid. lat -
Yesterday's lonj
Difll long. -

64°

5 . 48° 5> W.
2° 50* W.

Variation i Point B.

Long, in
Coarse and dist

48° 60 7 W.
. sailed 8. 56° W. 9C

5/. To find the bearing and dist of a place in Lat 63° 20' N.»

and long. 49° 10' W.

c e Lat. of place 63° 20'

J6 ' Lat. in 63° 10'

Diff. lat 10 miles.

Long, of place 49.10
Long, in 48 49

Diff. long. —21'

§ A. 21 being found in the dist. column (Table V.) over 63° the
lat, shows opposite to it 9.5 in the d. lat column ; 9.5 then is the dep.
between the ship and the place.

§ t. The diff. lat 10, and dep. 9.5 are found together under 43"
(Table V.) and opposite 14 ; the course and dist then from the
ship to the place, is N. 43° W. 14 miles.

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NAVIGATION.

$J. April 9, lat 89° 50' N., long. 70° 10' W. ; bound to New

April 10.

CotnptM Coursta

Cowmi Corrected.

Dirt.

DtttLst.

Dep.

h.

W.

W.N.W.

W.byN.JN.
W.byN.JN.

W.byN.
W.byN.JN.

W.|N.

W.JN.

W.byN.JN.

W.byN.JN.

W.byN.

W.JN.

W.JN.

W.JN.

West

30
85
18
89
86
9
5

8.7
6.1
8.5
2.8
3.8
A
.0

28.7
24.3
17.7
28.9
26.7

9

5

Yesterday's lat 89° W N.

26.3

139.3

Di£lat -

W 18" N.

Latin '

10° 16* 18"

Variation i Point W

Mid. lat. 40°
Yesterday's long
DifElong. -

. 70° 1C W.
3° * W.

Long, in

' Course and disti

78° 1* W.

■ailed N. 80° W. 142 miles.

$*. To find the bearing and diet of Sandy Hook Light House.
S. L. House, lat. 40° 28' N.
Lat ship - 40° 15' N.

Diff. long. 18'

Long. S. L. House, 74° 1' W.
Long, ship - 73° 12' W.

Diff. long. 49'

49' of long, in lat 40° lb equal to 37.5 miles. (Table V.)

$ /. 13 and 37.5 are found together over 71° (Table V.), and

opposite 40. The dist then of the Light House from the ship, is

N. 71° W. 40 miles.

E ■

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% 10 MERCATOIK'6 6A1UNG.

MERCATOR'S SAILING.

§ 611. Dep. : rad. : : diff. long. : sec. lat. (§587. $c). Upon
the principles involved in this proportion, the charts, called Merca-
tor's, are constructed.

§ a. In these charts, the meridians of longitude, after they cross
the equator, instead of approaching each other* till they reach the
parallel of any latitude, in the ratio (§ 486. $ 6.) of rad. to the cos.
of that lat., are drawn parallel to each other.

$ 6. And the parallels of lat., say at 1° a part, instead of being at
equal distances from each other, are drawn in the ratio of rad. to
the see* of their latitude* By which means, places on a sphere, are
represented on a plane with their proper relative positions*

§ 612. In this manner of representing portions of the surface of
a sphere upon planes, the parallels of latitude are lengthened out,
and the meridians of longitude are expanded.

fa* So that, if the 1st degree of lat-, from the equator, be divided
on the chart into 60 equal parts, every degree as they succeed each
other in numerical order will contain a greater number of these parts,
than the 1st, or than that which precedes it; thus, the 61°st of lat.
contains 126 of these parts, and the 31°st, 70 of the same parts.

$ 613. These parts are called meridional parts.

§ a. Table VI. shows the number of meridional parts from the
equator to any degree and minute within the parallel of the 84°th of
lat.

§ b. The number of meridional parts between any two parallels,
is called the meridional diff. lat of those parallels.

$ c. Thus, the mer. diff. lat between 13° 10' and 18° 10', is
312.

The mer. parts of 18° 10'=1109> rp . . VT
do 13°10'=797 5 TaWeVL

Mer. diff. lat= 312

§ 614* No,w, if any section of the earth's surface be represented
on a plane projected according to Mercator's plan, the true (nearly),
instead (§ 550.) of the approximate, distance, etc., between any two
places, may be determined by plane trigonometrical operations.

§ a. To do this, the Mer- diff- lat., as well as the actual diff. lat.
between places, must be used.

§615* Mercator's sailing is useful to the navigator, chiefly in
enabling him to find, by plane trigonometrical calculations, the num-
ber of miles between places that are at a great distance from each
other.

§ a. The several cases in loxodromic sailing, may also be accu-
rately solved, according to the principles of Mercator's sailing; but
the Mer. diff. lat. must be used, as well as the actual diff. lat.

§ 6. Methods of applying the principles of Mercator's sailing to
the solution of cases in loxodromic sailing, will be shown ; but the

Digitized by VjOOQIC

NAVIGATOR

tn

application of them to practice, will be left for the amusement of the
learner.

$616. ABO represents the relative position of Boston and New
Orleans in fat. and long., according to the principles of Mercator's
sailing.

DUE Long.

B-*42?23' N.
M.pta.-?813

O-*0« 57' Mer. pta.»1885.

§ a. And the triangle a b shows the bearing, dist., etc., accord-
ing to loxodromic sailing,-

$ b. The meridional fat. of Boston is 2813, and that of New Or-
leans, is 1885 (Table VL); and the difference between these (§ 613.
$ 6/) is (O A) the Mer. dinv lat. ; and* A B is the diff. long. ; A B
is the course from New Orleans to Boston,

§ c. According to the loxodromic plan of constructing the trian-
gle, a O is the actual diff. lat. between the two places, and a b is
the dep.

$ d. Whether tits problem proposed be solved according to the
loxodromic or Mercator plan, the actual diff. lat. between the two
places is the same.

$ «. But the dep. and disk determined by the former method
(§ 550.) are- not the true dep. and dist. between the two places.

§ 617. As the angles a b O and A B O (§ 616. § d.) are equal, a b
(§>30. § d.) is parallel to A B ; whence (§73. $ d.) arise the relation
between the sides of a triangle in loxodromic sailing when com-
pared with the sides of a similar triangle in Mercator's sailing.

$a. Diff. lat. : dep. : : Mer. diffllat. : diff. long. ; also inversely
and alternately.

$6. The relations between the other parts of the two triangles
may be established according to the principles derived from $ 72. &
§ 73. The learner may arrange them.

§ 618. If the Mer. diff. of lat. (O A) between New Orleans and
Boston be made radius, several sets of proportions ($ 75.) will ap-
pear among the different parts of the triangle involved*

Digitized by VjOOQIC

213

MERCATOR'S SAILING.
A Diff Long. B~3813. M. Parti.

O- 1886. M. Parts.

$ a. But the most useful proportion (and in fact almost the only
one which occurs in practice), in Mercator's sailing, is that by
which the true course and dist. between any two places are evolved.
I shall give the terms of this proportion, and leave the others to be
arranged by the learner.

§ 619. The Mer. diff. lat. (§ 618.*) being radius ; Mer. diff. lat. :
rad. : : diff. long. : tang, course.

§ a. The diff long, thus used, must in all cases be converted
into minutes, which must be used in the log. Table of Numbers, as
miles.

§ b. To find the course from New Orleans to Boston.
Boston lat. 42° 23' M. pts.— 2813
N. Orleans " 20° 57' " 1886

12° 26'
60

928— Mer. diff. laU

Diff. Iat.a746 miles.

N. Orleans, long. 90° 9' W.
Boston " 71° 4' W.

19° 6'
60

Diff. long.— 1145 minutes (').

^^^^^^^^ •

Now, ($ 619.) 928 : rad. : : 1145 : tang, course.
928 log. (Ar. co.) -=7.032452
1146 log. (§6.) =-3.058806

Tang, course— 0.091258 N. 60° 68' 84" E.

$ c. Then calling O A, the actual diff. lat., A B becomes dep., and
retaining O A, as rad., we have ($ 574. § d.) rad. : diff. lat. : : sec.
course : dist.

Digitized by VjOOQIC

SURVEYING. 213

Diff. lat. 746 miles log. =2.872739

Course N. 50° 58' 34" E. sec. =0.200906

Log. dist. =3.073645= 1184.8 miles.

§ d. Problems in Mercator's sailing may also be solved either by-
Tables IV. & V., or by the diagram (Plate 1). Thus the mer. diffV
lat, and the diff. long., being used on the diagram, or in the tables
as diff. lat. and dep., show the course. Then with this course the
dist. is found opposite to the actual diff. lat.

§ e. Philadelphia lat. 39° 57' N. M. parts - 2610
Washington city lat 38° 53' N. " - 2536

1° 4' M. diff. lat.— 83
60

Diff. lat.=64 miles.

Washington long. 77° 2' W,
Philadelphia " 75° 9' W.

l°5a*
6a

Diff. long. =113 minutes (').

1 13 in the dep. column stands opposite to 83 in the d. lat. column;
(Table V.) 54° is at the bottom of the page. The course then
nom Philadelphia to Washington is S. 54° W. On the same page
64 in the d. lat. column stands opposite to 109 miles, which is the
dist between the two places.

SURVEYING.

$ 620. In conducting the survey of a coast, harbour, etc., the first
object should be to ascertain the geographical position of some
point connected with the survey, and the next to establish a base
line.

$621. The most advantageous location for the base line, must
be determined by the surveyor himself; and in this, he should be
governed by circumstances, such as the nature of the ground about
the place to be surveyed, and the place itself.

§ a. A level piece of ground should be selected for the base line,
and the line should terminate at points, whence some of the promi-
nent points, headlands, etc., of the place (say a harbour) under sur-
vey can be seen.

§ 622. The length of the base line must be ascertained by actual
measurement, and the direction in which it lies must be established
by observations, and noted down.

$ a. Then, knowing the length of the base line, it serves as a

Digitized by VjOOQIC

214 SURVEYING.

given side of a triangle, either for determining the length *f other
lines, or for finding the distance between either end of it, anil any
point visible from eaeh end of the base.

§ b. For if the bearing of sueh point be taken from each end of
the base, a triangle may be formed, in which the angles and the
base are known, wherefore ($ 106. $ «.) the two sides are deter-
minable.

Plate 9. 5 $ ° 23, '^ ie fig"™ ia &* annexed plate is ike prqfiU
£ of a harbour, that is being surveyed. A B is the base
line, A C B, A D B, etc., are triangles constructed npon H; the
angles at A, B, C, D, etc., are measured with a sextant or a theo-
dolite, and the length of the lines B D, B C, A C, etc., is determined
by trigonometrical calculation. The principal triangles used, are
represented by the broken lines A X, A D, etc.

§ a. So that triangles may be constructed from one line to an-
other, until sufficient data are obtained for determining, by trigono-
metrical calculation, the position of every point of the survey.

$ b. The line (X Y) of verification is determined, as to its length,
both by trigonometrical calculation, and by measurement, in order
to establish the accuracy of the survey.

$ 624. The position of the prom in e nt points C, D, E, F, etc., is
determined to a greater degree of nicety in this way, than it can be
by taking cross bearings, etc., and the intersection of lines.

§ 625. After the triangulation of the survey is completed, and
the points C, D, E, F, etc., are laid down span ike chart to the
requisite scale; the intermediate spaces, GL, L H, etc., of the
shore, may be filled up by the eyev A little practice will enable
one to sketch these intermediate spaces <witk aU necessary accu-
racy.

§ «. The position though, of aye*y prosaineat point, sucmros tight
houses, rocks, hills, castles, etc., should be established by aofeoaf
calculation.

§ 6. In taking the angles, aU the angles of every triangle should
be measured whenever it can be done. This affords one means of
detecting errors during the work of triangulating; for if the sum
($ 82.) of the thsee angles of a triangle be not 180° according to ike*
measurement, an error is known to exist..

§ c. Also the sum. of all the angles around the same point f§ A&)
should be 360 ; and, the sum of all the angles which lie within- any
measured angle, should be equal ($ 24. $ a.) to the whole angle.

$ 626. After the work of triangulation is completed, or when it
is temporarily suspended, the trigonometrical calculations for the
part already triangulated, may be made.

§ 627. The soundings of the harbour should be accurately taken,
and eorrectly laid down upon the chart.

$ b. The most practicable way of doing this, is to take a row
boat, whieh will pull equal distances in equal times ; establish the
position of the point from which sheeets out, and let her pall directly
for a given point, sounding and noting the soundings at every inter-
val of one, two, three, or more, minutes, as she goes along. When

Digitized by VjOOQIC

SURVEYING.

910

she arrives at the end of Ikie line* establish the position of ttat end
also, draw the line on the chart, divide it into as many equal parts
aft there were sounding* taken* and write along the line every one
at its proper point*

§ 6. The sorest way of Basking a straight course in the boat, is to
bring two points on shore in the same line of bearing* and then poll
(or them, keeping them in that line. For instance; a boat at a,
would pull in a straight line to E, if she kept a tree, or a steeple,
or any object at d* in a line with £ ; then the soundings taken
would all be on the line a B; and the position at a being deter-
mined ; taking the angles B a E & E B a, the length of a £ may
be known, and the soundings laid down*

$ 638. The soundings should be given for low water; wherefore
the time of the tide when the soundings are being taken, should
be noted, in order to correct them for low water.

§ a. The corrections to be applied may be obtained, either accord-
ing to the rate of the rise and fall of the tide, as established by pre-
vious observation, or by means of a figure (§ «.), thus ;

§ 6. Say that the rise and fall of the tide is 8 feet, and that the
flood tide lasts 5£ hours, and the ebb, 7.

$ c. Let the diameter of a circle be divided into 8 equal parts, by
the chords 1, 2, 3, etc. Also let one semicircumference of this cir-
cle be divided into V.j equal parts or hours, to represent the time of
flooding ; and let the other semicircumference be divided into VII.
equal arcs to represent the hours during ebb.

§ d. The point in which each of these chords cuts the circle,
shows the correction in feet which is to be applied, when the tide
is that number of hours, flood or ebb. Thus, when the tide is lll.i
flood, the correction to be applied is 4 J feet; and when the tide is
III. hours ebb, the correction is 5 feet.*

$

x 5<

r

*l

\s

9j

a \e

x^>^

\

K

* These corrections are always subtracts.

Digitized by VjOOQIC

216 SURVEYING.

$ 629. The sort of bottom should also be noted down, i. e. mud,
rock, sand, etc.

$ 630. The position, extent, etc., of all the hidden dangers, such
as rocks, reefs, shoals, banks, and of every other object of import-
ance to the navigator, should be ascertained and laid down.

§ 631. Surveys of harbours are frequently taken by measuring
first, a base line, then drawing this line upon the paper at the pro-
portional length to the scale upon which the chart is to be con-
structed, and then establishing the position of the principal points
in the survey, by the intersection of the lines of their bearing from
different points.

§ a. But the difficulty of measuring with accuracy the proper
angles upon paper, makes this method of taking a survey more lia-
ble to inaccuracies, than the triangulating plan.

Digitized by VjOOQIC

Digitized by VjOOQIC

ftute ».

Bast- Lint'

Digitized by VjOOQIC

TABLE I.

LOGARITHMS OF NUMBERS

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE L

LOGARITHMS OF NUMBERS.

N.

Lof.

N.

Lo*.

N.

i* 1

W.

Log.

1

0.000000

95

1.397940

50

1.098970

75

1-875061

S

0.301030

98

1.414973

51

1.707570

76

1.880814

3

0.477121

97

1.431363 j
1.447158 '

59

1.711003

77

1.886491

4

oeoaotso

98

53

1.794976

78

1.8990*4

99

1.469398

54

1.739393

79

1.897697.

5

0.696570

6

0.778151

30

1.477191

55

1.740363

80

1.903090

7

0.8450*8

31

1491369

56

1.748188

81

1.906484

8

0.903090

39

1.505150

57

1.755875

89

1.913814

9

0.954949

33

1518514

58

1.763498

83

141L078

•

34

1.531479

59

1.770859

84

1.924979

10

1.000000

11

1.041393

35

1.544068

60

1.778151

85

1.999419

IS

1.07)181

36

155)309

61

1.785330

86

1.934499

13

1.113**43

37

1538909

69

1.7)9399

87

1.939519

14

1.146198

38

1.579784

63

1.799340

88

1.944483

39

1.591064

64

1.806180

89

1.949390

15

1.170091

IS

1.904190

4t

1.609060

65

1.819913

90

1.954943

17

1.930440

41

1.619784

66

1.819544

91

1951041

18

1.955379

49

1.693949

67

1.8Sf075

99

1.963788

19

1.978754

43

L633460

68

1.832509

93

1.968483

44

1.643453

09

1.838849

94

1.973128

90

1.301030

91

1399919

45

1.653919

70

1.845098

95

1.977794

99

1349493

46

1.669758

71

1.851958

96

1.989971

91

1.301798

47

1.6790P8

79

1.857332

97

1.966779

94

1.380811

43

1.681911

73

1J863393

96

1.991996

40

1690196

74

L869933

90

L995635

Digitized by VjOOQIC

TABLE I,

OF NOS.

N.

1

8

3

4

5

6

7

8

9

ICO

00

0000

0434

0868

1301

1734

3166

8598

3030

3400

3891

01

4331

4751

5181

5609

6038

6466

6894

7331

7748

8174

03

8600

9036

9451

9876

01

0300

0734

1147

1570

1903

9415

03

3837

3359

3680

4100

4531

4940

5360

5779

6197

0616

04

06

7033

7451

7868

8284

8700

9116

9532

9947

0361

0776

105

1189

1603

3016

8438

8841

3853

3664

4075

4486

4896

CO

5306

5715

6134

6533

6943

7350

7757

8164

8571

8978

07

9384

9789

03

0195

0600

1004

1409

1818

8216

9019

3091

08

3434

3836

4327

4CJ8

5028

5430

5830

6330

7098

09

04

7437

7835

8333

8080

9017

9414

9811

0807

066S

110

1393.

1787

8182

3576

8989

3368

3755

4148

22

4933

11

5333

5714

6105

6495

6885

7275

7084

8053

8830

IS

93J8

9608

9993

05

0380

0766

1153

1538

1984

9309

9*84

13

3078

3463

3846

4330

4613

4996

5378

57bl

6149

0594

14

08

6905

7386

7066

8046

8486

8806

9185

9563

9949

0390

115

0698

1075

1453

1839

8806

3583

8958

3333

3709

4083

16

4458

4832

5208

5580

5953

6336

6699

7071

7443

7815

17

8186

8557

8928

9298

9668

07

0038

0407

0777

1145

1514

18

1883

3350

8P17

8985

3353

3718

.4085

4451*

4816

5lHi

19

5547

5913

6276

6640

7004

7368

7731

8094

8457

btio

190

9181

9543

9905

08

0366

0686

0987

1347

1707

9087

9496

31

3785

3144

3503

3861

4319

4576

4934

5891

5647

0004

S3

63C0

6716

7071

7427

7781

8136

8491

8845

9196

9552

83

9905

09

0258

0611

0983

1315

1667

8018

9370

9791

3071

34

3433

3773

4123

4471

4820

5169

5518

5867

6915

titS

135

10

6910

7357

7004

7951

8897

8644

8990

9335

9081

0096

36

0370

0715

1059

1403

1747

3091

8434

8777

3119

3449

37

3804

4146

4487

4828

5169

5510

5851

61fl

6531

6b71

38

11

7310

7549

7888

8237

8565

8903

9241

9578

9916

39

0590

0986

1863

1598

1934.

3210

8605

8940

3875

3009

130

3943

4277

4611

4944

5278

5610

5943

6276

6608

0940

31

13

7271

7603

7934

8265

8535

8986

9256

9586

9915

0345

33

0574

0903

1833

15f0

1888

8216

8543

9871

3198

3595

33

3P59

4178

4504

4830

5156

5481

5? 07

6131

6456

6781

34

13

7105

7489

7753

8676

8399

8733

9045

9368

9690

0019

135

0334

0F55

0977

1398

1619

1939

8260

8589

9TO

3919

36

3539

3*58

4177

4496

4814

5133

5451

5709

€086

6403

37
38

6731
9879

7037

7354

7671

7987

8303

8618

B934>

9849

9564

14

0194

0598

0829

1136

1450

1763

9077

93T0

9709

39

3015

3327

3639

3951

4863

4574

4885

5196

5507

5bl8

140

6138

6438

6748

7058

7367

7676

7985

6394

8603

6911

41

9219

9587

9835

15

0148

0449

0756

1063

1370

1676

198

43

8368

3594

9900

3305

3510

3815

4130

4494

«98

5039

43

5336

5640

5943

6346

6549

6853

7154

7457

7759

fcOU

44

10

8368

8664

8965

9866

9567

9868

0108

0409

0769

10GB

145

1368

1687

1967

8866

8564

8863

3161

3460

3757

4055

46

4353

4650

4947

5844

5541

5838

6134

6430

6796

7099

47

7317

7613

7909

8803

8498

6793

9086

9381

9074

9968

48

17

0368

0555

0848

1141

1434

1786

8019

3311

9003

9895

N.

1

3

3

4

5

6

7

8

9

Digitized by VjOOQIC

TABLE I.— LOO. OF H08.

n.

1

9

3

4

5

7

8

•

149

17 3166

3478

3709

4060

4351

4641

4932

5222

5512

&S02

150

0091

6381

6670

6959

7248

7530

7825

8113

8401

8089

si

8977

9305

9558

9839

18

0126

0413

0699

0986

1272

1558

59

1844

9139

3415

3700

2J85

3270

3555

3839

4123

4406

53

4001

4975

525J

5543

5825

6108

0391

6674

61/56

7239

54

7531
19

7803

8064

8360

8647 .

, 8928

9210

9490

9771

0051

155

0333

0913

0893

1172

1451

1730

3010

3389

8568

2846

50

3135

3403

3681

3959

4238

4514

47i2

5069

5346

5693

57
58

5900

8657

6170
8932

6452
9260

672J
9481

7005
9755

7281

7556

7832

8107

8382

90

0039

0303

0577

0851

1124

50

1397

1070

1943
4669

3310

348S

2761

3033

3305

3577

3848

160

4190

4391

4934

5904

5475

5746

6016

6386

6556

01

6*30

70.16

7365

7034

7904

6173

8441

8710

8979

9347

08

9515

9783

91

0051

0319

0586

0853

1120

1388

1(554

1021

63

3188

3454

3730

2986

3252

3518

37d3

4049

4314

4579

04

4844

5109

5373

5038

5902

0166

6430

6694

6957

7221

165

7484

7747

8010

8373

8536

8798

9060

9323

9585

9846

00

S3 0108

0370

0631

0892

1153

1414

1675

1936

3196

3456

07

8717

2T76

3236

3496

3755

4015

4274

4533

4792

5051

08

5309

55'»8

5836

6084

6342

6 00

6858

7115

7372

7630

60

7887
S3

8144

8400

8657

8913

9170

9426

96d2

9938

0193

170

0449

0704

0960

1215

1470

1724

1979

3233

2488

3743

71

3990

3*250

3504

3757

4011

4264

4517

4770

5023

527.i

72

5528

5781

6033

62 J 5

6537

6789

7041

72;12

7544

77i»5

73

8046
34

0549

8397

8548

8799

9049

9300

9550

9800

0050

0300

74

0799

1048

1397

1547

1795

3044

3393

2541

37:4

175

3038

3386

3534

3782

4030

4977

4525

4772

5019

5366

70

5513

5753

6096

62J2

6409

6745

6091

7236

7482

7738

77

7973
85

0430

8319

8464

87U9

8954

9196

9443

9687

9932

0176

78

0664

CM»

1151

1335

1638

1882

2125

3368

3610

79

2853

3090

3338

3580

3822

4034

430V>

4548

47t»0

5031

190

5373

5514

5755

5996

6337

6477

6718

695?

7198

7433

81

7679

7919

8158

8398

8637

8877

9116

9355

95)4

9833

82

30 0071

0310

0548

0787

1025

12 a

1591

1738

1976

3314

83

3451

3lW8

2926

3162

33.19

3636

3*73

4101

4346

45-3

84

4818

5054

5290

5525

5761

5396

6333

6467

6702

0937

185

7173

74015

7641

7875

8110

8344

8578

8812

9040

9379

80

9513
IMS

9746

9980

0213

0446

0679

0913

1144

1377

1R09

87

3074

2303

253«

2770

3001

3233

3464

3696

3'27

88

4158

4339

4«.20

4&'j0

50 *\

5311

5542

5772

6002

62:s

89

0463

6693

6921

7151

7380

7609

7o38

8067

82J6

8525

190

8754

98

1013

8982

9310

9439

9667

9995

0123

0351

0578

0806

91

12*1

1488

1715

1942

2169

23*16

3622

3849

3075

93

3301

3527

3753

3<»79

4205

4431

4656

4882

5107

5333

93

5557

57*2

6007

6232

6457

6681

6905

7130

7^54

7578

94

7802

8025

8249

8473

8690

8920

9143

9366

9589

9813

195

39 0035
2356

0357

0480

0702

0925

1147

1369

1591

1813

3034

90

3478

3600

2'20

3142

333

3584

3804

4025

4246

97

44R6
W5

4<y*7

4"07

5127

5347

55S7

57"<7

£007

6226

644«

98

68*4

7104

7323

7542

77M

7U79

8138

8410

8635

99

8853
30

9071

9289

9507

9725

9943

0161

0378

05?5

0813

900

1030

1247

1464

1681

1898

2114

3331

3547

3764

30P0

01

3190

3412

3628

3844

4060

4375

4490

4706

4921

5136

N.

1

9

3

4

5

6

7

9

Digitized by VjOOQIC

TABLE I.— LOO. OF Mud.

N.

1

2

3

4

5

6

7

8

9

903

30 5351

5566

5781

5996

6211

6425

6639

6854

7068

7389

03

74J6

7710

7924

8137

8351

8564

6778

8991

9304

9417

04

9630

9643

31

0056

0268

0481

0893

0906

1118

1330

1549

305

1754

1966

2177

2389

8600

3813

3023

3334

3445

3959

06

3*7

4078

4289

4409

4710

4920

5130

5340

5551

5761

07

5370

6180

6&J0

6599

680J

7018

7227

7437

7646

7e54

06

8063

6273

8481

8689

8898

9106

9314

9532

9730

9H39

03

32 0146

0354

0502

0769

0977

1184

1391

1596

1896

991*

310

2219

2426

2633

2839

3046

3353

3458

3665

3871

4977

11

4263

4488

4694

4900

5105

5310

5516

5731

5936

9131

12

6336

6541

6745

6950

7154

7399

7563

7797

7973

8176

13

8360
33

8583

8787

8991

9194

9338

9601

9805

0008

93*1

14

0414

0617

0819

1022

1325

1437

1630

1833

9034

9336

215

2439

2640

2B42

3044

3246

3447

3649

3850

4051

4953

16

4454

4655

4856

5056

5257

5458

5657

5859

6059

394

17

6400

6660

68t»0

7000

72^0

7453

7659

7858

8058

6357

18

8457
34

8656

8855

9054

9253

9451

9650

9849

0047

9349

19

0444

0642

0841

1039

1237

1435

1633

1830

9038

9335

390

2423

2620

2817

3014

3213

3409

3606

3803

3990

4199

21

4392

4589

4785

4981

5178

5374

5570

5766

5969

9157

22

6353

6549

6744

6939

7135

7330

7525

7730

7915

8110

23

8305
35

8500

8694

8869

9083

9377

9472

9666

9800

9954

24

0248

0442

0636

0839

1023

1316

1410

1603

1799

I960

225

2182

2376

2568

2761

3954

3147

3339

3533

9794

39M

26

4108

4301

4493

4685

4876

5068

5260

5453

5643

5895

27

6026

6217

6408

6599

6791

6981

7172

7363

7554

7744

28

7935

8125

8316

8506

8696

8886

9076

9266

9459

9649

29

. 9836

36

0025

0215

0404

0593

0783

0972

1161

1359

1539

230

1728

1917

2105

2314

2483

8671

3859

3048

3339

3494

31

3612

3800

3388

4176

4363

4551

4739

4926

28

5301

32

5488

5U75

5862

6049

6236

6423

6610

6796

7170

33

7356

7542

7723

7915

8101

8387

8473

8659

8945

9939

34

9216

9401

9587

9772

9958

37

0143

0328

0513

0698

9989

235

1058

1253

1437

1622

1807

1991

2175

3390

3544

3739

36

2J12

30 J6

3380

3464

3647

3831

4015

4i98

4389

45u5

37

4748

4H32

5115

5398

54H1

5664

5846

6039

6912

63*4

3d

6577

6759

6942

7134

7308

7488

7670

7852

8034

6216

34

8398
38

8580

8761

8943

9124

9306

0487

9668

9849

0939

240

0211

0392

0573

0754

0935

1115

1296

1476

1656

1837

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2017

21'i7

2177

2557

2737

2 17

3097

3977

3456

3u36

42

3815

3195

4174

4353

4533

4712

4891

5070

5349

5498

43

5106

57P5

5364

6142

6321

6499

6677

6856

7034

7319

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7390

7568-

7746

7923

8.01

8279

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9166
39

9343

0521

9698

0875

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1112

1288

1464

1641

1817

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31l>9

9345

9531

47

2697

2873

3048

3234

3400

3575

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3936

4101

4377

48

4452

4627

4803

4977

5152

5336

5501

5676

fr-50

6935

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6199

6374

6548

6722

68U6

7071

7245

7419

7542

7796

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7940

8114

8287

8461

8634

8808

8981

9154

9337

9991

51

9674

9847

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0192

0365

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0711

0883

1059

1339

52

1400

1573

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1917

2089

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2433

2605

2777

3349

53

3131

3292

3464

3635

3807

3978

4149

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4493

40*3

54

4834

5005

5176

5346

5517

5'I88

5858

6039

6199

am

255

6540

6711

6881

7051

7221

7391

7561

773!

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1

9

3

4

5

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7

1 8

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Digitized by VjOOQIC

TAALK i

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Jf.

1

9

3

4

5

6

7

8

9

956

40 8940

8410

8579

8749

8918

9087

6257

9426

9515

9764

47

9933

41

0103

0371

0440

0608

0777

0946

1114

1283

1451

n

1690

1788

1956

9124

3293

9461

2628

27«6

2964

3132

59

3300

3467

3635

3b03

3970

4137

4305

4472

4639

4608

960

4073

5140

5307

5474

5641

5808

5974

6141

6308

6474

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0640

6807

6L73

7139

7306

7472

7638

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75*70

8135

69

8301

8467

8633

8798

8964

9129

9295

9460

9625

97bl

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9956

49

0191

0286

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0816

0781

0945

1110

1375

1439

64

1604

1768

1933

3097

3262

3426

2590

3754

2*18

3062

MS

3946

3410

3573

3737

3901

4064

4238

4392

4555

4718

66

4889

5045

5208

5371

5534

5697

5tC0

€023

6186

1349

67

6511

6674

6837

6999

7161

7324

7486

7648

7811

71.73

68

8135

8297

8459

8621

8783

8944

9106

9268

9429

65»1

60

9759

9014

43

0075

0236

0398

0559

0720

0881

1049

1203

970

1364

1535

1685

1846

2007

2167

2328

9488

9649

2809

71

9969

3130

33T0

3450

3610

3770

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40P0

4349

4409

73

4509

4738

4688

5048

5207

5367

5526

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5844

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73

6163

6323

6481

6640

6799

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7116

7275

7433

7512

74

7741

7909

8067

8226

6384

8542

6701

8859

6017

9175

975

9333

9491

9648

9808

9964

44

0122

0279

0437

0594

0752

76

0909

1066

1224

1381

1538

1615

1852

2009

21t6

2323

77

2480

9637

2793

2i»50

3107

3263

3420

3576

3732

3889

78

4045

4201

4357

4513

4669

4*25

4981

5137

52*3

5449

79

5604

5760

5915

6071

6226

63b2

6537

6693

6848

7003

980

7158

7313

7468

7633

7778

7933

8088

8242

8397

8552

81

8706
45

8861

9015

9170

9324

9478

9633

9787

9941

0095

89

0949

0403

0557

0711

0865

1018

1172

1326

1479

1633

83

1786

1940

2003

2247

2400

3553

8706

2859

3012

31C5

84

3318

3471

3694

3777

3930

4083

4235

4387

4540

46S»2

965

4845

4997

5150

5302

5454

5608

5758

5910

6062

6214

66

6366

6518

6670

6821

6973

7125

7276

7428

7579

7731

87

7889

8033

8184

8336

8487

8638

8789

8940

6091

9242

88

9399

9543

9694

9845

9995

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40

0146

0296

0447

0597

0747

89

0898

1048

1198

1348

1499

1649

1799

1949

2038

2248

990

9308

9548

9697

3847

2997

3146

3296

3445

3594

3744

91

3893

4042

4191

4341

4490

4639

4787

4936

5085

5234

99

5383

5533

5680

5829

5977

6126

6274

6423

6571

t>719

93

6868

7016

7164

7312

7460

7C08

7758

7! 04

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8200

94

8347

8495

8643

8750

8938

9065

9233

9380

S528

9675

995

9899

9969

47

0116

0963

0411

0557

0704

0P51

0998

1145

98

1438

1585

1733

1878

2025

2171

2317

2464

2610

97

9756

9903

3049

3195

3341

3487

3633

3779

3 25

4070

98

4916

4369

4508

4653

4799

4944

5OT0

5235

5381

5526

99

5671

5816

5962

6107

6252

6397

6542

6687

6832

6977

30ft

7191

7966

7411

7555

7700

7845

7989

8133

8278

8422

01

8967

8711

8855

8999

9143

9287

9431

9575

9719

98C3

69

48 0007

0151

03P5

0438

0583

0725

0869

1013

1156

12T9

03

1443

1586

1739

1879

3016

2159

2302

2445

2588

3731

04

9874

3016

3159

3303

3445

3587

3730

3873

4015

4157

905

4300

4449

4585

4737

4869

5011

5153

5995

5438

5579

66

5791

5863

6005

6147

6389

6431

6572

6714

6855

6997

07

7138

7380

7421

7563

7704

7845

7986

8128

8260

8410

68

8551

8099

8833

6974

9114

9255

9396

9537

9677

9618

69

9958

49

0099

0940

0369

0590

0661

0801

0941

1081

1223

no

-

1509

1649

1769

1999

9069

9909

9341

9481

9681

i*

1 •

1

9

1 •

4

5

7

a

9

Digitized by VjOOQIC

TABLE I. LOO. OF NOS.

N.

1

3

3

4

5

6

t'

.

6

311

49 3760

3900

3040

3179

3319

3458

3597

8737

3816

4815

12

4155

4294

4433

4573

4711

4850

4989

5138

5967

5406

13

5544

5683

5833

5960

6099

6337

6376

6515

6653

6791

14

6930

7068

7306

7344

7489

7631

7759

7897

8035

8173

315

8311

8448

8586

8734

8863

8998

9137

8875

8418

8558

16

9687

9835

9968

50

0099

0336

0374

0511

8017

0785

0989

17

1059

1196

1333

1470

1607

1744

1881

3154

8991

18

3437

3564

3700

8837

3973

3109

3346

3383

3518

3654

20

3791

3937

4063

4199

4335

4471

4607

4743

4878

5014

390

5150

5386

5431

5557

5693

5898

5964

6098

6834

6378

21

6505

6640

6776

6911

7046

7181

7316

7451

7586

TI81

S3

7856

7991

8136

8360

8395

8530

8664

8798

8934

SIMS

S3

9903
51

9337

9471

9606

9740

9674

0009

0143

0977

0411

34

0545

0679

0813

0947

1081

1315

1349

1488

1616

1750

385

1883

9017

3150

8384

3418

8551

8684

8818

9951

3884

SB

3318

3351

3484

3017

3750

3883

4016

4148

4888

4415

27

4548

4681

4813

4946

5079

5311

5344

5476

5609

*74J

SB

5874

6006

6139

6371

6403

6535

6688

6800

6838

1864

89

7196

7338

7460

7593

7734

7855

7987

8119

8351

83B8

330

8514

8646

8777

8909

9040

9178

8303

8434

8566

9897

31

9828

9959

53

0090

0331

0353

0483

0815

0746

0876

1807

33

1138

1300

1400

1530

1661

1793

1983

9053

3183

8314

33

3444

3575

3705

3835

3966

3096

3396

3356

3486

3616

34

3747

3877

4006

4136

4366

4396

4586

4656

4785

4915

335

5045

5174

5304

5434

5563

5683

5883

5951

6081

6910

36

6339

6469

6598

6737

6856

6085

7114

7343

7373

7301

37

7630

7759

7888

8016

8145

8374

8403

8531

8680

8788

38

8917
53

9045

9174

9303

9430

9559

9687

8815

9843

0978

39

0900

0338

0456

0584

0713

0840

0968

1086

1983

1351

340

1479

1607

1734

1803

1990

3117

9345

8378

9500

8887

41

3754

3883

3009

3136

3363

3391

3516

8645

3779

3889

42

4036

4153

4380

4407

4534

4661

4787

4914

5041

5168

43

5394

5431

5547

5674

5800

5937

6053

6180

6306

6433

44

6558

6685

6811

6937

7063

7189

7315

7441

7567

7683

345

7819

7945

8071

8197

8333

8448

8574

8699

8835

8851

46

9076
54

9303

9337

9453

9578

9703

8889

8954

0079

0804

47

0330

0455

0580

0705

0830

0955

1080

1305

1330

1454

48

1579

1704

1839

1953

8078

8303

8337

8458

8577

8701

49

8835

3950

3074

3199

3333

3447

8571

8096

3880

3944

350

4068

4193

4316

4440

4564

4688

4818

4936

5060

5183

51

5307

5431

5555

5678

5803

5935

6049

6173

6886

6418

53

6543

6666

6789

6913

7036

7159

7388

7406

7588

9658

53

7775

7898

8031

8144

8906

8389

8513

863S

8758

8881

54

9003
55

9136

9349

9371

9494

8616

9739

8861

8884

0186

.355

0338

0351

0473

0595

0717

0840

0968

HH4

1906

1388

56

1450

157S

1694

1816

1938

8060

8181

8303

9485

8547

57

3668

3790

2919

8033

3155

3376

3398

3519

3840

8768

58

3883

4004

4136

4347

4368

4489

4610

4731

4859

4874

59

5094

5315

5336

5457

5578

5699

5890

5940

6061

6188

360

6303

6433

6544

6664

6785

6805

7086

7146

7867

7387

61

7507

7698

7748

7868

7988

8108

8988

6348

6468

8588

63

8709

8898

8948

9068

9188

9308

8488

8548

8667

8787

63

9907

56

0036

0146

0965

0385

0504

0634

0743

0863

8988

64

1101

1331

1340

1458

1578

1687

1817

1836

8055

9174

365

9899

8419

8531

8688

3768

8887

3886

3185

3844

3888

N.

1

3

3

4

5

6

7

8

8

Digitized by VjOOQIC

TABLE I.— LOO. OF NO0.

If.

3

8

3

4

5

6

7

8

9

a*

50 3481

3600

3718

3837

3956

4074

4113

4311

4489

4S48

67

4666

4764

4903

5031

5139

5257

5376

5494

5613

5730

68

5848

5966

6084

6802

6320

6438

6555

6673

6791

6909

m

7026

7144

7363

7379

7497

7614

7733

7850

7967

8084

379

8303

8319

8436

8554

8671

8786

8905

9033

9140

9357

71

0374
57

0491

9608

0735

9843

9959

0076

0193

0309

0436

78

0543

0660

0776

0893

1010

1136

1343

1359

14"<6

1513

73

1709

1835

1943

3058

3174

831

3407

3533

3639

3758

74

3873

3988

3104

3330

3336

3453

3568

36b4

3800

3916

375

4031

4147

4363

4379

4494

4610

4736

4841

4957

5073

76

5168

5303

5419

5534

5650

57C5

5880

5996

6111

6836

77

6341

6456

6573

6687

6803

6917

7033

7147

73c3

7377

78

7493

7607

7723

7836

7951

8066

81fcl

8395

8410

8535

79

8639

8754

8868

8983

9097

9812

9326

9441

9555

9669

360

0784

9898

58

0013

0186

0340

0355

0460

0583

0697

0811

81

0935

1039

1153

1367

13P1

1415

1108

1723

1*36

1950

86

3063

8177

3391

3401

3518

3631

3745

3fc59

8973

3085

83

3199

3313

3436

3539

3652

37t5

3879

%92

4105

4218

84

4331

4444

4557

4670

4783

4896

5003

5122

5335

5348

385

5461

5574

5686

5799

5913

6034

6137

6250

6363

6475

86

6587

6700

6813

6935

7037

7150

7363

7374

7487

7599

87

7711

7833

7935

8047

8160

8373

8384

8496

a 08

6730

88

8833

8944

9055

0167

9379

9391

9503

9615

9736

9638

80

0050

50

0061

0173

0384

0366

0508

0619

0730

0643

0953

390

1065

1176

1887

1399

1510

1631

1733

1843

1955

8066

01

3177

3388

8399

3510

3631

3733

3843

8954

30t4

' 3175

03

3386

3397

3508

3618

3739

3840

3:50

40131

4173

4848

03

4393

4503

4614

4734

4834

4945

5055

5115

5376

5386

04

5496

5606

5717

5837

5937

6047

6157

63b7

6377

6487

395

6597

6707

6817

6987

7037

7146

7356

7366

7476

7586

06

7695

7805

7914

8034

8134

8343

8353

8463

8573

8681

07

08

8791
9863
60

8900
9993

9009

0119

9338

9337

9446

9556

9665

9774

0101

0310

0319

0488

0537

0646

0755

0864

00

0073

1083

1101

1399

1408

1517

1635

1734

1843

1951

400

9060

3169

8377

8386

8494

3603

3711

3810

8988

3036

01

3144

3353

3361

3469

3577

3686

3794

3603

4010

4118

03

4336

4334

4443

4550

4658

4766

4874

49b3

5KO

5197

03

5305

5413

5530

5188

5736

5843

5951

6059

6166

6374

04

6381

6480

6596

6704

6811

6910

7036

7133

7341

7348

405

7455

7563

7670

7777

7884

7991

8098

8805

8318

8419

06

8538

8633

8740

8847

8654

90C0

9167

9374

9381

9488

07

0594

0701

9808

9914

61

0031

0138

0334

0341

0447

0554

08

0660

0767

0873

0979

1086

11£8

13S8

1405

1511

1617

00

1733

1830

1936

8048

3148

8354

83€0

3466

8573

8u78

410

9784

8800

8996

3103

3307

3313

3419

3535

3630

3736

U

3843

3948

4053

4159

4864

4370

4475

4581

4686

4713

13

4897

5003

5108

5313

5319

5424

5539

5(335

5740

5845

13

5950

6055

6160

6365

6370

6476

6581

6686

6791

6815

14

7000

7105

7310

7315

7430

7535

7639

7734

7839

7943

435

8048

8153

8357

8368

6467

8571

8676

8780

8884

8989

16

0003

OS

0198

0303

9406

9511

0615

0719

9633

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0033

17

0136

0340

0344

0448

0559

0657

O7C0

0864

0968

1073

18

1176

1380

1384

1488

1593

1(95

1700

If 03

3007

8110

10

8314

8318

8481

8535

8638

8733

8636

8939

3048

3146

480

3849

3353

3456

3559

3663

3766

3969

3973

4076

4179

31

4888

43P5

4488

4591

46P5

47T8

4? 01

5004

5107

5310

S3

5313

5415

5518

5631

5734

5637

5i>30

6039

6135

6838

If.

1

3

3

4

5

6

7

8

9

Digitized by VjOOQIC

10

TABLS I.— LO0. OF HOS.

w.

1

9

3

4

5

6

7

8

•

483

09 6340

6443

6546

6648

6751

6853

6956

7059

7101

7963

94

7306

7406

7571

7073

7775

7878

7980

80ri9

8184

8987

433

8389

8491

8593

8695

8797

6000

0009

9104

9300

9109

90

9410
03

9519

9613

9715

9817

9919

0091

0193

0994

0096

97

0498

0530

0631

0733

0835

0930

1038

1139

1941

00

98

1444

1545

1647

1748

1649

1951

9059

9153

9955

9359

99

9457

9559

9660

9761

9809

9963

3064

3165

3906

3357

430

3409

3569

3670

3771

3879

3973

4074

4175

4976

4377

31

4477

4578

4079

4780

4660

4961

5081

5189

5963

5383

38

5484

5584

5885

5785

5866

5986

6080

6167

6987

4386

• 33

6488

6568

6688

6789

6889

0089

7080

7160

7990

7390

34

7490

7590

7690

7790

7890

7990

8090

8190

8999

8389

~*43S

8489

8589

8689

8789

8868

8968

9098,

9188

9967

9987

- 36

9487
64

9586

9686

9785

9865

9984.

0064

0383

6963

4069

87

0481

0581

0680

0780

0879

0978

1077

1176

1976

1375

38

1474

1573

1679

1771

1671

1970

9009

9168

9907

9306

30

9405

9563

9669

9701

9860

9959

3056

3157

3tt5

3354

440

3453

3551

3650

3740

3847

3940

4044

4143

4949

4340

41

4489

4537

4636

4734

4839

4931

5099

5197

5996

sm

49

5499

5591

5619

5717

5615

5913

0011

6110

6906

0906

43

6404

6509

6600

6096

6796

0894

6999

7069

7187

7995

44

7383

7461

7579

7676

7774

7879

7969

8067

8105

8999

445

6300

8458

8555

8653

8750

8648

8945

9043

9140

9937

40

9335

65

9439

9530

9697

9794

9898

9919

0010

0113

0UO

47

0307

0405

0509

0599

0090

0793

0690

0987

1084

U81

48

1978

1375

1479

1509

1600

1769

1859

1950

9053

9159

40

9946

9343

9440

9530

9633

9730

9690

9093

3019

3110

450

3919

3309

3406

3509

3598

3095

3791

3888

3084

0089

51

4177

4973

4369

4465

4509

4658

4754

4850

4940

9949

59

5196

5934

5331

5497

5593

5619

5715

5611

0009

58

6098

6104

6990

6386

6489

6577

0673

6769

6865

0900

54

7050

7159

7947

7343

7438

7534

7795

7891

7910

455

8011

8107

8909

8996

8393

8463

8584

8679

8774

6870

58

8065

9000

9155

9950

9346

9441

9536

9631

9796

9891

57

9916

06

0011

0106

0901

0996

0391

0486

0591

0076

•771

58

0886

0900

1055

1150

1945

1339

1434

1590

1693

1718

59

1813

1907

9009

9090

9191

9960

9380

9475

9599

480

9768

9859

9*47

3041

3135

3930

3394

3416

3513

3007

61

3701

3795

3B89

3983

4078

4172

4908

4360

4454

4549

69

4649

4736

4630

4994

5016

5119

5908

5999

5393

5487

63

5561

5875

5769

5669

5956

0050

6143

0937

6331

0494

64

6516

6619

6705

6799

0609

0966

7079

7173

7906

7300

465

7453

7546

7640

7733

7890

7990

8013

8106

8900

6993

60

8386

8479

8579

8605

8759

8859

8945

9036

9131

9994

67

9317
67

9410

9503

9596

9689

9789

9875

9907

0009

0153

66

0946

0339

0431

0594

0817

0710

0609

0695

0986

1080

09

1173

1965

1358

1451

1543

£638

1798

1891

1913

470

9098

9190

9963

9975

9467

9560

9659

9744

9837

9999

71

3091

3113

3905

3997

3390

3489

3574

3006

3756

3850

79

3949

4034

4196

4918

4310

4409

4494

4566

4677

4709

73

4861

4951

5045

5137

5938

5390

5419

5503

5595

5087

74

5778

5670

5969

0053

6145

6930

0398

0419

0511

475

6604

6785

0670

0968

7059

7150

7949

7333

7494

7510

70

7607

7606

7789

7661

7979

80*O

8154

8945

8336

8497

77

8518

660)

8700

8791

8689

6P73

9064

9155

9940

907

78

0498

08

0519

9610

9700

9791

9689

0973

0063

0154

0945

79

0380

0498

0517

0607

0698

0789

0879

0970

1000

1151

If.

1

9

3

4

5

6

7^

8

•

Digitized by VjOOQIC

TABU I.— LOO. OF NQS.

11

N.

1

8

3

4

5

6

7

8

480

08 1041

1333

1483

1513

1003

1093

1784

1874

1065

8055

61

S145

8335

8336

3416

8506

8596

8086

8777

8867

8057

61

3047

3137

3337

3317

3407

3497

3587

3677

3767

3857

a

3047

4037

4187

4817

4307

4306

4486

4576

4666

4756

84

4843

4035

5085

5115

5305

5304

5383

5473

5563

5658

405

37a

5331

5031

6010

6100

6180

6879

6368

6458

6547

8S

0830

6788

6815

6004

6094

7083

7178

7861

7351

7440

87

7330

7618

7707

7796

7886

7975

8064

8153

8848

8331

88

8400

8500

6508

8687

8776

8865

8053

9049

8131

9830

80

60

0308

0488

0573

9664

9753

0841

9030

0010

0107

400

0106

0885

0373

0468

0551

0630

0738

0816

0005

0003

01

1083

1170

1858

1347

1435

1534

1618

1700

1780

1877

03

1065

8053

8148

8830

8318

3406

3494

8583

son

8750

03

sen

8035

3083

3111

3190

3387

3375

3463

3551

3639

04

3737

3815

3003

3091

4078

4166

4854

4343

4430

4317

400

4805

4603

4781

4868

4056

5044

5131

5310

5307

5304

06

5483

5560

5657

5744

5833

5019

6007

6004

6188

0309

07

6356

6444

6531

6619

6708

6793

6880

6088

7055

7148

08

7880

7317

7404

7491

7578

7665

7753

7830

7986

8013

80

8101

6188

8375

8363

8440

8535

8683

8700

8796

8683

300

8070

0057

0144

0331

0317

9404

0401

9578

0664

0751

01

0838

0034

70

0011

0098

0184

0371

0357

0444

0531

0617

08

0704

0700

0877

0063

1050

1136

iy»

1300

1395

1483

09

1568

1654

1741

1887

1913

8000

8086

8178

8858

8344

04

8430

8517

8603

8680

8775

8961

8047

3033

3119

3305

305

3291

3377

3463

3540

3635

3781

3807

3803

3979

4065

08

4151

4836

4383

4408

4404

4579

4665

4751

4837

4033

07

3008

5004

5170

5365

5351

5436

5538

5607

5693

5778
0633

08

5864

5040

6033

6180

6808

6891

6376

6488

6547

00

6718

6003

6888

6074

7050

7144

7830

7315

7400

7465

310

7570

7655

7741

7886

7011

7906

8081

8166

8851

8336

11

84S1

8506

8391

8676

8761

8846

8931

0015

0100

0185

10

0370
71

0355

0440

0584

0609

0604

0770

0863

0048

0033

13

0117

0303

0387

0371

0456

0540

0685

0710

0704

0870

14

0063

1048

1133

1317

1301

1385

1470

1554

1636

1783

313

1807

1801

1076

8060

3144

8389

8313

8307

8481

8566

10

8650

8734

8818

8909

8986

3070

3154

3939

3383

3407

17

3401

3575

3658

3748

3910

3004

4078

4168

4846

18

4330

4414

4497

4581

4665

4749

4833

4916

5000

5084

10

5167

5851

5335

5418

5303

5586

5660

5753

5836

5030

300

6003

6087

6170

6854

6337

6481

6504

6588

6671

6754

SI

6838

6031

7004

7088

7171

7354

7338

7481

7504

7587

OS

7670

7754

7837

7930

8003

8086

8160

8353

8336

8410

83

8503

8585

8088

8751

8834

8917

9000

0083

8165

9848

84

0331
78

0414

0407

0580

9663

9746

9838

0911

0904

0077

333

0150

0848

0385

0407

0400

0573

0855

0738

0831

0003

SO

0086

1068

1151

1833

1316

1308

1481

1563

1646

1788

87

1811

1893

1075

8058

3140

8333

9305

8387

8460

8553

88

8034

S7M

3798

8881

3063

3045

3187

3300

3291

3374

80

3456

3538

3090

3708

3784

3866

3048

4030

4118

4104

330

4876

4358

4440

4583

4003

4085

4767

4840

4031

5013

31

5004

5376

5358

5340

5483

5503

55B5

5687

5748

5830

33

5013

5003

6075

6157

6838

6380

6401

6483

6564

6646

33

6787

6800

6800

6078

7053

7134

7816

7897

7370

7460

34

7341

7083

7704

7785

7860

7048

8030

8110

8101

8373

335

8354

8435

8516

8507

8676

8750

8341

8383

0003

0084

33

0365

0346

0337

0408

0480

0570

0351

0733

0813

0603

37

0074

73

0055

0136

0317

0308

0370*

0450

0540

0681

0701

K.

1

3

3

4

5

6

7

8

Digitized by VjOOQIC

IS

TABLE I.— LOO. Of N0«.

».

1

9

3

4

5

6

7

8

•

538

73 07^9

0863

0J44

1034

1105

1186

1366

1347

1488

1506

39

15c9

1609

1750

1830

1911

1991

8073

3153

8333

1.313

540

9394

9474

3555

8635

3715

3796

3676

8956

3037

3117

41

3197

3878

3358

3438

3S18

3598

3679

3759

3*39

3M9

43

31)99

4079

4160

4340

4330

4400

4460

4560

4140

4799

43

4£00

48fa0

49110

5040

5130

5900

5380

5360

5440

5599

44

5599

5679

5758

5638

5918

5998

6078

6157

6837

6317

MS

6397

6476

6556

6636

6715

6795

6874

6954

7034

7113

46

7193

7979

7359

7431

7511

7590

7670

7749

7839

7106

47

7*87

80o7

8146

8835

6*05

8384

8463

8543

8199

6701

40

8781

8960

8939

9018

6097

9177

9856

9335

9414

9499

49

9579
74

9651

9731

9610

9689

9968

0047

0186

0905

9984

450

0363

0449

0591

0600

0678

0757

0636

0915

0994

1073

SI

1159

1930

1309

13d8

1467

1546

1634

1703

1789

1K0

S3

1939

9018

8096

9175

3354

8333

3411

8490

8647

S3

9795

9804

8889

8961

3039

3118

3196

3174

3353

3431

54

9510

3588

3667

3745

3833

3908

3986

4058

4137

4915

555

4993

4371

4450

4538

4606

4684

4763

4840

4919

4997

56

5075

5153

S31

5309

5387

54C5

5543

5631

5699

5777

57

5855

5933

0011

6089

6167

6845

6333

6401

6479

6558

56

6634

6719

6700

6868

6945

7033

7101

7179

7356

7334

59

7419

7490

7587

7645

7733

7800

7878

7955

6033

8111

560

8188

8966

8343

8491

8496

8576

8653

8731

6808

6665

61

8963

9040

9118

9195

9873

9350

9437

9504

95ta9

9659

68

9736

9614

9891

9968

75

0045

0133

0300

0977

0354

0431

63

0506

0585

0663

0740

0817

0894

0971

1048

1185

1989

64

1979

1356

1433

1510

1587

1664

1741

1816

18k5

1679

565

9048

8195

8903

3979

8356

8433

3509

8586

9663

9740

68

9816

2*3

8970

3047

3133

3800

3377

3353

3430

3907

67

3583

3660

3736

3813

3899

3966

4043

4119

41M

4879

68

4348

4495

4501

4578

4654

4731

4807

4883

49C0

69

511?

5189

5365

5341

5417

5494

5570

5646

5789

5799

570

5875

5951

6097

6103

6180

6356

6333

6408

6464

6560

71

6636

6719

6789

6864

6940

7016

7009

7168

7344

7399

79

7396

7479

7548

7634

7709

7776

7851

7937

6603

f979

73

8155

8930

8306

8383

8458

8533

8609

8685

6761

8836

74

6019

8988

9063

9139

9314

9390

9368

9441

9517

9SU

575

9666

9743

9819

9894

9970

76

0045

0131

0196

0979

0947

76

0499

0498

0573

0649

0734

0799

0*75

0950

1635

1100

77

1176

1951

1396

1409

1477

1553

1697

1709

1777

If 53

78

1998

8003

9078

9153

3338

8303

3378

3453

3539

9(04

79

9679

8754

8899

8-03

8979

3953

3138

3803

3878

586

3498

3503

3578

3653

3737

3603

3877

3953

4097

4101

81

4176

4851

4386

4400

4475

4550

4634

4699

4774

4N8

89

4993

4998

5079

5147

5331

5396

5371

5445

5590

5594

83

5669

5743

5817

58P3

5906

6041

6115

61!

6864

6330

84

6413

6487

6569

6638

6710

6785

6659

0933

7067

HK3

585

7158

7930

7304

7378

7453

7537

7001

7675

7749

1f83

86

7898

7979

8046

8130

8194

8968

8343

8416

84*0

6964

87

fi<38

8719

8786

8860

8934

9606

9689

9156

9830

9894

88

9377
77

9451

0535

9599

9673

9747

9830

9694

9966

0049

89

0115

0189

0963

0336

0410

0484

0558

0831

0705

0776

596

0859

0C96

0999

1073

1146

1990

1893

1367

1441

1514

91

1587

1661

1734

1808

1661

1955

8038

9108

3175

9946

99

9399

9395

9468

8549

8615

8688

9768

9835

3*08

9 61

93

9055

3188

3801

3974

3347

3431

3494

3567

3640

3713

94

9786

3869

9933

4006

4979

4159

4371

4444

595

4517

4590

4663

4736

4809

4869

4955

5986

5100

5173

W.

1

S

3

4

5

6

7

8

t

Digitized by VjOOQIC

TABLE I.— LOG. Of NOf.

IS

!f.

1

2

3

4

5

6

7

8

9

596

77 5346

5319

5393

5465

5538

5610

5683

5756

5629

5902

97

5974

6047

6120

611.2

6265

6338

6411

6483

6556

6629

98

6701

6774

6846

6919

6992

7064

7137

7909

72b2

7354

99

7497

7499

7573

7644

m 7717

7789

7862

7934

8007

8079

600

8151

8334

8296

8368

8441

8513

8565

8658

8730

8802

01

8875

8947

9019

9091

9163

9236

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4533

4533

4648

4704

4753

4614

785

4870

4935

4980

5036

5091

5146

5302

5357

5318

5307

8tf

5433

5478

5533

5588

5644

5iW9

5754

5809

53*4

5919

87

5J75

6030

6085

6140

61J5

6251

630J

63J1

6416

6471

88

6534

6581

6636

6693

6747

6802

6857

6913

6*>7

7023

89

7076

7133

7187

7343

7337

7352

7407

7463

7517

7573

790

7637

7682

7737

7798

7847

7903

7957

8013

8067

8139

91

8176

8331

8386

8341

8336

8451

8506

8561

6616

8670

93

8735

8780

8835

8890

8944

89J9

9054

9103

9164

9316

93

9373

9328

9383

9438

9493

9547.

9603

9056

9711

9766

94

9831

9875

9930

9985

90

0039

0094

0149

0803

0858

6313

795

03S7

0433

0476

0531

0586

0640

0995

0749

0804

0859

96

0313

0368

1033

1077

1131

1136

1940

1-295

1343

1464

97

1458

1513

15 *7

1633

1676

1731

1735

1840

1894

1349

98

3003

3057

3113

8166

3331

3375

3339

8334

8438

84 3

99

8547

3601

3655

3710

3764

S818

8873

3337

8381

3934

800

3030

3144

3199

3353

3307

3361

3416

3470

3584

3578

01

3633

3687

3741

3795

3849

3304

3958

4013

40J6

4180

03

4174

4329

4283

4337

4391

4445

4499

4553

4607

4661

03

4716

4770

4834

4878

4933

4986

5040

50 »4

5148

5803

04

5356

5310

5364

5418

5473

5536

5580

5634

5698

5749

805

5796

5850

5904

5358

0012

6066

6130

6173

6887

6381

06

6335

6389

6443

6497

6550

6604

665*

6713

6766

6H99

07

6873

6937

6981

7035

7039

7143

7196

7350

7304

7356

03

7411

7465

7519

7573

7636

7680

7734

7787

7841

7805

09

7949

8003

8056

8110

8103

8817

8871

8334

8378

6431

819

8485

8539

8593

8646

8699

8753

8807

8860

8914

8967

11

9031

9074

9138

9181

9335

9339

9349

9398

9449

9503

13

9553
91

0030

9603

9663

9717

9770

9833

9877

9930

9984

0037

13

0144

0197

0351

235

0358

0411

0464

0518

0571

14

0634

0878

0731

0784

0891

0944

0996

1051

1104

815

1158

1311

1954

1317

1371

1434

1956

1477

1530

1534

1637

16

1690

1743

1797

1850

1903

8003

8063

8116

9169

17

S333

9375

2328

3383

3435

3488

8541

8594

8047

8700

18

3753

3866

3860

8913

8966

3019

3073

3125

3178

3331

19

3384

3337

3390

3443

3496

3549

3008

3655

3706

3761

830

3814

3867

3930

3*173

4036

4079

4133

4184

4837

4890

31

4343

4336

4449

4503

4555

4608

4660

4713

4766

4*19

33

4873

4935

4978

5030

5083

5136

5189

5849

avH

5347

33

5400

5453

5505

5558

5611

5664

5716

5769

5983

*«75

34

5337

5060

6033

6085

6138

6191

6843

6896

6349

6401

N.

1

8

3

4

9

6

7

8

9

t

Digitized by VjOOQIC

TABLE I.—- LOO. Of NOB.

17

N.

1

3

3

4

5

6

7

8

9

885

91 6454

6507

6559

6613

6665

6717

6770

6822

6875

6027

96

6980

7033

70*5

7138

7190

7243

7295

7348

7400

7453

87

7506

7558

7611

7663

7716

7768

7821

7873

7925

7978

98

8030

8083

8135

8188

8340

8293

8345

8397

8450

8502

29

8555

8607

8659

8713

8764

8816

8869

8981

8073

9026

838

9078

9130

9183

9835

9887

9340

9398

9444

9497

9549

31

9601
98

9653

9706

9758

9810

9863

9914

9967

0019

0071

a

0193

0176

0988

0980

0339

0384

0436

0489

0541

0593

33

0645

0697

0749

0801

0853

0906

0958

1010

1062

1114

34

1166

1818

1870

1323

1374

I486

1478

1530

1588

1634

835

1687

1730

1790

1843

1894

1946

1998

8050

8102

2154

36

8306

9858

9310

3368

8414

8466

3518

8570

3ti22

3674

37

8795

8777

2889

8881

8983

8985

3037

3989

3140

3198

38

3944

3906

3348

3400

3451

3503

3555

3607

3658

3710

39

3763

3814

3866

3917

3969

4031

4078

4124

4176

4328

840

4879

4331

4383

4434

4486

4538

4589

4641

4603

4744

41

4796

4848

4899

4951

5003

5054

5106

5157

5209

5361

49

5313

5364

5415

5467

5518

5570

5631

5673

5785

5776

43

5898

5879

5931

5983

6034

6085

6137

6188

6240

63M

44

6343

6394

6445

6497

6548

6600

6651

6703

6754

6805

845

6857

6908

6960

7011

7008

7114

7165

7816

7968

7319

46

7370

7493

7473

7594

7576

7637

7678

7730

7781

7833

47

7883

7935

7986

8037

8089

8140

8101

8843

8293

8345

48

8306

8447

8498

8550

8601

8658

8703

8754

8805

fce57

49

8008

8950

9010

9061

9118

9163

9814

9366

9317

9368

850

0419

9470

9531

9573

9683

9674

9735

9776

9887

9679

51

9930

9981

93

0033

0063

0134

0185

0836

0887

0338

0389

59

0440

0491

0542

0593

0643

0694

0745

0796

0647

0898

53

0949

1000

1051

1108

1153

1803

1354

1305

1356

1407

54

1458

1509

1560

1610

1661

1718

1763

1814

1865

1915

855

1066

9917

9968

3118

8169

8820

3871

3329

8378

8433

56

9474

8595

3575

3636

9677

2727

8778

2fc29

2879

3930

57

8961

3031

3*3

3133

3183

3934

3385

3335

3386

3437

58

' 3487

3538

3580

3630

3690

3740

3791

3843

3662

3943

.59

3093

4044

4004

4145

4195

4846

4296

4347

4397

4448

860

4499

4549

4590

4650

4700

4751

4801

4858

4902

4953

61

5003

5054

5104

5154

5305

5955

5306

5356

5407

5457

69

5507

5558

5608

5658

5709

5759

5810

5860

5910

5961

63

6011

6001

6111

6163

6313

6262

6313

6363

6413

6464

64

6514

6504

6614

6665

6715

6765

6815

6866

6916

6966

865

7016

7066

7116

7167

7217

7267

7317

7367

7418

7468

66

7518

7568

7618

7668

7718

7769

7819

7860

7919

7969

67

8019

8069

6119

8109

8219

8370

8330

8370

8480

8470

68

8530

8570

8620

8K70

8720

8770

8820

8870

8930

6970

69

9090

9070

9120

9170

9820

9870

9320

9370

9419

9469

870

9510

9569

0619

0669

9719

9769

0819

9869

0918

9968

71

94 0018

0068

0118

0168

0218

0267

0317

0367

0417

0467

79

0517

0566

0616

0G66

0716

0765

0bl5

0865

0915

0964

73

1014

1064

1114

1163

1213

1263

1313

1363

1412

1468

74

1511

1561

1611

1661

1710

17C0

1810

1859

1909

1958

875

9008

9058

2107

3157

9206

8356

3306

3355

8405

8455

76

9504

8554

3603

3653

2702

8753

8801

3851

8P01

9950

77

9000

3049

3090

3148

3198

3347

33J7

3346

3396

3445

78

3405

3544

3593

3643

3699

3743

3791

3841

3890

3940

79

3989

4038

4088

4137

4187

4336

4365

4335

4384

4433

880

4483

4539

4581

4631

4680

4739

4779

4888

4877

4987

81

4076

5035

5074

5134

5173

5322

5373

5331

5370

5419

89

5469

5518

5567

5616

5666

5715

5764

5813

5863

5911

83

5961

6010

6050

6108

6157

6207

6256

6305

6354

6403

N.

1

8

3

4

5

6

7

1 8

•

Digitized by VjOOQIC

18

TABLE I.— LOO. OF NOB,

N.

1

3

3

4

5

6

7

8

9

884

94 6453

6501

6551

6600

6649

6698

6747

6796

6845

0894

885

6943

6993

7041

7091

7140

7189

7338

7887

7336

7385

86

7434

7483

7533

7581

7630

7679

7738

7777

7836

7875

87

7934

7973

8083

8071

8119

8168

8317

6366

8315

8364

88

8413

8463

8511

8560

8609

8657

8706

8755

8804

ee53

89

8903

8951

8999

9048

9097

9146

9195

9344

9399

9341

890

9390

9439

9488

9536

9585

9634

9683

9733

9780

9899

91

9878

9936

9975

95

0084

0073

0131

0170

0319

0967

0316

93

0365

0414

0463

0511

0560

0808

0657

0708

0754

0603

93

0853

0900

0949

0097

1046

1095

1143

1193

1340

1389

94

1337

1386

1435

1483

1533

1580

1639

1677

1736

1775

895

1833

1873

1930

1969

3017

3066

3114

3163

8311

33£0

96

3308

3356

3405

8453

3503

3550

3599

3647

3696

3744

97

9793

3841

3889

3938

8986

3034

3063

3161

3180

3330

98

3376

3335

3373

3431

3470

3518

3566

3615

3663

3711

99

3760

3808

3856

3905

3953

4001

4049

4098

4146

4194

900

4343

4391

4339

4387

4436

4484

4533

4580

4638

4677

01

4735

4773

4831

4869

4918

4966

5014

5063

5110

5158

OS

5307

5355

5303

5351

5399

5447

5495

5543

5593

5640

03

5688

5736

5784

5833

5860

5938

5976

6034

6073

6130

04

6168

6316

6364

6313

6361

6409

6457

6505

6553

6801

905

6649

6697

6745

6793

6841

6889

6936

6984

7033

7680

06

7138

7176

7834

7378

7330

7368

7416

7464

7513

7559

07

7607

7655

7703

7751

7799

7847

7895

7943

7960

803B

08

8086

8134

8183

8339

8377

8385

8373

8430

8466

8516

09

8564

8613

8659

8707

8755

8803

8851

8898

8946

8994

910

9041

9069

9137

9185

9833

9880

9338

9375

9433

9471

11
IS

9518
9995

90

9566

9614

9661

9709

9758

9804

9653

9900

9947

0048

0090

0138

0185

0333

0380

0338

0376

0483

13

0471

0518

0566

0814

0661

0709

0756

0604

0851

0899

14

0946

0994

1041

1089

1136

1184

1331

1379

1336

1374

915

1481

1469

1516

1563

1611

1658

1706

1753

1801

1848

16

1896

1943

1990

3038

3085

9133

3180

3337

8375

3333

17

3369

3417

3464

3511

3559

3608

3653

3701

3748

37*5

18

3843

3890

3937

3985

3038

3079

3136

3174

3331

3968

19

3316

3363

3410

3457

3505

3553

3599

3646

3693

3741

990

3788

3835

3883

3939

3977

4034

4671

4118

4165

4313

SI

4860

4307'

4354

4401

4448

4495

4543

4560

4637

4f#4

SS

4731

4778

4835

4873

4919

4966

5013

5061

5106

5155

33

5303

5349

5396

5343

5390

5437

5484

5531

5578

5185

34

5673

5719

5766

5813

5660

5907

5954

6001

6048

€095

985

6143

6189

6936

6383

6339

6376

6433

6470

6517

6564

98

6611

6658

6705

6753

6799

G845

6863

6939

6986

7033

37

WO

7137

7173

7390

7867

7314

7361

7408

7454

7501

38

7548

7595

7648

7688

7735

7783

7839

7875

7993

7969

39

8016

8063

8109

8156

8903

8349

8396

8343

8360

8436

930

8483

8530

8576

8683

8670

8716

8763

8810

6856

6T03

31

8950

8996

9043

9090

9136

9183

9330

9876

9333

9369

39*

9416

9463

9509

9556

9603

9649

9695

,9748

9769

9635

33

9883

9938

9975

97

0091

0068

0114

0161

0307

0854

0300

34

0347

0393

0440

0486

0533

0579

0696

0673

0719

0765

935

0813

0858

0905

0951

0997

1044

1090

1137

1183

1339

36

1376

1393

1369

1415

1461

ism

1554

1601

1647

1ff3

47

1740

1786

1838

1879

K85

ltil

3018

8064

9110

3157

38

3903

3349

9395

3343

3388

3434

3481

3537

3573

8619

39

3666

3713

3758

3804

3851

3697

3943

3989

3035

3083

940

3188

3174

3290

3366

3313

3359

3405

3451

3497

3544

N.

1

3

3

4

5

6

7

6

•

Digitized by VjOOQIC

TABLE I. — LOO. OF NOS

19

N.

9

1

9

3

4

5

6

7

8

9

941

97 3S90

3630

3069

3798

3774

3890

3866

3013

3950

4005

49

4051

4097

4143

4189

493*

4981

4397

4374

4490

4466

43

4919

4558

4004

4650

4000

4749

4788

4834

4880

4996

44

4979

5018

5064

5110

5150

5909

5948

5904

5340

5380

945

9439

9478

5594

5570

5016

5669

5707

5753

5799

5845

46

9B91

5037

5063

6039

6075

6191

0167

6919

6958

6304

47

9330

0390

0449

6498

6533

6579

6671

6717

6763

48

8806

0854

6940

6008

7037

7083

7199

7175

7990

49

7900

7319

7358

7404

7449

7495

7541

7586

7639

7678

990

7794

7709

7815

7861

7900

7959

7068

8043

-8089

8135

91

8180

8990

8979

8317

8363

8409

8454

8500

8546

8591

m

8937

8083

8798

8774

8619

8865

8911

8950

9009

9947

a

9993

9139

9184

9930

9975

9391

9300

9419

9457

9503

54

9948

9594

9939

9085

9739

9776

9891

9667

9919

9958

989

99 0003

0949

0094

0140

6185

0931

0976

6399

0367

0443

90

0498

»

6549

0594

0640

0085

0730

0776

0891

0867

97

0919

1003

1048

1003

1130

1184

1930

1975

1399

98

1389

Mil

1450

1591

1547

1589

1637

1683

1798

1773

99

1819

1854

1909

1954

9000

9045

9000

3135

9181

9990

909

9971

9317

9399

9407

9453

9407

8543

8588

9633

9878

91

9793

9709

9814

9830

9904

9949

9995

3040

3085

3139

99

3179

3990

3965

3311

3359

3401

3446

3491

3530

3581

99

3999

3671

3716

3769

3807

3859

3897

3949

3887

4039

94

4977

4199

4107

4919

4957

4309

4347

4399

4437

4488

909

4597

4579

4017

4699

4707

4799

4797

4849

4887

4939

09

4977

5089

5007

5119

5157

5909

5947

5999

5337

5389

•7

9499

5471

5510

5561

5606

5651

5096

6189

5788

5631

09

9879

5890

5905

6010

6055

6100

6145

6934

6979

09

9394

6309

6413

6458

6503

6548

6593

6637

6089

6797

979

6779

6817

0801

6006

6951

6099

7040

7085

7130

7175

71

7919

7904

7309

7353

7398

7443

7488

7539

7577

7693

79

7699

7711

7756

7800

7845

7800

7934

7979

8094

8008

73

8113

8158

8009

8947

8991

8330

8381

8495

8470

8514

74

8599

8003

8048

8093

8737

8789

8896

8871

8916

8000

979

9009

9949

0094

9138

9183

9997

9979

9316

9361

9405

79

9490

9494

9539

9583

9099

9673

9717

9761

9800

9850

77

9999

9999

9064

99

0098

0079

0117

0161

0906

0950

0994

79

0839

0983

0498

0479

0510

0561

0605

0650

0694

0738

79

0783

0897

0871

0916

0960

1004

1049

1093

1137

1189

909

1995

1979

1315

1359

1403

1448

1499

1530

1581

1695

81

160)

1713

1758

1809

1846

1890

1935

1979

9093

9067

89

9119

9150

9900

9944

8988

9333

9377

9491

9465

9500

83

2554

9598

9649

9080

9730

9774

9819

9863

9007

9951

84

9095

3099

3083

3197

3179

3916

3960

3304

3348

3399

989

3430

3489

3994

9989

9913

3057

3701

3745

3789

3833

89

' 3877

3091

3065

4909

4053

4097

4141

4185

4999

4973

87

4317

4301

4405

4449

4493

4537

4581

4695

4608

4713

88

4797

4801

4845

4809

4933

4977

5091

5005

5108

5158

89

5199

5949

9984

5398

5379

5416

5490

5504

5547

5591

999

9935

9979

9793

9787

9911

5854

5898

5949

5880

6930

91

6974

6118

6191

6905

6949

6993

6337

6380

6494

6408

99

6919

6555

6909

6643

6087

6731

6774

6618

6889

6906

99

6949

0993

7037

7080

7194

7168

7919

7955

7999

7343

94

7389

7499

7474

7517

7581

7605

7649

7099

7730

7779

999

7893

7897

7919

7954

&

8041

8085

8198

8179

8916

99

8999

8347

8390

8477

8591

8999

8606

8699

97

8999

8799

8788

8890

8809

8919

8956

9999

9043

9087

99

9131

9174

9918

9991

9305

9348

9399

9435

9479

9599

99

9009

9999

9096

9739

9783

9899

9870

9913

9957

1899

69 9969

j 0049

0087

0130

0174

6917

0901

0304

0947

- 0391

1*

1 •

1 »

1 '

3

4

5

6

7

9

9
1

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE II.

LOGARITHMS OF SINES, COSEC'S, TANGENTS, &c.

Digitized by VjOOQIC

Digitized by VjOOQIC

Table n.— loo. tunco, tasq% Ac.

23

0»

0*

1'-

.15" !• —

15' 1»-150

170©

11*

Hours.

D»f.

L.Bin.

DUE

for 5"

L.Cotec

L.COS.

L.8ec

L.Tug.

DUE
for 5"

L.Cot.

D«

I-

Boon.

m

*

i

It

i

a

m

•

1

15

5.861666

100343
58697
41646

33303

4.138334

0.000000

0.000000

0461666

100343
58697

41646
38303

4.138334

45

00

53

3

30

6.168696

3437304

00

00

0.168696

8437304

30

58

3

45

.3387b7

.661813

00

00

438787

.661313

15

57

4

1

.463736

436874

00

00

463798

436874

59

58

5

15

.560636

36394
22316
19331
17051
15859

.438364

00

00

460636

86394
88316
19331
17051
15853

.439364

45

55

30

439817

460183

00

00

439817

.360183

30

54

7

45

.706764

493236

00

00

.706764

493836

15

53

8

3

.764756

435844

00

09

.764756

435844

58

58

9

15

415909

.184091

00

00

415900

.184091

45

51

10

30

461666

13798
18596

.138334

00

00

461666

13798
18596
11587
10788
9988

.138334

30

50

11

45

J03059

.096941

00

00

.903059

406941

15

49

12

3

.•40847

.059153

001

00

440847

.059153

57

48

13

15

475609 f*Si

.084391

00

00

475610

.084300

45

47

14

30

7407TC4

xurao
9988

0492306

00

00

7407794

9LQQM06

30

46

15

45

.037757

9343
8776
8375
7837
7485

.968843

60

00

..037758

9343

8776
8875

1Z

468848

15

45

16

4

465788

.934314

00

00

.065786

.934814

56

44

17

15

.093115

407885

00

00

.098115

.907885

45

43

18
19

30
45

.116939
.140430

483061
459580

00
00

00
00

.110939
.140490

JMW

99

15

43

41

90

5

.163696

7063
6735
6435
6161
1910

437304

00

00

.169696

7063

ss

6161
5910

437304

55

40

31

15

.183885

416115

0.000999

01

416114

45

39

33

SO

404089

.795911

99

01

404089

.795911

30

38

33

45

433394

.776606

99

01

483394

.776606

15

37

34

6

441877

.758183

99

01

441878

.758188

54

36

25

15

459606

5678
5464
5865
5010
4909

.740394

99

01

459607

5678
5464
5865
5989
4998

.740393

45

35

36

s

476639

.783361

99

01

.876640

.783360

30

34

37

493030

.708970

99

61

.708970

15

33

28

7

408834

491176

99

01

408835

491175

53

38

39

15

434064

475936

99

01

424065

475935

45

31

30

30

438787

4747
4596
4455
4383
4198

.661813

99

01

.338788

4747
4596
4455
4382

4196

461918

30

30

31

45

453088

.646979

99

01

.353029

.646971

15

39

33

8

466816

.633184

99

01

466817

.633183

58

38

33

15

480180

.619880

99

01

480181

.619819

45

27

34

30

.393145

.606855

99

01

493146

408854

30

86

35

45

.405734

4078
3966

3861
3760
3665

494866

99

01

405735

4078
3966
3861
3760
3665

404865

15

35

36

9

.417968

488038

99

01

417970

488030

51

34

37

15

.439867

470133

98

08

470131

45

33

38

30

.441449

458551

98

08

441451

458549

30

32

39

45

.453730

447870

98

02

.458732

447868

15

31

40

10

.463735

3575
3489
3406
3338
3853

430875

98

02

.463787

3575
3489
3406

333E
3853

436373

59

80

41

15

.474449

485561

98

08

.474451

435549

45

19

43

30

.484915

415085

98

03

.484917

415083

30

18

43

45

.495134

404866

98

02

.495136

404864

15

17

44

11

405118

,494831

98

08

405180

.494880

49

16

45

15

414878

3183
3113
3048
8985
8985

.485183

98

02

414880

3188
3113
3048
8985
8935

485180

45

•

15

46

30

434433

475577

98

02

484485

475575

30

14

S

45

433763

.466837

97

03

433766

.466234

15

13

13

443900

457094

97

03

448909

457091

48

18

49

15

451861

448139

97

03

451864

448136

45

11

50

30

460635

3867
3811
8758
8708
8656

.439365

97

03

460638

8887
8811
8758
8706
8656

.439368

30

10

51
53

13

45

469335
477668

.430765
.482338

97
97

03
03

469838

477671

430768

47

15

9

6

53

15

485941

414059

97

03

485944

414056

45

7

54

30

494059

.405941

9?

03

494063

.405938

30

6

55

45

.609038

8608

8568
8510
8475
8433

497979

97

03

,.608031

8608
8568
8518
8475
8433

497969

15

5

56

14

40P853

499147

96

04

.609857

490143

4«

4

57

15

417540

488460

96

04

.617544

468456

45

3

58

3°.

435093

474907

96

04

.685097

474903

30

3

59

4ft

433517

307483

98

04

.63858]

467479

15

1

60

15

7.939816

0469184

0.000096

O.00OO04

7.639820

83460180

45

00

Hoi

•
ire.

' i"

D6f.

L.0M.

Diff.
for 5"

L.8ec

| I* St*.

LCowe.

L. Got

Diir.

for 5"

L.Tmnf.

Bf.

m

Hon

*
r§.

*»

=

MP

l'-4«

lo«4*

00*

>

II*

Digitized by VjOOQIC

24

0» OO

tabue n.— log. bines, tang's, Ac*

l'-M" I-—]*' 1»— lfiO

17VO 11*

Hou

it.

De

i

I-

L.Bin.

Diff
for 5"

L.Ooiec.

L.CM.

L.S6C.

L.Tuf.

DUE

for 5"

L.Obt.

IX

*.

M

- 1-

1

15

7339816

8393
8354
8316
8380
8344

9360184

9.909096

O.OOOO04

7.639890

8393
8354
8316
8880
8844

9.360180

45

M ,*»

1

15

J646995

353005

96

04

.646999

353001

45

59

8

30

.654056

345944

96

04

.654061

345939

90

* 1

3

45

.661005

338995

96

04

361010

338990

15

57

4

16

.667845

338155

95

05

367849

333151

44

"I

5

15

.674578

8310
8177
3145
3113
8083

.385433

95

05

.674583

8810
8177
3145
3113
8083

385417

45

53

6

30

.661908

318793

95

05

.681913

318787

30

7

45

.687739

318861

95

05

387744

318856

15

8

17

.694173

.305887

95

05

.694178

305882

43

a2

9

15

.700513

399487

95

05

.700519

399481

45

51

10

30

.708763

8054
8085
1997
1970
1943

393838

94

06

.700768

8054
8035
1997
1970
1943

393838

30

50

11

45

.713933

387078

94

08

.718988

387078

15

49

IS

18

.718097

.881003

94

06

.719003

380997

48

4«

13

15

.734987

375013

94

06

.784993

375007

45

47

14

30

.730896

369104

94

06

.730903

369098

30

46

15

16

19

45

.736735

.743477

1917
1898
1868
1844
1821

363375
357523

94
93

06
07

.736738
.743484

1917
1898
1868
1844
1831

363866
357516

41

15

■ts

44

17

15

.748155

351645

93

07

.748161

351839

45

43

18

30

.753758

346843

93

07

.753765

346835

30

4?

19

45

.759391

340709

93

07

.759398

340703

15

41

90

30

.764754

1798
1776
1755
1734
1713

.835346

93

07

.764761

1798
1776
1755
1734
1713

335839

40

40

21

15

.770149

.839851

93

07

.770156

389844

45

39

39

30

.775477

334583

93

08

.775485

384515

30

3*

S3

45

.780743

319858

93

08

.780750

319850

15

37

94

31

.785943

314057

92

08

.785951

314049

39

45

36

25

15

.791083

1693
1674
1654
1636
1618

308918

92

08

.791090

1693
1674
1654
1636
1618

308910

35

36

30

.796162

.803838

93

08

.796170

303630

30

34

27

45

.801183

.198817

92

08

301191

.198809

15

33

38

39

.808146

.193854

91

09

.806155

.193845

38

39

39

15

.811053

.168947

91

09

311068

.188638

45
30

31

30

30

.815906

1599
1583
1565
1548
1533

.184094

91

09

315915

1599
1588
1565
1548
1533

.164085

30

31

45

.820704

.179396

91

09

330714

.179886

15

39

39

33

.825451

.174549

90

10

335460

.174540

37

2*

33

15

.830146

.169854

90

10

330156

.169844

45

87

34

30

.834791

.165309

90

10

.834801

.1C5199

30
15

86

35

45

.839386

1516
1500
1485
1470
1455

.160614

89

11

339397

1516
1500
1485
1470
1455

.160603

35

36

34

.843934

.156066

88

11

343944

.156056

36

84

37

15

348434

.151566

89

11

348445

.151555

45

23

38

30

.858889

.147111

89

11

358900

.147100

30

83

39

45

.857398

.148703

89

11

.857309

.148691

15

21

40

35

.861663

1441
1436
1413
1399
1385

.138338

89

11

361674

1441
I486
1418
1399
1385

.138386

35

90

41

15

365964

.134016

89

11

365995

.134005

45

19

42

30

£70368

.189738

88

12

370874

.135786

30

18

43

45

.874490

185501

88

18

.874511

.185489

15

17

44

36

.878695

* .121305

88

IS

37870S

.181398

34

16

45

15

382851

1373
1359
1346
1334
1333

.117149

87

13

389864

1378
1359
1346
1334
1338

.117136

45

15

46

30

.886968

.113032

87

13

.886P81

.113019

30

14

47

45

.891046

.108954

87

13

301059

.108941

15

13

48

37

395085

.104915

87

13

395099

.104001

33

ll 1

49

15

399088

.100912

87

13

399103

.100808

45

50

30

.903054

1310
1398
1387
1375
1364

.096946

86

14

.903068

1310
1398
1887
1875
1364

.096938

39

10

51

45

.908984

.093016

86

14

308098

.093008

15

9

53

38

.910879

.089121

86

14

.910894

.088101

38

8 l

53

15

.914740

.085360

86

14

314754

385346

45

7 i

54

30

.918566

.081434

85

15

.918581

.081419

30

6

55

45

.93335?

1853
1848
1338
1383
1811

.077641

85

15

.983374

1853
1348
1838
1838
1811

.077686

15

5

56

39

.oseiiB

373881

85

15

.996134

.073866

31

4

57

15

.930847

.070153

85

15

.989863

.070137

45

3

58

30

.933543

.066457

84

16

333559

.006441

30

8

59

45

.937308

.063798

84

16

.937884

.008776

15

1

1

60

30

7.940848

9.059158

0.090084

000016

7.940858

9.050148

39

08

m

I -

'

//

T\i^

L.Bac

L.Bin.

L. Gowc

L.Oot.

Diff.
for 5"

L. TftDf.

'

if

•

Hoan.

Dog.

L- 00 * IfwJ'

D6f.

Horn.

_

6*

— 1

H>o

1'— 4»

IC-.4"

1

100

&•

Digitized by VjOOQIC

TABLE II. — LOtL «N£0, TANG't, <fcc.

35

!•— 15" 1-— 15' 1* — IS©

1W>

11*

Houn

o

i

3

4

5

6
7

I

10
M
i:
U

U

.15
1 •
17
U
1J

a i
11

19

21

m

25
- "'

*3

•j-

S.i

30
31

32
:y
M

38
35
37
'«-

:n

.in
u
ii

!'
II

45

-in

■IT

40

r.«i

.54

53

.-»■!

.':
>

Dix

IS

JO
45

15
3(1
45

J+. Bin.

m

M

45

L5
,.W

15

I:".
HI

IS

15
311

15

15
»

15

I."

15
30
15

L5
45

15
TO
15

45

15

10

45

I.".
1 5

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13(14

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12*1

Biowa

75

35

.3^02it3

1313
1306

121>7
J 280
12.+2

.619737

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73

87

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70

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.38549*

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30

,3§frtR!3

,611908

te

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.009330

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1251
1244

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1200
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.596602

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1237
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1223
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.4II.V3-

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1331
1224
1217
1210

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0.090Nol

0OOO149

8.41*04^

1^1938

M)

541

u

i

•

■

It

L.COR.

DtlT
for
15"

Ufa

L- Sin.

L. Ctwc.

UCot.

Din:

Oir
15"

LTaD(t

1

■'

-

1

ttouri

Peg.

o*r

Haiti*. 1

orl*

91 Y

G*

V

IV

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Digitized by VjOOQIC

TABLE n

suret, tang's, &e.

.15" !•— 15' 1* — 150

1TSO 11»

Dog.

L.Bin.

Diff.
for
15"
orl*

L.Coeec.

L. Co*.

L.8ec.

L. Tang.

Diffi

for
15"
orl«

L.CM.

Deg.

30
30

30
30
30

30
30

30
3D
30

30
30

30
30
30

30
30

30

30

30

i
30

30

►

30

\

30
I

30

5

30
5

30

7

30

8
30

8.417910
.4520335
.42*717
.4250 Mi
.427402

.420616
.432156
.4344*4
.430800
.439103

.441394
.443074
.445941
.448190
.450440

.452672
.454803
.457103
.459301
.461489

.463665
.465830
467985
.470129
.472263

•474386

.476498
.478601
.480(593
.482775

.484848
.486910
.488003
.491001;
.493040

.495064
.497079
.499084
.501080
.503067

.505045
.507014
.506974
51025
.512807

.514801
516796
.518643
.520551
.522451

.524343

.52622**.
.528102
5291:69
.531828

53367T

537?5<

53!'18f

.54U>07

8.542819

Deg.
9l5~

1203
11%
1JW*

1183
1176

1170
1164
1158
1151
1145

1140
1133
1127
1122
1110

1110
1105
1099
1094
1088

1082
1077
1072
1067
1062

1056
1051
1046
1041
1030

1031
1020
1021
1017
1012

1007
1003
998
993

989

984
980
075
971
967

902
958
954
950
94(i

041
938
933

92!>
925

922

918
913
910
90C

1.582081
.570675
.577283
.574J04
.5T253*

570184
.507844
•50551t>
.503200

5606y*

.558600
.556320
.554059
.551804
.549500

.547328
.545107
.542807
.540699
.538511

.536335
.534170
.532015
.5 29871
.527737

.525614

Diff.

I. Co* |£, r ,

orl«

9.999851

50
48
46
44

43

41
39

.52131*
.51930'
.517225

.515152
.513090
.511037
.508994
.505960

.504936
.502921
5009U!
.498020
.496933

.494955
499U8t
49J02<
489075
.487133

.485199
.483274
.481357

.47044H
.477549

.475657
.473774
.471808
.470031
.468172

.466321
.464477
.462641
.400814
.458993
1.45718J

Ii* BCC

34
33
31

99
27

25

as

22
20

18

16
14
13
11
09

07
05
03
01
.9997*)

97
95
93
92
M

88
86
84
82
80

78
76
74
72

57
55

53
51

48

46
44
42
40
38
9.999735

0.099149

50
52
54

56

57
59
61
62
64

06
67
69
71
73

75
77
78
80
82

84
86
87
89
91

93
95
97
99
.000901

03
05
07
08

10

8.418068
.420475

422e09

L.Bin.

33
35
37
39
41

43
45

47
49

54
56
68

60

62

0*000965

L. Coaec.

.427618

.499973
.432315
.434645
.430902
.430267

.441560
.443841
.446110
.44830'
.450613

.459847
.455070
.457281
.459481
.461671

.463849
.466010
.468172
.470318
.472454

.474579
.476693
.478798
.480892
.482976

.485051
.487115
.480170
.491214
.493250

.495276
.497203
.499300
•501298

.305267
.507238
.509200
.511153
.513096

.515034
.516961
.518880
.52O7S0

.524586

.526471
.528349
.530218
.532080

.533033
.535779
.537616
.539447
.541209
8.543084

L.Cot.

1904
1197
1190
1184
1177

1171
1105
1150
1152
1146

1141
1134
1198
1123
1117

1111
1100
1100
1005
1089

1083
1078
1073
1068
1063

1057
1052
1047
1042
1037

1032
1027
1022
1018
•1013

1008
1004
999

994
990

985
981
976
972
968

963
959
955
951
947

942
930
934
930
996

923

919

914

91

907

1.581922
57T525
.577131
.574750
•5723*2

53

Diff.
for
15"
orl

30
30

.570087
.567*85
.565355
.563038
580733

558440
.55815ii
553810
551033
549387

547153
.544930|22
542711
54051!

30
30

30
30

536151
533984
531828

599682
527544,

523307
521902
51910b
517024

514949
512885
510830

508786
506750

504794
502707

.498702
.496713

.494733
4927€2
490800
488847

.484966
483039

481120
479210
477308

475414
473599
471651
409782
467990

466067
464221
402384
4T0554
458731
1.456916

30
30

30
30
30

30
30

30
30
30

30
30

30
30
30

ss

5*

56
54

50

48
46
44
42

40
38
36
34
32

26
24
22

18
16
14
12

10

6
6

4
2

58
56
54
52

L. Tang.

30
30

30
30
30

59

48
46
44
42

40

38
36
34
39

30

28

22

18
16
14
12

10

8

6
4
2

•

Deg.

Boa

1'— 4- lo— 4-

M6 H

Digitized by VjOOQIC

TABLE II.— LOO. SINKS* TAMO's, &C.

Ok 90

!• — IS" 1- - IS' 1* - 150

1TTO

Dirf

[>itr

9t<Hin>

i**.

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1.6ee.

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for
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0* fto

1" —

15" 1« —

15' 1»«* 15©

177°

11*

1

Diffi

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1

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. 8970

1030

. 6704

4

32J6

. 2206

. 7734

57

4-

4

. ft)J0

. 0010

. 6688

4
4

. 3312

. 3302

6696

56

I 44

5

.091008

251
253
252
252
251

.908993

6673

. 3327

. 4335

358
357
350
353
355

. 5665

55

L

6

. 2024

. 7976

. 6657

4
4

. 3343

. 5307

. 4633

54

'3

7

3037

, 6903

. 6641

. 3359

. 6390

. 3u04

53

it

8

. 4047

. 5953

. 6625

4

. 3375

. 7422

. 2570

52

jr

9

. 5050

4944

. 6610

4
4

. 33J0

. 8446

1554

51

1

10

. 60(12

251
250
250
244
248

. 3938

. 6594

. 3406

. 9468

355
354
354
353
352

. 0533

50

"J»

11

. 7005

. 3935

. 6578

4

. 3422

.100487

.89.4513

49

It

12

80J6

. 1934

. 6563

4

. 3438

1504

. 84'JO

48

_ 12

13

. 9005

. 0335

. 6546

4

. 3454

351*

. 74H1

47

«•

14

.100002

.89J938

. 6530

4
4

. 3470

. 3532

. 6468

46

«

15

. 1056

248
247
247
246
245

. 8944

. 6514

4

. 3486

. 4542

852
251
851

350
350

. 5458

45

31

15

. 3048

. 7953

. 6498

. 3502

. 5550

. 4450

44

i>

17

. 3037

. 6963

. 6481

4
4
4
4

. 3519

. 6556

3444

43

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18
19

. 4025
. 5010

. 5975
. 4990

. 6466
. 6449

. 3534
. 3551

. 7559
. 8501

. 3441
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43
41

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44

1

20

. 5992

245
244
244
243
243

. 4008

. 6433

4

4
4
4
4

. 3567

. 9550

349
349
348
347
347

. 0441

40

40

21

. 6973

. 3027

. 6417

. 3583

.110556

.889444

39

j.

22
23

. 7951
. 8927

. 3049
. 1073

. 6400
. 6384

. 3600
. 3616

. 1551
. 3543

. 8443
. 7457

38
37

j^

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24

. 9901

. 0099

. 6368

. 3633

. 3533

. 6467

36

24

25

.110872

342
242

241
241
340

.889138

. 6351

4

4
4
4
4

. 3649

. 4531

346
346
345
845
844

. 5479

35

20

26
27

2d

. 1842

880J

. 3774

. 8158
. 71J1
. 6220

. 6335
. 6318
. 6302

. 3665

3683

. 5507
. 6491

. 7473

. 44J3

. 3500

8528

34
33
38

lo '
12

24

. 4737

. 5203

. 6885

'. 3715

. 8452

. 1548

31

4

30

. 5698

339
339
339
338
337

. 4303

. 6369

4

4
4
4
4

. 3731

. 9489

344

343
343
342
342

0571

30

SO

31

. 6650

. 3344

. 6252

. 3748

.190404

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29

5.

32

. 7612

. 8388

. 6235

. 3765

. 1377

8633

28

j£

33

. 8507

. 1433

. 6219

. 3781

. 3348

7652

87

4f

34

. 9519

. 0481

. 6302

. 3798

. 3317

. 6683

36

M (

35

.1310469

837
836
336
335
335

.879531

. 6185

4
4
4
4

4

. 3815

. 4384

841
840
340
339
839

. 5716

85

40

30

1417

. 8583

. 6168

. 3832

. 5349

4751

34

*> '

37

. 2362

. 7638

. 6151

. 3849

. 6311

378.1

23

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38
34

. 3306
. 4248

. 6694
. 5753

. 6134
. 6118

. 3806
. 3882

. 7172
. 8130

. 3828
. 1870

28
21

24

40
41

. 5187
. 6125

5*
334
833
333
832

. 4813
. 3875

. 6100
. 6084

4

4

4
4

4

. 3900

. 3916

. 0087
.130041

838
838
837
837
336

. 0913
.86945

20

19

30 '
In .

42
43
44

. 7030
. 7993
. 8925

. 3940
. 3007
. 1075

. 6066
. 6049
. 6032

. 3934

. 3951
. 3988

. 0994
. 1944
. 88J3

. 9606
. 805i
. 7107

18
17
16

u :

B
4

45

. 9854

832

331
331
330
330

. 0146

. 6015

4
4
4

4
4

. 3985

. 3839

336
8J6
335
834
834

6161

15

99

40

.130741

.863219

. 5998

. 4002

. 4783

5317

14

*»

47

. 170$

. 83M

. 5980

. 4020

. 5730

4274

13

52

48

. 2.130

. 7370

. 5983

. 4037

. 0667

3333

13

4"*

49

. 3551

. 6449

. 5946

. 4054

. 7605

. 3395

11

44

50

. 4470

839
329

828
828
827

. 5530

5988

4
4
4
4

4

. 4073

. 8543

833
833
833
833
833

1458

10

40

51

. 5387

. 4613

. 5911

. 4089

. 9476

. 0524

9

3.1

52

. 6303

. 3697

. 5894

. 4106

.140409

.8535)1

8

32

53

. 7216

. 3784

. 5876

. 4134

. 1340

6660

7

**

54

. 8128

. 1872

. 5859

. 4141

. 8369

. 7731

6

24

55

. M37

827
826
326
325
835

. 0963

. 5841

4
4
4
4
4

. 4159

. 3196

331
331

£2
330

839

. 6804

5

20

56

. 9945

. 0055

. 5824

. 4176

. 4121

. 5879

4

10

57

.140850

.859150

. 5806

. 4194

. 5044

4956

3

12

58

1754

. 8246

. 5788

. 4312

. 5966

40fc

8

5J

. 2050

. 7344

. 5771

. 437)

. 6885

. 31 IS

I

4

60

9.143555

0.856445

9.995753

0.001347

9.147802

O.S52I *

as

•

*

Diff
for

Diff
for

for
15"

'

■

J7"

L.GM.

15"

L.See.

L.Bin.

15"

L-Oomc

L-Cot.

L.TAinf

Oef.

Hoot

orl'

orl'

orl"

*

r>

1' -

-4* lo

— 4»

•

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r

Digitized by VjOOQIC

TABLE II. — LOO. SINES, TANg's, &C.

Oft So

P — 15" 1-—15'

1»-150

171

1

Diff.

Diff.

"MTi

1

Hours.

Deg.

L.Sin.

for

15"
orl»

L.COMC

L.Coi.

orl«

L.Sec

L. Tang.

for
15"
orl*

L. Cot. i

Deg.

-

•

/

'

3*

0.143555

224
224
223
223
222

0.856445

0.005753

0.004247

0.147*02

229

0.85215 8

00

4

1

4453

. 5547

. 5735

. 42c5

. 8718

. 12*2

59

8

2

. 534u

. 4051

. 5717

. 4283

. 9032

228

. 03U8

5*

12

3

. 6243

. 3757

. 5699

. 4301

.150544

228
227
227

.849450

57

10

4

. 7130

. 2864

. 5682

. 4318

. 1454

. 8540

56

30

5

. 8026

222
222
221
221
220

1974

. 5663

. 4337

. 2363

220
226
220
225
225

. 7637

55

24

6

. 8915

1085

. 5646

. 4354

3269

. 6731

54

26

7

9H>2

. 0198

. 5628

. 4372

4174

. #20

53

32

8

.15068L

.849314

. 5609

. 4391

. 5077

. 4923

52

3b

9

. 1569

. 8431

. 5591

. 4409

. 5978

4022

51

40

10

. 2451

220
219
219
2J8
218

. 7549

. 5574

. 4426

. 6877

224

224
223
223
222

. 3123

50

44

11

3330

. 6670

. 5555

. 4445

. 7775

. 2225

49

48

12

. 420b

. 5792

. 5537

. 4403

. 8671

. 132 •

48

52

13

. 5084

. 4910

. 5519

. 4481

. 95(5

0435

47

56

14

. 5957

. 4043

. 5500

. 4500

.100457

.839543

40

33

0'

15

. 6829

218
217
210
216
216

. 3171

. 5482

. 4518

1347

222
222
221

. 8653

45

4

10

. 7700

. 2300

. 5464

. 4530

. 2230

. 7764

44

6
12

17
18

. 8509
. 9435

1431
. 0565

. 5446
5427

. 4554
. 4573

. 3123
. 4008

. 6877
. 5992

43

42

16

19

.100301

.839699

. 5409

. 4591

. 4tti2

221
220

. 5108

41

20

90

. 1164

215
215
214
214
213

. 8836

. 5390

. 4610

. 5774

220

219
219
210
218

. 4226

40

24

28

21
32

2026
. 2885

. 7974
. 7115

5372
. 5353

. 4628
. 4647

. 6654
. 7532

. 3340

. 2408

39
38

32
30

23
24

. 3743
. 4600

. 6257
. 5400

. 5334

. 5310

. 4666
. 4684

. 8409
. 9284

. 15!)]
. 0710

37
30

40

25

. 5454

213
213
212
212
211

. 4546

. 5297

4
5
5
5
5

. 4703

.170157

218
217
217
217
210

.889843

35

44

20

6307

. 3693

. 5278

. 4722

1029

. 8971

34

48

27

. 7159

. 2841

5200

. 4740

. 1899

. 8101

33

52

28

800c

1992

. 5241

. 475.

. 2767

. 7233

33

50

29

8850

. 1144

. 5222

. 4778

. 3634

. 6366

31

34

30

. 9702

211
211
210
210
20:i

. 0298

5203

5
5
5
5
5

. 4797

. 4499

210
215
215
215
214

. 5501

30

4

31

.170546

.899454

5184

. 481(

. 5362

. 4038

29

8
12

33
33

. 1389
. 2230

. 8611
. 7770

. 5105
. 5146

. 4835
. 4854

. 6224
7084

. 3770
. 2»10

28
27

10

34

. 3070

. 6930

. 5127

. 4873

. 7943

. 2057

20

20

35

. 3908

209
208
208
208
207

. 6092

. 5108

5
5
5
5
5

. 4892

. 8800

214

213
213
213
212

. 1200

25

24

30

. 4744

5256

5089

. 4911

. 9655

. 0345

24

28
32
30

37
38
39

. 5578
. 6411
. 7243

. 4422
. 3589
. 2757

. 5070
. 5051
. 5032

. 4930
. 4949

. 4968

.180508
. 1300
. 2211

.819402
. 8040

. 7789

23
22
21

40

40

. 8072

207
207
206
206
205

. 1928

. 5013

5
5
5
5
5

. 4987

. 3059

212
211
211
210
210

. 6941

20

44

41

8900

1100

4993

. 5007

. 3907

. 60!)3

19

48

42

. 9727

. 0273

.. 4974

. 5020

. 4757

. 5247

18

52

43

.180551

.819449

. 4954

. 5046

. 5597

. 4403 17

50

44

. 1374

. 802;

. 4935

. 5065

. 643i>

. 3501

16

33

45

. 2196

205
204

204
204
203

. 7804

. 4910

5
5
5
5
5

. 5084

. 7280

210
20(1
20.)
20'
208

. 2720

15

4

46

. 3016

. 6984

. 4890

. 5104

. 8120

. 18K)

14

8

47

. 3834

. 6166

. 4877

. 5123

. 8057

. 1043

13

12

48

. 4051

. 5349

. 4857

. 5143

. 97P4

. 0200

12

10

49

. 5467

. 4533

. 4838

. 5162

.100029

.803371

11

30

50

. 6280

203
203
202
202
201

. 3720

. 4818

5
5
5
5
5

. 5182

. 1462

208
207
207
207
206

. 8538

10

24 51

. 70i»2

. 2908

. 4798

. 5202

. 2234

. 7700

9

'26 ! 52

. 7903

. 20»7

. 4779

. 5221

. 3124

. r*7i;

8

32 53
36 j 54

. 87J2
. 9519

. 1288
. 0481

. 4759
. 4739

. 5241
. 5261

. 3953
. 4780

. f047
. 5220

7
G

40 ' 55

.100325

201
201
200

2W>
It*

.809675

. 4719

5
5
5
5
5

. 5281

. 5606

200
200
205
205
204

. 4304

5

44 5tf

. 1130

. 8870

. 4700

5100

. 6430

. 3.570

4

48 57

. 1933

8007

4680

. 5«0

7253

. 2747

3

52 58

. 2734

. 7206

4660

. 5V(

. 8071

. V. 2f5

50 50

. 35^4

. 6406

. 4040

. 5300

. 8894

1105 1

35 00 (»

1043:i2

0.805608

0994620

O.OO5380

0.103712

o.^oo.^!

•t 1 • f

Diff.

Diff

Diff.

1 '

Lours. Deg.

L. Co*.

for
15"
orl«

L. Sec.

L. Sin.

for
15"
orl«

L.Cotec.

L. Cot.

for
15"
orl*

*-*■■* -5*7

O*

08

V —

*• IO-

r4"»

81

I

Digitized by VjOOQIC

TABLE 11. — LOO. 8INB8, TANo's, <fcc.

90

V

-15" 1--

-15' 1» — 150

170O

11*

bfifc

Dilf.

Dift

eg-

L-Sin.

for
15"
orl'

L-Cosec

L.Coi.

for
15"
orl*

L,Sec

L.Tang.

for
15"
orl«

L-Cot.

Dec.

Ho«n.

9.194332

199
199
198
198
198

0.805668

0.994620

5
5
5
5
5

0.005380

9.19i712

204
204
203
203

203

O.8O0288

60

23 6'

1

. 512s)

. 4871

. 4600

. 5400

.9O052J

.70J471

59

5

2

. 5925

. 4075

. 4580

. 5420

1345

. &J55

58

£_

3

. 6719

. 34*1

. 4560

. 5440

. 215J

. 7em

57

4*

4

. 7511

. 2489

. 4540

. 5460

. 2971

. 70*>

56

4*

5

. 8302

197
197
197
196
196

. 1698

. 4519

5
5
5
5
5

. 5481

3783

302
202
202

201
201

. 6217

55

4"

6

. 9091

. 0909

4499

. 5501

. 4592

. 5408

54

j*

7

. 9879

. 0121

4479

. 5521

. 5400

. 4600

53

32

8

.800666

.70J334

. 4459

. 5541

. 6207

. 3793

58

2*

9

. 1451

. 8549

. 4438

. 5562

. 7013

. 2*7

51

h

10

. 2235

195
195
195
194

194

. 7765

. 4416

5
5
5
5
5

. 5582

. 7817

900

. 2183

50

2P

11

. 3017

. 6983

. 4398

. 5602

. 8619

200
200
19J
199

13H1

4'J

K

12

. 3797

. 6203

. 4377

. 5623

. 9420

054)

48

12

13

. 4577

. 5423

. 4357

. 5643

.910220

.789?oO

47

14

. 5354

. 4646

. 4336

. 5664

. 1018

. 8*£

48

4

15

. 6131

194
193
193
192
192

. 3869

. 4316

5
5
5
5
5

. 5684

. 1815

199

198
198
198
197

. P185

45

S3

16

. 6906

. 3094

. 4295

. 5705

. 2611

. 73**

44

*

17

. 7679

. 2321

. 4274

. 5726

. 3405

. 65J5

43

iJ

L8

. 8452

1548

. 4254

5746

. 4198

. 5802

42

4.- "

19

. 9222

. 0778

. 4233

. 5767

. 4989

. 5011

41

44

»

. 9991

192
191
191
191
191

. 0009

. 4212

5

1

5788

5779

197
197
196

. 4221

40

40

21

.910760

.789240

. 4192

. 5808

. 6568

. 3432

30

3*

22

. 1526

. 8474

. 4170

. 5830

. 7356

. 9644

38

32

23

. 2202

. 7708

. 4150

. 5850

. 8143

196
196

. 1858

37

2*

24

. 3055

. 6945

. 4129

. 5871

. 8926

. 1074

36

24

25

. 3818

190
190
190
189
189

. 6182

. 4108

5

5
5
5

. 5892

. 9710

195
195
195
194
194

. 0290

35

«*»

26

. 4579

. 5421

. 4087

. 5913

.830492

.779508

34

lr>

27

. 5338

. 4662

. 4066

. 5934

1272

. 8728

33

12

28

. 6097

. 3903

. 4045

. 5955

. 2052

. 7948

32

«.

23

. 6854

. 3146

. 4024

. 5976

. 2830

7170

31

4

30

. 7609

189
188
186
187
187

. 2391

. 4003

5
5
5
5
5

. 5997

. 3600

194
193
193
193
193

. 6394

33

a*

31

. 8364

1636

. 3982

. 6018

. 4383

. 5618

29

5i

32

. 9116

. 0884

. 3960

. 6040

. 5156

. 4844

28

52 .

33

. 9868

. 0132

. 3939

6061

. 5929

. 4071

27

4*

14

.3*0618

.779382

. 3918

. 0082

. 6700

3300

26

,44 .

35

. 1367

187
186
186
186
185

. 8633

. 3896

5
5
5
5
5

. 6104

. 7471

193
192
192
191
191

. 2539

25

40

»

. 2115

. 7885

. 3B75

. 6125

. 8240

1760

31

3tf

37

. 2861

. 7139

. 3854

. 6146

. 9007

. 0393

23

32

38

. 3606

. 6394

3832

. 6168

. 9774

. 0226

22

2*

39

. 4350

. 5650

. 3811

. 6189

.330539

.769461

21

24

10

. 5092

185
185
185
184
184

. 4908

3789

5
5

5

. 6211

. 1303

190
190

. 8697

93

20

11

. 5833

. 4167

. 3768

. 6232

. 2065

. 7935

19

16

12

. 6572

. 3428

. 3746

. 0254

. 9836

190
190

. 7174

18

12

13

. 7311

. 2689

. 3725

. 6275

. 3586

. 6414

17

8

14

. 8048

. 1952

3703

. 0397

. 4345

189

. 5655

16

4
1

15

. 8784

184
183
183
182
182

. 1216

. 3681

5

5
5
5

. 6319

. 5103

183
189
188
188

. 4897

15

»1

16

. 9519

. 0481

. 3660

. 6340

. 5859

. 4141

14

5*

17

.330252

.769748

. 3638

. 6362

. 6614

3386

13

52

t8

. 0984

. 9016

. 3616

. 6384

7368

9633

19

4«

19

. 1714

. 8286

. 3594

. 6406

. 8190

188

. 1880

11

44

50

. 9444

182
182

181
181
181

. 7556

. 3572

1

5
6
5

. 6428

8872

187
187
187
186
186

. 1198

10

!<o

51

. 3172

. 6828

. 3550

. 6450

. 9692

. 0378

9

36

52

3899

. 6101

. 3528

. 6472

.340371

.759629

8

32

53

. 4625

. 5375

. 3508

. 6494

1119

. 8881

7

29

54

. 5349

. 4651

. 3484

• 6516

. 1865

. 6135

\ U

55

. 6072

180
180
180
179

179

. 3998

. 3462

5
5
5
5

6

. 6538

. 9610

186
186
185
185
185

7390

5

1
30

56

. 6794

. 3206

. 3440

. 6560

. 3354

. 6646

4

16

57

. 7515

. 2485

. 3418

. 6582

. 4037

. 5903

3

12

58

. 8235

. 1765

. 3398

. 6004

. 4839

. 5161

9

i *

59

. 8953

1047

3374

6626

. 5579

. 4431

1

1 4

UO

0.939670

0.760330

0.003351

0.006649

0.346319

0.753681

so

'

Diff

for

-Dilf
for

Diff
for

' l • '

teg.

L.C<m.

15"

L.See.

L. Sin.

15"

L.COMC

L.C0L

15"

LTatif.

De«.

Hours.

orl'

orP

orl*

1

I

»90

r-

-4' I©

— 4«

80/

»

5*

Digitized by VjOOQIC

TABLE II.— LOG. SINKS, TANo's, &C.

41

0*

10O

l» — 13" 1-

-15'

1» — IS©

160© '

11»

UiC

Did:

JJilt

Hoar*

Deg.

L.0U.

for
J5"
orl-

L.Oo«e.

L.CM.

for
15"
orl'

L.8K.

L.Tftaf.

for
15"
orl*

L.Cot.

Peg.

/

Hour*.

•

•

i

«

•

40

O.939670

179

179
178
178
178

O.760330

0.003351

5

5
5

003049

0.046319

184

0.753081

00

10

00

4

1

.940380

.759014

. 332J

. 0071

. 7057

184

. 2143

59

50

H

2

. 1101

. 88J9

. 3307

. 0693

. 7794

184

. 2203

53

52

14

3

. 1814

. 8180

. 3264

. 0710

. 8530

183

1470

57

48

16

4

9590

. 7474

. 3202

. 0738

. 9304

183

. 0730

53

44

M

5

. 3338

177
177
177
170
170

6703

. 3340

. 0700

. 9998

183

. 0002

53

40

21

6

3947

0033

. 3217

. 0783

.960730

1H3

182

.74>270

54

30

.N

7

. 4656

. 5344

. 3195

0805

. 1461

. 8539

53

32

Si

8

. 5303

. 4037

. 3172

. 0828

. 2191

182

. 730 •

52

28

33

9

. 6009

. 3931

. 3149

. 0851

. 3930

132

7080

51

24

40

10

. 6775

170
170
175
175
175

. 3325

. 3127

. 6873

. 3648

181
181
181
181
180

. 6352

50

20

41

11

. 7473

. 3522

. 3104

. 6890

. 4374

. 5626

49

10

48

12

. 8181

. 1819

. 3081

. 6919

5100

. 4900

48

12

52

13

. 8883

. 1117

. 305)

. 6941

. 5324

. 4176

47

8

SO

14

. 0583

. 0417

. 3036

, 6904

. 0547

. 3453

40

4

41

15

.950232

174
174
174
173
173

.749718

. 3013

. 0967

. 7369

180

. 2731

45

10

4

10

. 093U

. 9020

. 9990

. 7010

. 7990

180
180
179
179

. 2010

44

50

8

17

. 1677

. 8323

. 3967

. 7033

. 8710

. 1290

43

52

13

Id

9373

. 7027

. 3344

. 7050

. 9429

. 0571

43

43

16

19

3067

. 0933

. 3921

. 7079

.960140

.739854

41

44

3D

90

• 3701

173
173

179
172
179

0339

. 9398

. 7103

0603

179
178
178
178
178

. 9137

40

40

24

31

. 4453

. 5517

. 9875

. 7125

. 1578

. 8492

39

30

28

82

. 5144

. 4850

. 9852

. 7148

. 8933

. 7708

38

33

33

23

. 5334

. 4100

. 9829

. 7171

. 3005

. 0995

37

38

36

94

0533

. 3477

. 9800

. 7194

. 3717

. 0383

30

34

40

25

. 7911

179
171
171
171
170

9789

9783

. 791)

177

. 5572

35

90

44

96

7898

. 2102

9700

. 7940

! 5138

177
177

170
176

. 4802

34

10

48

97

. 8533

. 1417

9730

. 7304

. 5347

4153

33

19

32

98

. 9988

. 0732

. 9713

. 7287

. 6555

. 3445

32

8

36

99

. 9951

. 0049

. 9090

. 7310

. 7201

. 8739

31

4

44

30

.960333

170
170

.739307

• 9900

. 7334

. 7907

176
176
175
175
175

. 8033

30

18

4

31

. 1314

. 3643

. 7357

. 8071

. 1321

39

50

8

32

. 1994

.' 8000

. 3319

. 7381

9375

. *»025

38

52

13

33

9673

170
109
109

. 7327

. 3590

. 7401

.9 70077

.799923

37

48

16

34

. 3351

. 0049

. 3572

. 7428

. 077J

. 9321

90

44

90

35

4097

109
138

. 5973

. 9548

. 7452

. 1479

175
174
174
174
174

. 8521

35

40

94

36

. 4703

. 5217

. 3523

. 7475

. 3173

. 7322

24

30

•18

37

5377

. 4021

. 3501

. 7410

. 8*W

. 7124

23

32

32

38

0051

168

3>41

. 847*

. 7522

. 3573

. 0427

22

28

36

33

. 6723

108
108

. 327?

. 3454

. 7540

. 4239

. 5731

21

24

40

40

7394

. 230~i

. 2430

. 7570

. 4904

174
173
173
173
172

. 5030

20

20

44

41

. 8035

108

. 1933

. 240'i

. 7514

. 535)

4341

19

18

4S

42

. 8731

187

. 12V»

. 23*2

. 701*

. 0352

. 3043

18

12

52

43

. 9402

107

. 05T*

. 2*59

. 7041

7043

. 3157

17

8

53

44

.970039

107
100

.793331

3333

. 7303

. 7734

. 8300

10

4

43

45

• 0735

. 9205

. 9311

7089

. 8434

172
172
172
171
171

. 1570

15

17

a

4

46

1400

166

. 8090

. 92*7

. 7713

. 9113

. 0887

14

50

8

47

9064

100

. 7930

. 3213

7737

. 9801

. 0199

13

52

IS

48

. 973S

105

. 7274

. 8213

7702

980438

.719512

13

48

16

49

. 3388

105
165

. 6013

. 22II

. 7786

. 1174

. 8386

11

44

20

50

4043

105

. 5951

• 9190

. 7810

. 1859

171
J71
170
170
170

. 8141

10

40

24

51

. 4708

. 52T2

. 9100

. 7834

. 3542

. 7453

9

30

W

52

5187

105

164

. 4533

. 9149

. 7358

. 3325

. 6775

8

32

32

53

0025

. 3975

. 9118

. 7882

. 3907

. 0013

7

28

36

54

. 6081

104
104

. 3319

. 9033

. 7907

. 4588

. 5412

24

40

55

7337

103

9033

. 9009

. 7931

5308

J70
109
109
109
109

. 4732

5

90

44

50

7991

. 9991

. 9044

. 7090

. 5947

. 4051

4

10

48

57

8645

103

. 1355

. 9090

. 7980

. 6095

. 3375

3

13

52

58

9997

163

0701

1990

. 8004

. 7301

. 8099

9

8

36

53

9948

103

. 0052

1971

. 8099

. 7977

. 8033

1

4

43

60

60

0.960599

103

O.719401

0.001947

0009053

0988052

0.711348

16

1-

L.OM.

DiC
for

15"

L.SM.

L.flin.

Diflt
for

15"

L.COMC.

L.COC

Diff.
for
15"

L.T0Bff.

t

p

»

Hoara.

Tleg.

Ham.

orl*

orl*

orl»

6*

10

0*

r —

P lo.

-4*

P UPl

too

6*

Digitized by VjOOQIC

42

TABLE II.— LOG. 81**8, TANo'g, &C.

0>

11°

!•-

- 15" 1- -

15* 1»

-150

108O

11

Diff

dht.

Dift.

Hour*.

Deg.

L.Bin.

for
15"
orl*

L.Cotec.

L.Cbe.

for
15"
orl*

L.8ec

L.l*ng.

for
15"
orl*

L.OM.

Dog.

Bowk.

•

'

i

-

•

44

9.880599

163
MS
163
163
161

O.T19401

9.991947

• 008053

9*88653

168
168

168
168

K8

9.711348

69

18

60

4

1

. 1348

. 8753

1933

. 8078

. 9336

. 0674

59

56

8

3

. 1897

. 8103

. 1898

. 8103

. 9999

. 0091

58

53

IS

3

. 3544

. 7456

. 1873

. 8137

.990671

.79933U

57

48

16

4

. 3191

6809

. 1848

. 8151

1343

. 8658

56

44

90

5

3836

161
161
160
160
160

. 6164

. 1833

. 8177

. 8013

167
167
167
167
166

• 7987

55

40

34

6

. 4480

. 5530

. 1798

. 8393

. 3683

. 7318

54

36

38

7

. 5134

. 4876

. 1774

8336

. 3350

. 6650

53

33

33

8

* . 5760

. 4334

. 1749

. 8351

. 4017

. 5983

S3

38

,

38

9

. 640H

. 3593

. 1734

. 8376

. 4684

. 5316

51

34

40

10

. 7048

160
160
159
159
159

. 3953

. 1699

. 8391

!. 5349

106
186
106
MS
M5

. 4651

50

SO

44

11

. 7687

. 8313

. 1674

. 8396

. 0013

. 3967

49

16

48

18

. 8336

1674

. 1649

. 8351

. 6677

. 3333

48

13

S3

13

. 8984

. 1036

. 1634

. 8376

. 7340

. 3660

47

8

56

14

. 9600

. 0400

. 1599

. 8401

. 8091

. 1999

46

4

**

15

.890339

159
158
158
158
158

.709764

, 1574

. 8436

. 8693

MS
M4
164
M4

164

. 1338

45

18

4

16

. 0871

. 9139

. 1549

. 8451

. 9333

. 0678

44

*

8

17

. 1504

. 8496

. 1534

• 8476

. 9980

. 0039

43

53

13

18

. 3137

. 7863

. 1499

• 8591

.800638

43

tf

16

19

. 8768

. 7333

. 1473

• 8537

. 1395

. 8705

41

M

90

90

. 3399

157
157
157
157
156

. 6601

. 1448

. 8553

. 1951

M4
163
M3
163
M3

. 8049

40

10

94

31

. 4039

. 5971

. 1433

. 8578

. 8697

. 7393

39

36

38

S3

. 4658

. 5343

. 1397

. 8693

. 3861

6739

38

»

33

83

. 5386

. 4714

. 1373

. 8688

. 3914

. 6066

37

38

3d

34

. 5913

. 4087

. 1346

. 8654

. 4567

. 5433

36

94

40

35

. 6539

156

156
156
155
155

. 3461

• . 1331

. 8679

. 5318

163
163
163
163
163

. 4783

35

30

44

86

. 7164

. 8836

. 1395

. 8705

. 5869

. 4131

34

16

48
58
56

37
88
39

. 7788
. 8413
. 9034

. 8313
. 1588
. 0966

1369

1344

. 1318

. 8731
. 8756

. 8782

. 6519
. 7lfi8
. 7816

. 3481
. 8833
. 3184

33
33

31

13
6
4

40

30

. 9655

155
155
155
154
154

. 0345

. 1193

8807

. 8489

163
161
161
161
161

. M38

30

14

O

4

31

.390376

.699734

. 1167

. 8833

. 9109

. 0891

39

56

8

33

. 0895

. 9105

. 1141

. 8859

. 9754

. 0346

38

53

19

33

. 1514

. 8486

. 1115

. 8885

.310399

.889091

37

48

16

34

. 8133

. 7868

. 1090

. 8910

1043

. 8958

36

44

30

35

. 3749

154
153
153
153
153

. 7351

. 1064

i 8936

. 1685

109
160
160
160
159

. 8315

35

40

34

36

. 3365

. 6835

. 1088

8963

. 3337

7673

34

36

38

37

3979

. 6031

. 1013

8988

. 3967

7033

33

33

39

38

. 4593

. 5407

. 0986

! 9014

. 3607

. 6393

33

38

36

39

. 5307

. 4793

. 0960

. 9040

. 4347

. 5753

31

34

40

40

. 5819

159

153
153
153
153

. 4181

. 0984

. 9066

. 4885

159

IS

. 5115

SO

90

44

41

. 6430

3570

. 0908

m

i . 8099

. 5533

. 4478

19

16

48
53
56

43

43
44

. 7041
. 7650
. 8359

. 3959
. 3350
. 1741

. 0883
. 0855

. 0839

. 9118
. 9145
. 9171

. 6159
. 6795
. 7430

. 3841
. 3395
. 3670

18
17
16

13

8

4

47

45

. 8867

153
151
151
151
151

. 1133

0808

. 9197

. 8064

158
158
158
158
157

. 1936

15

18

4

46

. 9474

. 0536

0777

jj

. 9393

. 8097

. 1303

14

56

8

47

.310080

.089930

. 0750

. 9350

9380

. 0670

13

53

13

48

. 0685

. 9315

. 0734

2

• !55

! 9961

. 0089

13

48

16

49

. 1389

. 8711

8097

9303

.380893

.889408

11

44

SO

50

. 1893

150
150
150
150
150

. 8107

0871

. 9339

. 1339

157

£
157
156

8771

10

40

34

51

. 8495

. 7505

. 0844

. 9356

. 1851

. 8149

9

36

98

53

3097

. 6903

. 0618

. 93B9

. 8479

. 7531

8

33

33

53

. 3698

. 6303

. 0593

. 9408

. 3108

. 6894

7

38

36

54

, 4398

. 5703

. 0565

. 9435

3733

. 0367

6

34

40

55

. 4896

149
149
149
149
149

. 5104

. 0538

. 0463

. 435B

156
156
156
153
155

. SMS

;

SO

44

56

. 5495

. 4505

. 0513

. 9488

. 4983

. son

4

16

48

57

. eon

. 3908

. 0485

. 9515

. 5007

. 4393

3

13

58

59

6689

. 3311

. 0458

. 9549

0331

3701

9

9

- F

59

. 7384

. 3716

. 0431

. 9569

. 6853

. 3147

1

4

47 (flO

60

9-31 7879

0.083181

9.990404

0.009598

9.387478

0-873534

'

18

O

. 1 .

r

Ltd*

Diir.

fhr
15"

L.8ee.

USin.

DMT
for
15"

L.Cotet.

I* Cot.

Diff
for
15"

L.Ttnf.

1 '

•

•

Hourr

Dm.

P*

Bow-

Of V

orl'

orl»

f 1

8»

141

Ho

r-

4* lo

-4»

78

a <

*

Digitized by VjOOQIC

tails n^-ioo. ones, tang's, tic.

43

O*

1*>

r—

15" !■ — 15' 1»-

-ISO

16*0

11»

Bi*

DilE

bitt.

Ooun.

D«*

UBU.

for
15"
orV

L.OM6S.

IftCot.

for

15"

L.Bee.

I*.TYnf.

for
15"
orl'

I* Cot.

Deg.

Hoarv-

•

t

i

/
60

m
"ll

•
60*

18

e

9.317929

148
148
148
148
147

3.333131

9.339494

9.039596

9.837475

155
155
155
155
154

04172525

4

i

„ 8473

. 1537

. 0378

. 9633

. 8095

. 1905

59

<»

8

3

. 9086

. 0934

. 0351

. 9649

. .8715

. B85

58

13

3

. 9658

. 0343

. 0334

. 9676

. 9334

. 0666

57

48

16

4

.3*0350

.379750

. 0397

• 9703

. 9053

• 0047

56

44

90

5

. 0840

147
147
147
147
146

. 9160

0970

9730

.330570

154
154

154
153

.669430

55

40

34

6

. 1430

. 8570

0343

. 9757

. 1187

. 8813

54

36

38

7

. 3019

. 7981

. 0310

. 9784

. 1803

. 8197

53

32

33

8

. 3806

. 7394

. 0188

. 9813

► 9418

. 7583

53

28

36

9

. 3194

• 0161

9839

3033

. 6967

51

24

40

10

• 3780

146
146
146
146
145

. 0134

9860

. 3640

153
153
153
153
153

. 6354

50

30

44

11

• 4366

I 5634

.. 0107

. 9893

. 4351)

. 5741

49

16

48

13

. 4950

. 5050

. 0079

J

. 9981

» 4871

. 5139

48

13

53

13

. 5534

. 4466

. 0059

J

. 9948

1 • 5488

. 4518

47

8

56

14

. 6118

3889

. 0035

. 9975

6093

1907

46

4

*9

15

6700

145
145
145
144
144

3300

.989997

.910003

6703

153
153
153
151
151

3397

45

11

4

16

. 7381

> , 9719

. 9970

7

0030

. 7311

. 9689

44

50

8

17

. 7863

. 9138

* 9949

7

. 0058

7930

. 9080

43

53

13

Id

. 8443

. 1559

. 9915

. 0085

. 8537

. 1473

43

48

16

19

. 9030

. 0980

, • 9887

. 0113

. 9133

. 0867

41

44

30

90

. 9599

144

144
144
144
143

. 0401

. 0140

9739

151
151
151
151
150

. 0361

40

40

34

31

.330176

•669B34

1 9839

*

. 0168

.340344

•659656

39

36

2*

33

. 0753

. 9347

. 9804!

. 0196

. 0949

. 9051

39

33

32

31

. 1339

. 8671

.. 9777

. 0233

. 1558

8448

37

28

36

34

. 1904

. 8096

- 9749

. 0351

~ 9155

.' 7845

36

24

40

35

. 3478

143
143
143
143
149

. 7533

. 9731

. 0379

. 9757

150
150
150
150
149

. 7343

35

20

44

36

. 3051

, 0949

. 9693

. 0307

. 3358

. 6643

34

16

48

37

. 3034

. 6376

. 9666

. 0334

. 3958

.• 6042

33

12

53

38

. 4195

. 5P05

. 9637

.. 0363

. 4558

. 5442

33

8

56

SO

► 4767

. 5333

- 9810

. 7

• 0390

.. 5157

* 4843

31

4

50

30

. 5337

143
143
143
143
141

. 4663

. 9583

. 0418

/ 5755

149
149
149
149
: 148

. 4345

30

10

4

31

. 5906

. *094

. 9553

: . 0447

. 6353

. 3647

39

56

8

38

. 6475

. . 3525

. 9535

1 .. 0475

. 0950

. 3050

38

52

13

33-

. 7043

. 3957

. 9497

. 0503

. 7546

. 3454

37

48

16

34

. 7610

• 3390

. 9469

.. 0531

. 8141

. 1859

96

44

30

35

. 8176

141
141
141
141
140

. 1834

. 9441

. 0559

. 8735

148
148
148
148
148

• 1335

35

40

34

36

. 8743

. 1358

. 9413

. 0587

. 9339

. 0671

34

36

38

37

. 9307

. 0693

. 9385

. 0615

. 9932

. GOTO

23

32

33

38

. 9870

. . 01X

. 9356

. 0644

.330514

.649486

33

23

36

39

.340434

•559566

» 9338

. 0673

. 1106

. 8894

31

24

40

49

. 0996

140
140
140
140
139

. 9004

9399

. 6701

. 1697

147

IS
147

147

147

. 8303

90

30

44

41

. 1558

• 8443

! 9371

. 6799

~ 9987

. 7713

19

16

48

43

. 9119

. 7881

~ 9343

0751

. 9876

. 7134

18

13

53

43

3679

. 7331

. '9314

67811

. 3465

• 6535

17

8

56

44

. 3239

• 6761

, 9186

. 0614

• 4053

» 5947

16

4

51

45

3797

139
139
139
139
139

6303

*. 9157

. 0843

. 454C

147
146

5300

15

9

4

46

. 4355

. 5645

. 9138

. 6873

• 5997

4773

14

56

8

47

• 4913

. 5067

. 9100

'

0900

. 5813

. 4187

13

53

13

48

. 5469

. 4531

. 9071

0999

. 63H

. 3603

IS

48

16

49

. 0034

3976

. 9049

• 0958

. 3018

11

44

30

50

. 6580

138
138
138
138
138

. 3490

. 9014

0986

. 7506

iS

145
145
145

. 9434

16

40

34

51

7134

S8M

. 8989

! 1015

. 8149

. 1851

9

36

38

53

. 7687

. 9313

8956

. 1044

. 8731

. 1369

8

33

33

53

. 8340

. 1760

. 8937

. 1073

. 9313

. 0687

7

38

36

54

j . 8793

• 1908

« 1109

. 9894

. 0106

6

34

40

55

' . 9343

137
137
137
137
137

. 6057

. 8809

. 1131

.360174

145
145
144

144
144

.339536

5

SO

44

56

9893

. 0107

. 8840

. 1160

. 1053

. 8947

4

16

48

57

i .350443

.€49557

v 8811

J

. 1189

. 1633

. 8368

3

13

53

58

. 0993

• 9008

8783

7

. 1918

. 3310

. 7790

9

8

56

59

. 1540

. 8460

.. 8753

. 1347

. 9787

. 7313

1

8

•

4

31

60

60

3.338088

6)4*7912

9.386734

9:011378

9.333364

0.636036

6

•

•

t

i

L.GCW.

DUE

for
15"

. i.Sec

1 L.BUI.

DUE
for
H"

L.C0Me.

L.Oot.

Diffi
for
15"

L.Ting.

i

Hours. iDag.

Do*

Hours

j-*^

ml'

orl*

orl*

0*

14

»6 '

r-

-4? IP

— 4F

7

|P

8*

Digitized by VjOOQIC

44

TABLE II.— LOO. SINES, TANO'fl, &C.

130

V — 15" 1" — 15'. 1» — 150

1660 11*

Diff

Diff.

Diff

Hourt.

Deg.

L.Sin.

for
15"
orl*

L.COMC.

L.C0*

for
15"
orl*

L-Soc

L.Tanf.

for
15"
orl*

L-Coc

D*.

Hoen.

•

f

i

t

-

a

$*

9.359088

137
136
136
136
136

0.647912

9.988724

7
7

7
7
7

0.011276

9.363364

%

143
143
143

0.636636

69

T

m

111 4

1

. 9635

. 7365

' . 8695

1305

. 3040

. 6960

59

56

W 8

2

. 3181

. 6810

. 8665

1335

. 4516

. 5484

58

■Si

12

3

. 3726

. 6274

. 8636

. 1364

. 5090

. 4910

57

4f

16

4

. 4971

. 5729

8007

. 1393

• 5664

. 4336

58

44

90

5

. 4815

136
136
135
135
135

. 5185

. 8578

7
7
7

7

. 1422

6937

143

143
143

143
142

3763

55

40

24

6

. 5358

. 4642

. 8548

. 1452

. 6810

. 3190

54

36

28

7

. 5901

. 4099

. 8519

1481

. 7389

. 9618

53

ae

32

8

. 6442

. 3558

. 8489

. 1511

. 7953

. 904?

58

** ■

36

9

. 6964

. 3016

. 8400

. 1540

. 8524

. 1476

51

24

40

10

. 7524

135
135
135
134
134

. 2476

. 8430

7
7

7
7
7

. 1570

. 9094

142
142
142
142
141

. 0906

50

90

44

11

. 8064

. 1936

. 8401

. 1599

. 9663

0337

49

16

48

12

. 8603

. 1397

. 8371

. 1699

.890932

.690768

48

12

52

13

. 9141

. 0859

. 8342

. 1658

0799

. 9901

47

8

56

14

. 9679

. 0321

. 8312

. 1068

. 1367

. 6933

46

4

53

15

.360215

134
134
134
133
133

.639785

8982

7

7

7
7

. 1718

. 1933

141
141
141
141
141

8087

45

7

4

16

. 0751

. 9249

. 8259

. 1748

. 9499

. 7501

44

56

8

17

. 1287

. 8713

. 8292

. 1778

. 3065

. 6935

43

52

12

18

. 1822

. 8178

. 8193

. 1807

3629

6371

49

48

16

19

. 9350

. 7644

. 8163

. 1837

. 4193

5807

41

44

20

20

. 9889

133
133
133
133
132

. 7111

. 8133

7
7
7

7

. 1867

4756

141
140
140
140
140

. 5944

49

49

24

21

. 3422

. 6578

. 8103

. 1897

. 5319

. 4681

39

36

28

99

. 3954

. 6046

8073

. 1997

. 5881

. 4119

38

3S

32

23

. 4485

. 5515

. 8043

. 1957

. 6442

. 3558

37

*

36

94

. 5016

. 4984

. 8013

. 1987

7003

. 9997

36

24

40

95

. 5546

132
139
132
132
131

. 4454

7983

7
8

7
7
7

. 9017

. 7563

140
140
139

130
139

. 9437

35

99

44

90

. 6075

. 3995

. 7953

. 9047

. 8129

. 1878

34

16

48

97

. 0603

3397

7992

. 9078

. 8681

. 1319

33

12

52

98

. 7131

. 9869

. 7899

. 9108

9939

. 0761

39

8

56

99

. 7659

. 9341

. 7862

. 9138

. 9797

0903

31

4

54

30

. 8185

131
131
131
131
131

. 1815

7839

8

7

?

8

. 9168

.380354

139
139
139
138
138

.619646

39

6

4

31

. 8711

1989

. 7801

. 2199

. 0910

9090

99

56

8

39

. 9230

. 0764

7771

9999

. 1465

. 8535

98

52

12

33

. 9761

0239

. 7740

. 9900

. 9021

7979

97

48

16

34

.370985

.69J9715

. 7710

. 9990

. 8575

. 7495

96

44

20

35

. 0806

130
130
130
130
130

. 9199

. 7679

7
8
7
8
8

. 9391

. 31-20

138
138
138
138
138

. 6871

95

40

24

36

. 1330

8670

. 7649

. 9351

. 3061

. 6319

94

36

28

37

. 1859

. 8148

. 7618

. 9382

. 4934

5766

93

32

38

. 9374

. 7626

. 7588

. 9412

. 4786

. 5914

29

2*

36

39

. 9894

. 7106

. 7557

. 9443

5337

. 4983

91

24

40

40

. 3414

130
130
199
199
129

. 6586

. 7596

8
7
8
8
8

. 9474

5988

137

137
137
137
137

. 4112

99

90

44

41

3933

0067

. 7495

. 9505

. 6438

. 3569

19

16

48

49

. 4459

. 5548

. 7465

. 9535

6987

i 3013

18

12

52

43

. 4970

5030

. 7434

. 9566

. 7536

. 9464

17

8

56

44

. 5487

. 4513

. 7403

. 8597

. 8984

. 1916

16

4

55

45

. 0003

199
199
198
128
128

. 3997

7379

8

8
8
8
8

. 9098

. 8931

137
137
136
136
136

. 1369

15

5

4

46

. 6519

. 3481

. 7341

. 9659

. 9178

0899

14

56

8

47

. 7035

. 9965

. 7310

9690

. 9795

. 0975

13

52

12

48

. 7549

. 9451

7979

. 9791

.390270

.609730

19

48

16

49

. 8063

. 1937

. 7948

9759

. 0815

. 9185

11

44

20

50

. 8577

198
198
128
128
197

. 1493

. 7917

8

8
8
8

9783

. 1300

136
136
135
135
135

8840

10

49

94

SI

. 9080

. 0911

. 7186

. 9614

. 1903

. 8097

9

36

28

59

. 9009

. 0398

. 7155

. 9845

. 9447

. 7353

8

32

32

53

.380113

.619887

. 7194

9676

. 9989

. 7011

7

2ft

36

54

. 0094

. 9376

. 7093

. 9907

. 3531

. 6469

6

24

40

55

. 1134

197
197
127
127
197

8960

'. 7061

8
8

8
8
8

. 9939

4073

135
135
135
135
134

. 5987

5

90

44

56

. 1644

. 8356

7030

. 9970

. 4614

. 5396

4

16

48

57

. 2159

. 7848

. 6998

. 3009

* 5154

. 4846

3

12

52

58

. 9661

7339

. 6967

9033

. 5694

. 4306

9

8

58

59

. 3168

6839

. 0935

. 3065

6933

3767

1

4

55

00

69

9.383675

0.616395

9.986904

0.013096

9.398m

0.603999

6

■

«

#

L.ON.

IMC
for
15"

L.8M.

L.81&.

Biff
for

15"

IhCoate.

L,Oot

Diff
for

LTtlf.

i

•

Hour*.

Deg.

Deg.

Hoar*

orl*

orl*

orl*

6»

10

0O

r-

-4«lo

— 4»

i

r*V>

O*

Digitized by VjOOQIC

TABLE II.— LOG. 8INB8, TAMo'g, &C.

45

0>

HP

V

-15" 1«-

- 15' 1» — 150

105O <

- 11*

M

Diff.

Diff.

Hours.

Dei-

L.SiA.

for
15"
orl*

LCoeec.

IfrOoe.

for
15"
orl*

L.8ee.

L.Tang.

for
15"
orl»

L-Cot.

Deg.

i

Houn.

-

•

i

50

9.383675

137
126
126
126
136

0.010325

9.986004

8
8
8

O.O13096

9.396771

134
134
134
134
134

0.603229

60

360

4

1

. 418.»

. 5818

. 6873

. 3127

. 7309

. 2691

59

56

8

s

. 4687

. 5313

. 6841

. 3159

. 7846

. 3154

58

53

12

3

. 5199

. 4808

. 6609

. 3191

. 8383

. 1G17

57

48

16

4

. 5697

. 4303

. 6778

. 1PW

. 8919

. 1081

56

44

90

5

. 6901

136
196
125
125
125

. 3799

. 6746

8

. 8

8

I

. 3354

. 9455

134
133
133
133
133

. 0545

55

40

24

8

. 6704

. 3296

. 6714

. 3286

. 9990

0010

54

36

26

7

7207

. 8793

. 6683

. 3317

.400524

.599476

53

33

32

8

. 7709

. 3391

. 6651

. 3349

1058

. 8943

53

38

36

9

. 8210

. 1790

. 6619

. 3381

. 1501

. 8409

51

34

40

10

.. 8711

135
135
135
125
134

. 1389

. 6587

8

8

8
8

. 3413

. 8134

133
133
133
133
133

. 7876

59

20

44

11

. 9311

. 0780

. 6555

. 3445

. 8656

. 7344

49

16

48

13

. 9710

. 0290

. 6533

. 3477

. 3187

. 6813

48

13

52

13

.390200

.009791

. 6491

3509

. 3718

. 6282

47

8

50

14

. 0708

. 9292

. 6459

. 3541

. 4849

. 5751

48

4

57

15

. 1203

134
124
124
124
133

. 8794

. 6437

8
8
8
8
8

. 3573

. 4779

138
133
133
139
138

. 5331

45

3

4

16

. 1703

. 8297

. 6305

. 3605

. 5308

. 4698

44

56

8

17

. 9199

. 7801

. 6363

3637

. 5836

. 4164

43

53

152

18

. 9695

. 7305

. 6331

. 3669

. 6364

. 3636

48

48

lt>

19

. 3191

. &5<)<>

. 6899

. 3701

. 6898

. 3108

41

44

20

530

. 36*5

133
123
133
123
133

. 6315

6366

8
8
8
8

. 3734

. 7419

131
131
131
131
131

. 8581

40

40

24

SI

. 417.)

. 5831

. 6334

. 3766

. 7945

. 8055

39

36

28

22

. 4673

. 5327

. 6308

. 3798

. 8471

. 1539

38

33

32

23

. 5166

. 4Kn

. 6169

. 3831

. 8997

. 1003

37

38

36

24

. 5658

. 4u..

. 6137

. 3863

. 9531

. 0479

38

34

40

95

. 6150

133
123
133
133
139

. 3850

. 6105

8

8
8
8

. 3895

.410045

131
131
131
139
130

.589955

35

30

44

96

. 6641

. 3359

. 6073

. 3988

. 0569

. 9431

34

16

48

97

. 7131

. 3869

. 6039

. 3981

. 1093

. 8906

33

13

52

98

. 7689

. 8378

. 6007

. 3993

. 1615

. 8385

39

8

56

99

. 8111

. 1889

. 5974

. 4086

. 3137

. 7863

31

4

58

30

. 8600

139
139
139
189
131

. 1400

. 5948

8
8
8
8
8

. 4C58

. 8658

130
130
130
130
130

. 7343

30

51

4

31

. 9088

. 0918

. 5909

. 4091

. 3179

. 8831

89

56

8

39

. 9575

. 0435

. 5876

. 4134

. 3699

. 6301

88

53

12

33

.400063

.599938

. 5843

. 4157

. 4319

. 5781

87

48

16

34

. 0549

. 9451

. 5811

. 4189

. 4738

. 5868

96

44

20

35

. 1035

131
131
121
121
121

. 8965

. 5778

8
8
8
8
8

. 4333

. 5357

189
139
139
139
139

. 4743

85

40

24

36

. 1520

. 8480

. 5745

. 4355

. 5775

. 4985

84

36

28

37

. 9005

. 7995

. 5713

. 4388

. 6393

3707

83

33

32

38

. 3489

. 7511

. 5679

. 4331

. 6810

. 3190

88

38

*j

39

. 3D7S

. 7028

. 5646

4354

. 7330

. 8874

91

34

40

40

. 3455

121
180
190
130
130

. 6545

. 5613

8
8

!

. 4387

. 7848

139
139

138
138
138

. 3158

90

SO

44

41

. 3938

. 6063

. 5580

. 4430

. 8358

1042

19

16

18

49

. 4430

. 5580

. 5547

. 4453

8873

. 1137

18

13

52

43

. 4001

. 5099

. 5514

. 4486

. 9387

. 0613

17

8

56

44

. 5381

. 4619

. 5480

. 4530

. 9901

. 0999

18

4

59

45

. 5863

130
190
130
119
119

. 4138

. 5447

8
8
8
8

. 4553

.430415

138
138
188
, 138
138

.579585

15

1

4

46

. 6341

. 3659

. 5414

. 4586

. 0987

. 9073

14

56

8

47

. 6830

. 3180

. 5380

. 4630

. 1440

. 8560

13

53

12

48

. 7399

. 8701

. 5347

. 4653

. 1953

. 8048

18

48

16

49

. 7777

. 3333

. 5314

. 4688

. 3463

. 7537

11

*4

20

50

. 8254

119
119
119
119
119

. 1746

5380

8

1

8

. 4730

. 8974

187
137
127
137
137

. 7086

10

40

24

28

51
59

. 8731
. 9307

. 1369
. 0793

. 5347
. 5313

. 4753
. 4787

. 3484
. 3994

. 6516
. 6006

9

8

36
32

32

53

. 9689

. 0318

. 5180

. 4830

. 4503

. 5498

7

28

36

-54

.410157

.583843

. 5146

• 4854

. 5011

. 4989

6

34

40
44

55
56

. 0633
1106

118
118
118
IIP
118

. 9368
. 8894

. 5113
. 5079

8
8
8
8

. 4887
. 4981

. 5519
. 6037

137
137
137
136
136

. 4481

. 3973

5
4

30

16

48

57

. 1579

. 8431

. 5045

. 4955

. 6534

. 3466

3

13

52

58

. 9059

. 7948

. 5011

. 4989

. 7041

. 8959

3

8

59

56
iO

59
60

. 3525
9.413996

. 7475
O.587004

. 4978
9.984944

. 5033
0.015056

. 7547
9.4*8052

. 3453

0.571948

1

O

4
0-

m

f

/

L-Ooe.

Diff.
for
15"

USee.

L.8in.

Diff.
for

15"

L.Comc.

L. Cot.

Diff.
for
15"

L.Tang.

, ,. 1.

Houn.

Deg.

Dei. Houn.

orl»

orl«

.

orl -

".I

•

* 1

040

r-

• 4 1 1°

-4«

*

*3 s»

5*

Digitized by VjOOQIC

46
l*—lSo

TABLE H. — LOO. SINKS, TANo'fl, &C
!• — 15" 1«— M' V — 150

1640 IO»

Hours.

De g .

4
8
13
16

30

34
38
33

40
44

48

4
8
13
16

30

34
39
38
36

44

48
53
56

4
8
IS
16

34

38
33
36

44

48
53
50

4
8
13
16

30
34
SB
33
36

40
44
48
53
56

40
41
43
43
44

45

46
47
48
49

50
51
53
53
54

55
56

57
58
59

■Mir*. Deg.

L.8in.

Diff.
for
15"
orl"

0.41SJ98

. 3407

. 3438

. 4403

. 4878

. 5347

. 5815

. 6383

. 6751

. 7318

. 7684

. 8150

. 8015

. 9079

. 9544

.490007

. 0470

. 0933

. 1395

. 1857

. S318

. S778

. 3338

. 3697

. 4156

. 4615

. 5073

. 5530

. 5987

. 6443

. 6899

. 7354

. 7801

. 8717

. 9170

. 9031
.430975

. 0517

. 0378

. 1434

. 1971

. 812*

. 877*

. 3333

. 3675

. 4132

. 45*>

. 501*

. 5463

. 5108

. 6151

. 0793

. 7343

. 7686

. 8131

. 8573

. 9014

. 945*

. 0817
0.440338

L.Cofl.

118
118
117
117
117

117
117
117
117
116

116
116
116
116
116

116
116
115
115
115

115
115
115
115
115

114
114
114
114
114

114
114
113
113
113

113
113
113
113
113

113
113
113
113
113

113
113
113
Ml
111

111
111
HI
111
111

111
110
110
110
110

»-105«

Diff.

fhr

15"

orl«

L.Comc L. Cob.

0-587004

0533

6063

. 55D3

. 5133

. 4653

, 4185

. 3717

. 3349

. 8783

. 8316
1850
1335
0)31

. 0456

.579993
. 9530
. 9067
. 8005
. 8143

6763
6303
5844

. 4470
. 4013
. 3557

. 3101
. S646
. 8H1
. »1737
1383

. 0830
0377
.569935
. 0473
. 0033

. 8573

. 8131

. 7673

. 7381

. 6764

. -3378

. 5131

. 4984

. 4538

. 4013

. 3047

. 3303

. 8758

. 8314

. 1871

. 1438

. 0186

. 0544

. 0101
0.559603

L.8«e.

0.084944

4910
. 4876
. 4643
. 4808

. 4774
. 4740
4708
. 4673
. 4638

. 4601

. 4569

. 4535

. 4500

4431
4397

4394

4334
4189
4155

4130

4035
4050
4015
39H1
3946

3911
3875
3840
3805
3770

3735
3700
3664

3594

3553

3531
34R7
3453
3416

3381
3345

3309
3373

. 3166

. 3130

. 3014

. 3058

. 3033

. 8986

. 8150

. 8914

. 8378
0.003843

L. Sin.

Diff.
for
15"
orl«

Diff.
for
15"
orl»

L-Soc.

0.015056

5040
5134
5J58

5193

5394
5338
5363

5397
5431
5465
5500
5534

5569

5003

5637
5673
5708

5741
5776
5811
5845
5880

5915

6019
0054

0135
6100
6195

0365
6300
6336
6371

6477
6513
6548
•584

§619
•655

0091

8737

•798
6834

0078
7014
7050
7080
7133
017156

L. Ootcc

L. Tanf.

9.4*8053

. 8557

. 90(33

. 9560
.430070

. 0573

. 1075

. 1577

. 3079

. 8580

. 3031

. 3581

. 4080

. 4570

. 5078

. 5578
0073

. 0570

. 7007

. 7563

8554

. 0543
.440030

1033
1515

8497

3479
3900
4458

4947

5435
5933
6411
6898
7384

7870
8356
8841
0336
0810

0777
1300
1741

3187
3088
4148

5107
5586

. 0543

. 7019

O.457406

L. Oot.

out

for
15"
orl«

L.Cot.

Deg.

136
136
186
136
136

135
135
135
135
135

135
135
185
135
134

134
184
134
134
134

184
184
183
183

133

183
133
133
133
183

133
133
133
133
133

133
133
133
131
131

131
131
131
131
131

131
131
131
130
130

ISO
130
130
130
130

130
119
119
119
110

Diff.

15"
orl*

0.571948
. 1443
. 0938
. 0434

943i
8935
8433
7931
7430

0919
•419
5J30
5431
4933

3937
3430
8933
8437

1941

1440
0951
0458

9470
897P
84*3
7994
7503

7013
0581
6031
5543

. 4565

. 4077

. 3580

. 3103

. S616

. 3130
1644
. 1159
. 0674
. 0190

.549708
. 0331
. 8740
. 8357
7776

7394
6813

5373

. 4414
. 3036
. 3458
. 818!
O.543504

L.Ttnf.

!'_* 10-4*

>6D
56
53
4*
44

36
33
38
34

30

16
13

8

4

60

50

156

38
34

M

13

8

4

•
56
53
48
44

40
36

33
38
34

57

56

38
34

Def.

V4o 4*

Digitized by VjOOQIC

TABLS n.— LOO. 8INBS, TANo's, &C.

47

1*

16o

1'-

- 15" 1- -

. 15' 1>

— 150

1630

10*

Uiff

Diff

Dili

El alt M

ft*

L. Sin,

Got

11"

LG«k

L-Co*.

far
15"

L-Sec'

U Tftftf .

far

15' h

L. Oat.

1 l* X

Kauri

*

1

'

or 1'

orl*

url*

f

m

•

4

»

o t*o:U-

no
no
ira

(CM

O.aieaoa

U UN^Ji

9

S

<M)1715*

»4&74O0

llJ
119
UJ
ULp
US

u VU^l

•ii

d>

tiO

1

i

1798

, 99*3

. 3*5

i ruts

7H73

p 30ii

59

50

a :

3

. 1318

. 8783

. nm

i 7331

B440

. 1551

58

53

13

3

losa

. 8313

. wm

. ms

8935

1075

57

48

w

4

. 9OJ0

7J04

. Sfrtt

. 7301

, 9400

OiiUO

50

44

zu

5

3535

100
KM

Ift-J
10J

7405

. -j'- <•

9

9

. 7310

. 9875

11*
11
11-
llr

11*T

, 0135

55

40

M

1

, 2^73

. 7037

. 3 m

. 7370

.46031 lp

,33n,jj

ji

JU

>

7

3410

t 65'JO

, S547

. 7413

- 0*H

. 0177

53

33

;■

a

. 3*17

. 6253

, 3550

7450

„ Wit?

. 8703

53

38

30

■

+ 4^1

- 5716

3514

. 748*

. 177U

, mm

51

34

iU

10

. 4730

1011

109

lff.1

109
108

♦ 5390

. 3477

9

9

. 7533

. 331'

118
lift
118

Ufl

118

n 7757

50

30

n

11

, 5155

■ 4845

. 3441

. 755i

- 2714

v:-.

16

r

13
LI

5540

6035

4410

. 3075

. 3404

. mm

. 7596
, 70-13

. 31*

. sua-

0H14
0343

48
47

I.'

56

||

645'j

. 3541

. 3331

. 7609

. 4138

sen

46

4

5

n

15

6H>3

103

m-

104
103

. 3107

. 3304

9

. 7706

IPS

117
117
117
117

w

5401

45

55

i

16

. 73*;

- 3674

. S237

. 7743

5ii r.i

- 4i*31

11

5ti

17

18

. 7750
. 81<"1

. 3341
. 180-3

2183

* 7780
. 7817

- 553

. 44i.l

, 3093

43
42

I-

i'i

id

, 8623

. 1377

+ 3146

, 7&W

. 6477

. 3533

41

li

31

30

. 0054

109

107
107
107
107

. OHO

. 3109

9
9

9

. TB9I

» 69^5

117
117

n;

1.7

in

. 3055

40

40

N

31

0435

■ 0515

. -,niT-j

> 703? 1

- 7413

i 35*7

39

30

H

aa

. 6015

. 0085

i 3035

. 7063

7^H1

i 913U

38

;«3

fj

33

430315

>r

, 1WI8

. 8003

8347

. 1053

37

3*

w

34

. 0775

. 0335

. 1961

- B03J

k 8814

. 1186

30

n

10

35

. 13iH

107
103

]»;

107
107

. 8790

MM

9
9
9

. mn

t 9-2R1

lit)

Mi
lit)

Ui

. 07-J

35

3o

11

1.-

n

37

1039
1 . 3000

, 8368
. 7040

. 1^0

. 8114

8 151

. 974li

,47oan

0354
,3tl97Hy

34
33

16
13

59

3d
30

. 34**
, 3915

. 7513
. 7085

. 1K13
, 1774

. ei88

p 0676
. 1*41

. 1»334
. B85»

33

JJ

'I

30

. 3343

106

ii'.
100
106
100

. 0658

. iri7

9

, 8303

, 1605

lie

116

lib
ill

115

i 8395

30

04

1

:n

. 3768

. 5313

* 1IEI9

, 8301

. 90tm

. 7tJ31

■.".'

56

H

33

. 4104

5806

, 1003

* E138

. 351*3

. 7408

38

52

ft

n

. «ua

. 538]

. vm

- 837t :

»): 5

. 7005

37

48

If,

31

, 5044

. 4050

1587

. B413

i 3457

. 0543

30

44

u

-15

. s*en

100

w\

106
106
105

4531

. 1550

. §450

. 3910

115

li.
J15

i •:

, 6081

25

40

.' 1

36

, 5*13

. 4107

. 1513

8488

. 43-1}

, MIS

34

W

,._ M

17

, 61 lf»

. 3584

. 1474

. 853(1

4tm

. 5l5f

33

n

aj

3d

* 6730

. 3311

. i4:m

- 8564

. 530-.I

. 4697

33

38

M

&

. 7 UK

. 3838

1300

, 8601

571^

. 4337

31

«

10

iU

. 7584

105
105
105
105
105

. 94W

. 13H1

e

9
9

• 86351

, 0333

115
115

m

114
114

♦ 3777

39

30

M

41

-I'M.

, 1004

. 1333

. BR77

0683

. 3317

19

10

14

43

. 6437

* 1573

13«5

, 8715

. 7142

. L".V-

18

13

SI

P

. 8848

. 1152

. 1347

. -T.Vr

. 7*01

. 311! HI

17

*

10

44

. BQ68

0732

mm

. 8701

. 8053

. 11H1

16

4

T

ii

4

45

40

. 9688

.400 JO*

105
105
in",
104
101

. 0313

. 1171
. 1153

9
9
9

i

. 8890
. 6807

. 8517
. 8975

1M

in

114

111

1L4

. 1483
1035

15
14

6$

5ii

! rt

47

. 0527

p 0473

. 10^5

i 8f»05,

g 9439

. 0568

13

53

ta

48

. t\w,

0054

. 1057

. im

t 9880

. oin

J -J

IK

tfl

49

■ 1304

. 8630

* 1010

t 8U8I

.480345

.919055

11

11

.-■1

50

. 17*3

104
104
104
104
104

. 8319

- 0081

9
10

10

9

. 0016

, 0801

114
114
114
114

114

. 0100

10

40

N

SI

. 3199
. 3010

. 7801

. 7384

. 0043

. OOSm
, 90.Ni

. 1357
. 1713

. 8743

- 8388

1

8

36

8

53

, 3n-«

, 60«8

t 0866

, 9134

, «llJd

. 7834

7

3-"

w

54

i 344a

- 6553

- 0697

, 9173

. 3631

. 7379

34

m

55

. 3*H

104
IM

!01
103
103

. 6136

. 07W

10
9
10
10

, 0311

, 3075

ll.l
113
113
113
113

■ 0035

5

30

.!

M

. 437-1

- 57*1

. 0751

. 0349

. 353-*

6471

4

16

i-

57

. 40M4

♦ £Hfti

* 0713

. 03*8

, 3083

. 00!8

3

13

33

58

5108

4809

. H.17I

, 0336

p 4134

■ 5506

f

8

n

30

- 55-23

. 447H

. 0635

. 9305

4RB7

. 5113

1

4

7

iti

AD

i* m. -,-►■■..:.

">3«i>:,

o.9S05:m;

0,010404

O.i^.VIL.

O51460I

39

i

a

•

F

Dfff.
fur
15 J '

Diir

for
15"

Diff
f«r
15"

'

Ifoun.

Dtf,

L.Co§.

L. Set.

L, Bin.

L. Cowa U Oot.

LTfcDi,

Def

1 1 . iij r- [

or 1*

«r

1

orl"

|

i*

IOC

|o

1*-

4' 10.

-4*

f3«

4

fc»

Digitized by VjOOQIC

48

TABLE II.— LOCI. SINKS, TAKg's, &C.

1T>

1' — 15" !• — 15' 1»-150

109)0

ia»

Hoars.

De*

4
8
12
16

30
24

40
44

48
32
56

4
8
12
16

20
24

28
32
36

10

11

44

48
52
56

4
8
12
16

20
24
98
32
36

40
44
48
52
56

4

8
12
16

20
24

28

1
2

3

4

5

6
7
8
9

10
11
12
13
14

15

16
17
18
10

20
21
22
23
24

25

26
27
28
20

30
31
32
33
34

35

36
37
38
30

11

41
42
43
44

45

46
47
48
49

50
51
52
53
54

55

56
57
58
59

Hoars. Deg.

L. Bin.

DitT
for
15"
orl«

9.465935
6348
6761
7173
7585

7900
8407
8817
9227
9637

.470046
0455
0863

1271

1071'

9065

2492
2899
3304
3710

4115
4519
4923
5327
5731

6133
6536
6936
7340
7741

8142
8542
8949
9342
9741

.480140

. 0539

. 0937

1334

1731

. 2128

. 2525

. 292)

. 3316

. 3712

4107
4501

6975
6467
6859
7251
7643

. 8814

. 9204

. 9593

9.409962

L.O0S.

103
103
103
103
103

102
102
102
102

102
102
102
102
102

102
102
101
101
101

101
101
101
101
101

101
100
100
100
100

100
100
100
100

100

99
99
99

99

98
97
97
97
97

3% 10 1*

Diff

15"
orl-

L. Goose.

0.534065

2415

2004

1503
1183
0773

.530054
. 9545
9137
. 8729
. 8321

. 7915

. 7508

. 7101

. 6690

5481
5077
4673
4269

3867
3484

1458
1058
0658

.019660
. 9461
. 9063

8666

7872
7475
7079
6684

. 5893

. 5499

. 5105

. 4711

. 4318

. 3925

. 3533

. 3141

. 2749

. 2357

. 1967

. 1576
1186

. 0796

. 0407
O.S100I8

L.Bec

L.OM.

9*9805 W

. 0558

. 0519

. 0480

. 0442

. 0403

. 0304

0247

0208
0169
0130
0091

. 0912
.•79973
. 9934
. 9894
. 9855

. 9816

9776

9737

. 9697

. 9658

. 9618
. 9579

9340

9180
9140
9099

9019

8979
8939

8898

8818

fc 8777

*8736

8696

8655

8615
8574
8533
8493
8452

8411
8370

TOT

for
15"
orl«

. 8247
0.978266

9
10
10

9
10

10
10
10
JO
10

10
10
10
10
10

10
10
10
10
10

10
10
10
10
JO

10
10
10
10
10

10
10
10
10
10

10
10
10
10
10

10
10
10
10
10

10
10
10
10
10

DMT

15"
orl«

L.Sec

0.019404

9442
9481
9520
9558

9597
9636
9675
9714
9753

9792
9831
9870

9988
.030027

0066
0106
0145

0184
0224
0263
0303
1042

0382
0421
0481
0501
0541

0740
0780

0901
0041

0981
1021
1061
1102
1142

1182
1223
1264
1304
1345

1385
1426

1467
1507
1548

1589
1630
1671
1712
1753
.0911794

L.O0M6.

r — * io— 4»

L.TMf-

9.485330

. 5791

. 6242

. 6693

. 7143

. 7593

. 8043

. 8499

. 8941

.490286
. 0733
. 1180
. 1627

2073
. 2519

3410

4743
5186

6515

7399
7841

9169

0481

0920
1359
1797
2S35

2672

3109
3546

3982
441*

5724
6150
6593

74611
7893

8756
9191

.510054
. 0485
. 0916
. 1346
9511776

L.OM.

DRT
for
15"

orl-

L.O*.

113
113
113
112
112

112
112
112
112
112

112
112
112
112
HI

111
111
111
111
HI

111
111
111
111
110

119

110
110

lie
no

no
ue
no
no
no

no

100
169
109
109

109
100
109
109
100

109
109
108
108
108

108

108
108
108

108
108
108
107
107

0.5140*1

. 490U
. 3758

9857

2407
1957
1508
1059
0GM>

.0O9714

DUE

15"
orl»

7927
7481
7035
6590
6145

5701
5357
4814
4370

3485

3943

2159
1718

1278

9519

9089
8641

8803
7765

6454
60IH
55H2

5146

4711
4276
3841
3407
9973

2540

9107
1074

.489046
. 9515

L.T*off

IJH-I'

51

27

21

Oof

•let

.12

Oil*

52
4-

1\

Digitized by VjOOQIC

1» 18©

Hours.

13

14

15

15

L.8U.

9.459983
.490371
0759
1147
1530

1923

3306
3695
3061
346b

3851
4336
46S1
5604

5773
6154
6537
6919
7301

7683
8063
8444

9584
9983
.500343
0731
1099

1477
1854
3331
3608
3984

3735
4111
4485

5334
5608
5081
6354

6737

7471
7843

8314

310065

0434

1173
1540
1907
3375
94113643

I* Cos.

for
15"
orl*

TABLE II. — LOO. 8TKE8, TANo's, &C

l'-iU" ^tM' 1»~>150

49

96
98
96

96
96
95
95
95

95
95
95
95
95

95
95
95
94

94
94
94
94
94

94
94
94
94
93

93
93

93
93
93
93

03
93

03
93

93
93

n 149Q

DtC
for
15"

orF

L.COMC

510018

509639
9341
8853
8465

8078
7693
7305
6919
6534

8149
5764
5379
4996
4613

3846
3463
3081
3699

8318
1937
1556
1175
0795

0416
0037
499658
9379
8901

8146
7769
7393
7016

6640
6365
5889
5515
5140

4766
4393

4019
3646
3373

3901
3539
3157
1786
1415

0874
0304
,489935

9197

7735
0.467398

L. Bee

L.Co&

9.938906

. 8165
. 8134
. 8083
. 8043

7959
7918
7877
7835

7794
7753
7711
7669

7586
7544

7503
7461
7419

7377
7335
7393
7353
7310

7167
7135
7083
7041

6957
6914
6873
6830

6787

6745
6703
6660
6617
6575

6533

6489
6446
6404
6361

6.11

6189
6146

6017
5974

. 5887
. 5844
. 5801
. 5757
. 5714
9.973070

for
15"
orl-

Di<F
for
15"

orl-

0.091794
. 1835

. 1876
. 1917
. 1958

1999
. 3041
. 3083
. 3133
. 8165

83F9
3331
3373

8414
3456
3497

3707

3748
3790

3875
8917
3959
3001

3043
3086
3138
3170
3313

3355
3398
3340
3383
3435

3468
3511
3554

3596

. 3735

. 3768

. 3811

. 3854

3897
. 3940
. 3983
. 4036
. 4070

. 4113
. 4156
. 4199
4343
. 4386
O.O44330

L.CMM.

L. Tmng.

9.511776

. 3306

. 3635

3064

3931
4349
4777
5904

6057
6484
6910
7335
7760

8186
8610
9034
9458

.590305
. 0738
. 1151
1573
. 1995

. 3417

3660
4100

4530
4940
5359
5778
6197

6615
7033
7451

7868

8703
. 9119
. 9535
. 9950
.530366

0781

1196

. 1611

9439

. 3366

3679

. 4001

. 4504

. 4916
. 5338
5739
. 6150
. 6561
9.536973

I» Cot.

V-* JO — 4*

O

BiftT
for
15"
orl*

101O io»

107
107
107
107
107

107
107
107
107
107

107
100
106
106
106

106
106
106
106
106

106
106
105
105
105

105
105
105
105
105

105
105
105
105
104

104
104
104
.104
104

104
104
104
104
104

104
104
103
103
103

103
103
103
103
103

103
103
103
103
103

Diff
for
15"
orl*

I* Cot

7794

7365
6936
6507

6079
5651
5333
4796
4369

3943
3516
3090
8665
3340

1814
1390
0966
0543
0118

479695
9373
8849
8437
8005

7583
7163
6741
6330
5900

5480
5060
4641

3365
8967
3549
3133
1715

1398
0881
0465
0050
.469634

9819
8804
8389
7975
7561

7147
6734
6331
5909

5084
4673
4361
3850
3439

L.T»nf.

Deg.

D€f.

Hours.

47 €

47

48
44

40
36
33

30

16
13

8

4

56
53

48
44

40
36

46

45

30

16
13
8

4

56
53
48
44

40
36
33
38
34

30

16
13
8

4

56
53
48
44

34

16
13

8

4

44

Horn*

**> 4-

Digitized by VjOOQIC

TABLE TL—10Q. 8IN1B, TANO's, Ac.

1*>

V

— 15" 1«-

-15' 1» — 150

1000

16*

biff

Diff

ftiff

Pel-

L.Sin.

/or
15"
orl«

L-Oomc.

L.OM.

for
15"
orl*

L.866.

L.Tanf.

for
15"
orl*

L-Ooc

»*

How*.

/

t

m

•

9.513648

93
91
91
91
91

0.487358

9.975670

11
11

\\

11

0.0*4330

9.536918

109
103
103
103
103

0.463038

60

43

61

1

. 3009

. 6991

. 5637

. 4373

7389

. 9618

58

56

S

. 3375

. 6635

. 5583

. 4417

. 7793

. 9308

58

J3

3

. 3741

. 6359

. 5539

. 4461

. 8303

. 1798

57

48

4

. 4107

. 5893

. 5496

. 4504

. 8611

. 1389

56

144

5

. 4473

91
91
91
91
91

. 5538

. 5453

11
11
11
11
11

. 4548

. 9030

103
103
103
103
103

. 0980

55

|o#

6

. 4837

. 5163

. 5408

. 4593

. 9439

. 0571

54

7

. 5303

. 4798

. 5365

. 4635

. 9837

. 0163

53

8

. 5566

. 4434

. 5331

. 4679

.540345

.459755

59

l

9

. 5030

. 4070

5977

. 4793

. 0653

. 8347

51

--i

10

. 6394

91
91
91
91
90

. 3706

5333
. 5189

11

8
ii
ii

4767

. 1081

109
103
103
103
101

. 8939

58

&

11

. 6657

. 3343

. 4811

. 1468

. 8533

40

'

13

. 7030

. 9960

. 5145

. 4855

. 1875

. 8135

48

IJ

13

7383

. 9618

. 5101

. 4899

. 9381

. 7719

47

4

14

. 7745

. 9355

. 5057

. 4943

968G

. 7313

46

4

15

. 8107

90
90
90
90
90

. 1893

. 5013

ii
ii
ii
ii
ii

. 4987

. 3094

101
101
101
101
101

. 6808

45

43

Q

16

. 8468

. 1533

. 4060

. 5031

. 3499

. 6501

44

51 !

17

. 8830

. 1170

. 4035

. 5075

. 3005

. 6089

43

H j

18

. 9190

. 0810

. 4880

. 5130

. 4310

49

*!

19

. 0551

. 0449

. 4836

. 5164

. 4715

.' 5983

41

44

90

. 9911

90
90
90
90
90

. 0089

. 4793

ii
ii
ii
ii
ii

. 5908

. 5119

101
101
101
101
101

. 4881

40

m

21

.090371

.479739

. 4747

. 5853

. 5594

. 4476

38

M

S3

. 0631

. 9369

4703

. 5997

. 5098

. 4073

38

33

33

. 0990

. 9010

. 4659

. 5341

6331

. 3669

37

*

34

. 1349

. 8651

. 4614

. 5386

. 6735

. 3965

36

*€

35

. 1708

89
89
89
89
89

8393

. 4570

ii
n
ii
ii
ii

. 5430

. 7138

101
100
100
100
100

. 9803

35

■

36

. 3066

. 7934

. 4535

. 5475

. 7541

. 9459

34

Ii

37

. 3434

. 7576

. 4481

. 5519

. 7043

. 9057

S3

li

38

. 9781

. 7319

. 4436

. 5564

. 8345

. 1655

38

-

39

. 3138

. 6863

. 4391

. 5609

. 8747

. 1953

31

-i

30

. 3495

89
89
89
89
89

. 6505

. 4346

ii
ii
ii
ii
ii

. 5654

. 9149

100
100
100
100
100

. 0851

39

14

g

31

. 3853

. 6148

. 4303

. 5698

. 0550

. 0450

98

'■

33

. 4308

5793

. 4357

. 5743

. 0051

. 0049

98

33

. 4564

. 5436

. 4313

5788

.550359

AAQftlfl

97

t-

34

. 4919

. 5081

. 4167

. 5833

0759

.' 8848

96

41

35

. 5375

89
89
89
88
88

. 4735

. 4133

ii
ii
ii
ii

5878

. 1153

100
100
100
100
100

. 8847

95

M0 '

36

5630

. 4370

. 4077

. 5933

. 1553

. 8447

94

hi

37

. 5984

. 4010

. 4033

. 5968

. 1059

. 8048

93

—

38

. 6339

. 3661

. 3987

. 6013

. 9359

. 7648

99

3d

39

. 6693

3307

. 3943

. 6058

. 9751

. 7949

91

94

40

. 7046

88
88
88
88
88

. 9954

. 3897

ii
u
ii

:;

. 6103

. 1140

100
09
99

89

6851

90

M '

41

. 7400

. 9600

. 3853

. 6148

. 3848

. 6453

19

M !

43

. 7753

. 9347

. 3807

. 6193

. 3946

. 6054

18

13

43

. 8105

. 1895

. 3761

. 6939

. 4344

. 5656

17

81

44

. 8458

. 1543

. 3716

. 0984

. 4749

16

4 ,

45

. 8810

88
88
88
88
87

. 1190

. 3671

8

it
ii
n

. 6339

. 5130

89
90
90
80
90

. 4861

15

41

46

. 9161

. 0839

. 3695

. 6375

. 5536

. 4464

14

56

47

. 9513

. 0487

. 3580

. 6480

5033

. 4067

13

59

48

. 9864

. 0136

. 3535

. 6465

. 6399

. 3671

19

48

49

.580315

.459785

. 3489

. 6511

6796

. 3974

11

41

50

. 0565

87
87
87
87
87

. 9435

. 3444

ii
ii
ii
ii
u

. 6556

7191

90
90
90
90
90

. 9878

18

40 ,

51

. 0915

. 9085

. 3398

. 6603

. 7517

. 9483

36

53

. 1965

. 8735

. 3359

. 6648

. 7913

. 9087

8

33

53

. 1615

. 8385

3307

. 6603

. 8308

7

98

54

. 1964

. 8036

. 3361

• '6739

8703

! 1997

6

94

55

. 9313

87
87
87
87
87

7688

. 3915

n
n
ii
ii
u

6785

. 0007

89
88
88
88
88

. 0903

5

90

56

. 9661

7339

. 3169

. 6831

. 0409

. 0508

4

16

57

. 3009

. 6091

. 3194

6876

. 8885

. 0115

3

19

58

. 3357

. 6643

. 367«

. 6993

.500979

.498731

9

8 '

59

. 3705

. 6995

. 3039

. 6968

. 0673

. 8397

1

4

60

9 594660

0.465946

9.973986

0.997614

0.561086

9.498834

49 •

/

L.O«.

Diff
for

15"

L-Bcc

L.8in.

Die

for

15"

L-Oone.

L.OOC

DiC

for

15"
orr I

L.T*of.

i

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orl'

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tot
IS

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4000

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it

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H
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, 7945

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. 73T>1

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. 0,01

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49

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52

TABLE n. — LOO, SINES, TANo's, &C,

i* aio

V

-=15" 1-

-= 15' 1* — 150

15SO lO

Diir

Diff

Dirt

M..U- -

M"ff.

L Bin,

orl'

L.Co«c

L-Cm.

for
15"
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for
IS'

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■

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m
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0.1 1... 1

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12
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12

12

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04
04
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- 5342

. 0101

9H97

. 4535

5445

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h

2

. 4«1*7

. 5013

, OU55

0045

. 4932

50U8

58

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li

1

. 5315
. 5643

. 40*5
- 4357

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. 0904
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. 4314

.77
96

1

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5

, 5971

82
82
82
82
81

, 4039

. 9900

12
12

12
12
12

„ 0001

. 6062

04
04
94
04
04

3038

55

M

24

6

, m

. 3701

. 9860

Oi-10

. 6430

' 3561

54

>

-

7

. eon

. 3:rr4

. Wll

, 0189

. lk-J5

, 3185

53

as

I

. 6U53

* 3017

. 07O3

. 0237

. 71 WJ

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m

j

73M*

- 2720

9714

. 02cNj

. 79$

, 2434

51

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in

Ki

, 760ii

ei

81

ei

81
81

. 2304

. 9665

12
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12
12
12

. 0335

. 7041

04
04

•4 1

03

1C

, 90ft

m

11

fl

. TB99

- 3068

. '.Mm

Q3#|

, 83149

; 1B84

m

U

13

. &35P

. 1742

. 9567

, 0433

. WBSl

. 1309

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a

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. 8584

. 14 Ifi

. 951^

. 0482

. POtiH

0994

47

■

m

14

; fflffe

- 10B

. 04fi0

0531

i 0440

. 0500

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i

jft

t

15

P234

81
81
SI
81
81

. 07QE

: 0420

12
12
12
12
12

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03
03

93

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. 0442

. 0370

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81

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12
12

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1681

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93

. H3rJ 40

m

21

. nra

, 8822

. 9J24

, 0870

SU54
, 9420

. 794f*i 30

•

■^

t»2

, 1501

. 8499

, !KI75

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, 7574 »

a

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12
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. 2799

, 71W1 -T7

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, 21411

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. 8070

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80

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12
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03

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. 8777
. 872H

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; 1273

. 4656
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32
31

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80
80
80
80
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. 5025
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80
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. 8429
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. 7247
. 7616

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. mm

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. 8723

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70
70

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. 7977

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. 9172

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79
31

70
79

1144

. 7927
. 7870

13
12
13
13
13

, 2073
9124

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t I20fi

92
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, 7725

. 2225
, 2275

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17
11

m

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70
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51
53

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, 7522

. 237«

2427

. 2478

. 3127
* 3858

4 0873
. 4307

9

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7

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. 7171

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, 4223

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78

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7*

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to

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7677

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g. ©07 166

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Digitized by VjOOQIC

TABLE II. — LOO. 8INE8, TANGOS, <fcc.

53

1*

««o

!• — 15" !• — 15' 1»—150

15TO

10*

Dtff

J > k J 1 .

UiJ-

Hours.

llfljf-

L. sin.

1 fir

U Cotec.

1 l - c ^ s»

L. Sec.

L. Tanf ,

S-"i '

L Col,

Dkjt.

Houri^

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1

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31

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CO

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0.5 13571

! 2

0.iaU2l

0.9&7 tM

13
13
13

13
13

0,03V 1

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1 Hi

U.3IM.7H.

60

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9

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13
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55

10

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. 4553

, 6H3J

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* 193

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54

30

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7

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p «ast

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53

:!2

fU

s

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. anai

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. 037*

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51

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. 0UO«

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. ttflO-2

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, 9m

. 65S0

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13
13

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. 3553

♦ lite

, 89JU

46

4

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n

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. 6234

77
77

s

1764

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13

, 3605

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90
00

, 815^

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. (044

. 3U56

. 2301

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. 07 in

41

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2

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, 3S04

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s

80

. 0350

40

40

1

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13
13
13

, 3LM5

. 400(1

, maa

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38

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. 0301

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, saw

. -IX.

t 5l>4l

34

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13
13

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. 5077

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25

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76

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76
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13
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13

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, 5435

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, 7405

. 560d

. 433t

. mi

. 3133

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70
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74
76

. 7160

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80
83

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. 4437

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56

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* aji5

. 4S7*

. 5121

. 2J07

1 T;va

lit

4

Si

1

45
in
47

, 73*7
, 7>H

75
75

, sin
. sfiii

, 4773
. 4730

13

ia

13

a

13

. 5174,
. 52*7

, sm

. 2561
- ftl|5
. 3 J II

88

88

n

S8

, 70 <5
* 0731

11
14
13

as

5i

51

\2

■H

. S20

75

- 1711

. 1'-'^

. 5334

. S0tt3

: an

12

J-

1 .

43

- B3*J

75
75

- 1411

. 4iii:i

. 53*7

. 3V76

, coji

11

4^1

-'►

Mi

t^W

. 1116

, «SM

13
11
13
13
13

, 5440

. 43m

»4

88

88

sr,7o

10

40

.'1

si

0*10

75

IWlft

Sj ,:

, 54 :*:i

* 4iw;»

' 33J7

K

52

5*

53

, 94*0
. 07*1

75
75

ihri t

. 44^11

, 5546

. 5035
i 53-*h

4*5
4$tl

8
7

t;

54

.v.i-y.--

75

:

71
74

.40 ^M

. 4347

. 3 ■:> \

. 5741

4253

6

M

to

55

03*7

. "--r"

. irM

570fi

. B033

88

88

m

88
88

. 3TO7

:»

20

II

51

. OW

, gnu

. 4^*0 ;5

. 41*7i }ij

■ *«>

. 40^0 Jj

0010>H; L1

57H0

A446

3154

4

16

f-

57

. 0>M

. 001 1

. 5H13

. m n

3303

3

12

I*

53

, 12*1

. HT71*«

i 5^17

, 7140

. M»

3

8

1}

3>

. 15*0

sin

5^120

n 7500

, 25110

1

4

&i

-.■'

60

o.sohto

74

40 J |-»

035:174

O.037«SJ

37-T'

■is

m.

-*

"J '

L,Co*.

WIT
fhf

15"

Ute

L.Kim.

Jiitr
I5 rt

L. Covet.

L. Cot

n>ir

f.r
15"

L.Ting, |

'

Himrj. ,Def.

**

Hours.

1

nrl'

rtr 1*

r.irl*

J'

114

a

1 J u

e io

-4*

01

ro 1

1*

Digitized by VjOOQIC

54

TABLE U. — LOO. 8INB8, TANg'b, &C

1» 830

1

• — 15" 1"

— 15'

1»-.150

I66O 10*

Diff

Diff

DUE

Houn.

Deg.

L.8in

for
15"
orl*

L.Coaec.

L.Cof.

for
15"
orl'

L.8ec.

L.Taag.

for
15"
orl'

L.Cot.

De*.

Hoar*

M

•

i

i

m

»7

•

3a

9.591878

74
74
74
74
74

0.408122

9.964026

13
13
13
13
13

0.035974

9.6*7852

88
68
88
88
88

0.372148

60

4

1

. 2175

. 7825

. 3972

. 6028

. 8203

. 1797

59

56

8

2

. 2473

. 7527

. 3919

. 6081

. 8554

. 1446

58

52

12

3

. 2770

. 7230

. 3865

. 6135

. 8905

. 1095

57

4*

16

4

. 3066

. 6934

. 3811

. 6189

. 9255

. 0745

56

44

20

5

. 3363

74
74
74
74
74

. 6637

. 3757

13
13
13
13
13

. 6243

. 9606

87
87
87
87
87

. 0394

55

40

24

6

. 3659

. 6341

. 3703

6297

. 9956

. 0044

54

36

28

7

. 3956

. 6044

. 3650

. 6350

.630306

.369694

53

33

32

8

. 4251

. 5749

. 3596

. 6404

. 0655

. 9345

52

28

30

9

. 4547

. 5453

. 3542

. 6456

. 1005

. 8995

51

24

40

10

. 4842

74
74
74
73
73

. 5158

. 3488

14
13
14
13
13

. 6512

. 1354

87
87
87
87
87

. 8646

50

20

44

11

. 5137

. 4863

. 3433

. 6567

. 1704

. 8296

49

16

48

12

. 5432

. 4568

. 3379

. 6621

. 2053

. 7947

48

12

52

13

. 5727

. 4273

. 3325

. 6675

. 2402

. 7598

47

6

56

14

. 6021

. 3979

. 3271

. 6729

.. 2750

. 7250

46

4

33

15

. 6315

73
73
73
73

73

. 3685

. 3217

14
14
13
14
14

. 6783

. 3098

87
87
87
87
87

. 6902

45

97

4

16

. 6609

. 3391

. 3162

. 6838

. 3447

. 6553

44

56

8

17

. 6903

. 3097

. 3108

. 6892

. 3795

. 6205

43

52

12

18

. 7197

. 2803

. 3054

. 6946

. 4143

. 5857

42

4*

16

10

!. 7490

. 2510

. 3000

. 7000

. 4490

. 5510

41

44

20

20

. 7783

73
73
73
73
73

. 2217

. 2945

14
13
14
14
14

. 7055

. 4838

87
87
87
87
87

. 5162

40

40

24

21

. 8075

. 1925

. 2890

. 7110

. 5185

. 4815

39

36

28

22

. 8366

. 1632

. 2836

. 7164

. 5532

. 4468

38

32

32

23

. 8660

. 1340

. 2781

. 7219

. 6879

. 4121

37

*

36

24

. 8952

. 1048

. 2726

. 7274

. 6226

. 3774

36

24

40

25

. 9244

73
73
73
73
73

. 0756

. 2672

14
14
14

14
14

7328

. 6572

87
86
86
86
86

. 3428

35

30

44

26

. 9530

. 0464

. 2617

7383

. 6919

. 3081

34

16

48

27

. 9827

. 0173

. 2562

. 7438

. 7265

. 2735

33

12

52

23

.600118

.399882

. 2507

. 7493

. 7611

. 2389

32

8

56

29

. 0409

. 9591

. 2453

. 7547

. 7956

. 2044

31

4

34

30

. 0700

72
72
72
72
72

. 9300

. 2398

14
14
14
14
14

. 7602

. 8302

86
86
86
86
86

. 1698

30

96

4

31

. 0990

. 9010

. 2343

. 7657

. 8647

. 1353

29

56

8

32

1280

. 8720

. 2288

. 7712

8992

. 1008

28

52

12

33

. 1570

. 8430

. 2233

. 7767

.' 9337

. 0663

27

48

16

34

. 1860

. 8140

. 2178

. 7822

. 9682

. 0318

26

44

20

35

. 2150

72
72
72
72
72

. 7850

. 2123

14
14
14
14
14

7877

.640027

86
86
86
86
86

.359973

25

40

24

36

. 2439

. 7561

. 2067

7933

. 0372

. 9628

24

36

28

37

. 2728

. 7272

. 2012

. 7988

. 0716

. 9284

23

32

32

38

. 3017

. 6983

. 1957

. 8043

1060

. 8940

22

28

36

39

. 3305

. 6695

. 1902

. 8098

. 1403

. 8597

21

24

40

40

. 3593

7-2

. 6407

. 1846

14
14
14
14
14

. 8154

. 1747

86
86
86
86
86

. 8253

20

29

44

41

3882

. 6118

. 1791

. 8209

. 2091

7909

19

16

48

42

. 4170

72
72
72

. 5830

. 1736

. 8264

. 2434

. 7506

18

12

52

43

. 4457

. 5543

. 1680

. 8320

2777

. 7223

17

8

56

44

. 4745

. 5255

. 1625

. 8375

. 3120

6880

16

4

35

45

. 5032

72
72
72
71
71

. 4968

. 1569

14
14
14
14
14

. 8431

. 3463

86
85
85
85
85

. 6537

15

95

4

46

. 5319

. 4681

. 1513

. 8487

. 3806

. 6194

14

56

8

47

. 5606

. 4394

1458

. 8542

. 4148

. 5852

13

52

12

48

. 5892

. 4108

. 1402

. 8598

. 4490

. 5510

12

48

16

49

. 6179

. 3821

. 1347

. 8653

. 4832

. 5168

11

(4

20

50

. 6465

71
71
71
71
71

. 3535

. 1291

14
14

14
14
14

8709

. 5174

85
85
85
85
65

. 4826

16

10

24

51

. 6751

. 3249

. 1235

. 8765

. 5516

. 4484

9

K

28

52

7036

. 2964

. 1179

. 8821

. 5857

. 4143

8

32

32

53

. 7322

. 2678

. 1123

. 8877

. 6199

. 3801

7

28

36

54

. 7607

2393

. 1067

. 8933

. 6540

. 3460

6

14

40

55

. 7892

71
71
71
71

71

. 3108

. 1011

14
14
14
14
14

. 8989

. 6881

85
85
85
85
85

. 3119

5

»

44

56

. 8177

. 1823

. 0955

. 9045

. 7222

. 2778

4

16

48

57

. 8461

. 1539

. 0899

. 9101

* 7562

. 2438

3

12

52
56

58
59

. 8745
. 9029

. 1255
. 0971

. 0842
. 0786

. 9158
. 9214

. 7903
8243

. 2097
1757

2

1

8

4

35

60

60

9.609313

0.390687

9.960730

O.039270

9.648583

0.351417

94

m

•

i

L.C«.

Diff
for
15"

orl*

L. Bee.

I* Sin.

DilT
for
15"

orl*

L.Cotee.

L. Got.

Diff
for
15"
orl*

L.Tuif.

i

«

1

Hour*. jDeg.

Def.

Hot

in.

T»

11

,30

r-

-4* 10

«. 4F

6

W> 4

I* "

Digitized by VjOOQIC

TABLE II. — LOO. SINES, TANo'8, &C.

55

i»

MO

; Hours.

Deg.

L. Sin.

"BIT
for
15"
orl»

P—13" !•— 15' 1> — 150

L.Cotee.

L-Cos.

Diff.

for
15"
orl»

L.Sec

L. Tang.

for
15"
orl*

1550 10*

L. Cot

Deg.

4
8
13
10

90
34
38
33
36

40
44

46
53
56

4
8
13
16

30

34
38
33
36

40
44

48
53

1
3
3

4

5
6

7
8

10
11
13
13
14

15
16
17
18
10

30

31

4
8
13
16

30
34
38
33

40
44

48
53
56

4

8
13
16

30
21

:«

36

40
44
48
53
5\
iO

30
31
33
33
34

35

36
37
38
30

40
41
43
43
44

45
46

47
48
49

50
51
53
53
54

55

56
57
58
59
60

9.609313

. 9597

. 9880
.610163

. 0447

. 0739

. 1013

. 1394

. 1576

. 1858

. 3140

. 3431

. 3703

. 3983

. 3364

. 351

. 3±!3

. 4105

. 4385

. 4665

5503
5781
6060

. 7173

. 7450

7737

. 8004

. 8381

. 8558

. 8834

. 9110

. 9386

. 9603

. 9938
.690313

. 0488

. 0763

. 1038

. 1313

. 1587

1861

. 9135

. 3409

. 9883

. 3356

. 3503

. 3774

. 4047

. 4319

. 4591

. 4863

. 5134

. 5406

. 5*177
9.695948

71
71
71
71
71

71
70
70
70
70

70
70
70
70
70

70
70
70
70
70

70
70
70
70

69
69

68
68

68
68
68
68
68

68
68
68
68
68

68
68
68
68

68

O.390J87

. 0103

. 0130
.389837

. 9553

. 9371

. 8988

. 8706

. 8434

. 8143

. 7860

. 7579

. 7396

. 7017

. 6736

. 6455

. 6175

. 5895

. 5615

5056
4777
4498
4319
3940

. 3384

. 3106

. 9838

. 9550

. 9373

. 1996

. 1719

. 1443
1166

. 0890

. 0614

. 0338

. 0083
.379787

. 9519

. 9937

. 8963

. 8687

. 8413

. 8139

. 7865

. 7591

. 7318

. 7044

6771

. 6498

. 6936

. 5953

. 5681

. 5409

. 5137

. 4866

. 4594

,. 4333
0.374052

9.960730

0674

. 0617

. 0561

. 0505

. 0448

. 0393
0335

. 0379

. 0333

. 0166

. 0109
0053
.959995

. 9938

. 9883

. 9835

. 9768

. 9711

. 9654

. 9596

. 9539

. 9483

. 9435

. 0368

. 9310

. 9353

. 9195

. 9138

. 9081

. 9033

. 8965

. 8908

. 8850

. 8793

. 8734

. 8677

. 8619

. 8561

. 85i):i

. 8445

. 8386

. 8339

. 8371

. 8913

. 8154

. 8096

. 8038

. 7979

. 7931

. 7863

. 7804

. 7745

. 7687

. 7570
. 7511
. 7453
. 7333
. 7334
9.957270

14
14
14
14
14

14
14
14
14
14

14
14
14
14
14

14
14
14
14
14

14
14

14
14
14

14
14
14
14
14

14
14
14
14
14

14
14
14
14
14

14
14
14
15
15

14
14
15
14
14

15
15
15
15
14

15

15
15
15
)5

O.O39370

. 9336

. 9383

. 943U

. 9495

. 9553

. 9608

. 9665

. 9791

. 9778

. 9834

. 9891

. 9948
.040005

. 00(33

. 0118

. 0175

0346

0404
0461
0518
0575

. 0690

. 0747

. 0805

. 0863

. 0919

. 0977
1035

. ion

. 1150
1308

. 1366
. 1333
. 1381
1439
. 1497

. 1555
1614
. 1671
. 1739
. 1788

. 1846
. 1904
1963
. 9031
. 9079

. 9137

. 9196

. 9355

. 9313

• 9373

. 9430
. 9489
9548
. 9607
. 9660
1.049734

9.648583

. 8333
. 92*
. 9603
. 9943

.650381
. 0630
. 0959
. 1997
. 1636

. 1974

9313

. 3350

. 3988

4000
4337
4674
5011

. 5684

. 6030

. 6356

. 6693

. 7098

. 7363

. 7699

. 8034

. 8369

. 8704

. 9039

. 9373

. 9708
.660843

. 0376

. 0709

. 1043
1377

. 1710

. 9043

. 9377

. 9701

. 3043

. 3375

. 3707

. 4039

. 4371

. 4703

. 5035

. 5366

. 5698

. 6039

. 6360

. 6691

. 7031

. 7353

. 7683

. 8013

. 8343
9.668673

85
85
85
85
85

85
85
85
85
84

84
84
64
84
84

84
84
84
84
84

84
84
84
84
84

84
84
84
84
84

84
84
84
83

0.351417

. 1077

. 0737

. 0398

. 0058

.349719

. 9380

. 9041
8703

. 8364

. 8036

. 7688

. 7350

. 7013

. 6674

6337

. 6000

. 5663

. 5330

. 4989

. 4653

. 4316

. 3960

. 3644

. 3308

. 9973

. 9037

. 9301

. 1966

. 1631

. 1396

. 0961

. 0637

. 0399
.339958

. 9694

. 9391

. 8957

. 8633

. 8990

. 7957

7633

. 7391

. 6958

. 6993

. 5961

. 5130

. 5237

. 4965

. 4634

. 4303

. 3971

. 3640

. 3309

. 9979

. 9648

. 9318

. 19P7

. 1657
0.331338

60
59
58*

57
56

55

54
53
53
51

50
49
48
47
46

45
44

43
43

41

40
39
38
37
36

35
34
33
33
31

30

37

Hours.

r>*

L.Cos.

Diff.
for
15"
orl«

L.8ec

L-8in.

Diff
for
15"
orl*

L. Cosec.

L.Cot.

Diff.
for
15"

orl*

L.Tang.

Deg. Hours.

Hi"

1' — 4' lo =

6JO

Digitized by VjOOQIC

56

TABLE n. — LOO. SINES, TANG'S, &C.

1»

950

1» — 15" 1"

—15'

1» — 150

1540

IO*

Diff.

Diff.

Diff.

Hours.

Dog.

L. Sin.

for
15"
or V

L. Cosec.

L.Cos.

for
15"
orl»

L. Sec.

L. Tang.

for
15"
orl*

L-Coc.

Deg.

Hours.

■

i

'

40

9.635948

ft
68

V-

0.374052

9.95727f

15
15
15
15
15

0.043734

9.668673

83
82
82
83
83

0.33 32*

60

19 u» !

4

1

. 6319

. 3781

. 7317

. 2783

. 9003

09!*

5J

Oil

8

3

. 6490

. 3510

. 715h

. 2843

. 9332

061b

58

53 |

13

3

. 6760

. 3340

. 7099

. 3901

. 9661

033!

57

4-

16

4

. 7030

. 3970

. 7040

. 3960

. 9990

. 0010

56

44

30

5

. 7300

67
67

. 3700

. 6980

15
15
15
15
15

. 3030

.670320

83
83
83
83
83

.399680

55

40

24

6

. 7570

. 3430

. 6931

. 3079

. 0649

. 9351

54

3.;

38

7

. 7840

. 3160

. 6863

. 3137

. 0977

. 9033

53

:a '

33

8

. 8109

. 1°91

. 6803

. 3197

1306

. 8694

53

2* '

36

. 8378

1632

. 6744

. 3256

. 1634

. 6366

51

1

40

10

. 8647

67
67
67
67
67

. 1353

. 6684

15
15
15
15
15

. 3316

. 1963

82
83
83
83
69

. 8037

50

»'

44

11

. 8916

1084

. 6635

. 3373

S291

. 770:

49

J" !

48

13

. 9185

. 0815

. 656U

. 3434

. 2619

. 7381

48

12

53

13

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67
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15
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82
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15
15
15
15
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for
15"

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TABLE n. LOO. 81NE8, TANo'8, &C.

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67

1530 10*

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08

TABLE II. — LOO. 8INZS, TANO'l, <fec.

1»

»7°

!• — 15" 1» — 15' 1*— 150

Ultf.
fi.r
IS"

Of l'

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I» 980

1* — 15" 1«

— 15'

1»-»150

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DiflT

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59
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59
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17
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76

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4

15
16

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59
59

59
59
58

. 4845
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. 4922
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17
17
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42
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. 4376

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. 5214

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12

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76

. 8859

16

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. 3906

. 4650

17
17

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76
75

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20

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58

58
58
58
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. 3672

. 4582

17
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75
75

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. 4514

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. 2048

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39

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. 444(5

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. 7650

38
37

32

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. 2970

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17
17
17

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. 2C53

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. 7347

36

24

. 7264

. 2736

. 4309

. 5691

. 2955

75
75

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36

40

25

. 7498

58
58
5-

58

. 2502

. 4241

17
17
17
17
17

. 5759

. 3257

. 6743

35

44
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26
27

. 7731
. 7964

. 2260
. 2036

. 4J73
. 4104

. 5827

. 5896

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75
75

. 6442
. 6140

34
33

52

29

. 8197

. 1803

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. 4161

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32

36

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. 8430

58

. 1570

. 3967

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75
75

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58

1337

. 3899

17
17
17

. 6101

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5236

30

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4

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58
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75

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. 0872

. 376J

. 6239

. 5367

75

4633

28

12

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58
58

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. 3692

. 6308

. 5668

75

4332

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16

34

. 95J2

. 0408

. 3624

17
17

. 6376

. 5968

75
75

. 4032

96

20

35

. 9824

• 58
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. 0176

. 3555

17
17
17

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. 6269

75
75
75

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24

30

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. 348(i

6514

. 6570

3410

94

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37

. 0288

58
58
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. 9712

. 3417

. 6583

. 6871

. 312T

93

32

38

. 0519

. 9481

. 334*

. 6652

7171

. 282''

22

36

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. 0750

. 9250

. 3279

17
17

. 6721

. 7471

75
75

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21

40

40

0982

58
58
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58

. 9018

. 3210

17
17
17
17
17

6790

. 7772

75
75
75
75
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2228

20

44

41

. 1213

. 8787

. 3J41

. 6859

. 8072

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19

48

43

1443

. 8557

. 3072

. 6928

. 8371

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18

52

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. 1674

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. 6997

. 8671

. 132T

17

56

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. 8035

. 2934

. 7066

. 8971

. 102T

16

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57
57
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. 7865

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. 2795

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. 9570

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. 7175

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. 2656

. 7275
. 7344

. 9870
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13
12

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49

. 3055

. 0945

. 2587

. 7413

. 0468

. 9532

11

20
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50
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57
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. 6716
. 6486

. 2517
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17
17
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75
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75

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. 8934

10
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32
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52
53
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. 3743
. 4201

. 6*257
. 6028
. 5799

. 2378
. 2308
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. 7622
. 7692
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1365

. 1664

1969

. 8635
. 833P
. 8038

8

7
6

40

55

. 4430

57
57

57
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17
17
17
17
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. 7831

. 92*1

75
75
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41

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57
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4*7

. 5115

. 5342
. 5113
. 4885

. 2099

. 2029

1959

. 7901
. 7971
. 8041

. 2559
. 2858
. 3156

. 7441

. 7142

6844

4
3

3

55

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9.041819

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9.743752

74
74

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0.256248

1

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for
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54
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54
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18
18

18
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72
72
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53
53
53
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18
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72
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. 2820

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. 5277

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21

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40

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53
53
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53
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. 2394

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19
19
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19
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72
72
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. 5502

. 3321

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53
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. -4199

19
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19
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72
72
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4

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. 1118

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. 5877

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71
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1480

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. 6902

. 2U09

. 7391

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46

a

52

13

. 4561

5439

. 2075

. 7925

24M

. 7514

47

t

56

14

. 4769

. 5231

. 1998

. 8002

. 2771

722*

46

4

5

4

15

16

. 4D77
. 5186

52
5-'
52
52
52

. 5023
. 4814

. 1921
1845

19
19
19
19
19

. 8079
. 8155

. 305b
. 3341

71
71
71
71

. 6944
. 6659

45
44

55

56

8

17

5394

4C06

. 1768

. 8232

362b

. 6374

43

52

12

18

. 5f01

. 4399

. 1691

. 8309

. 3910

(jfc.0

42

48

16

10

. 5009

4191

1614

. 8386

. 4195

5t05

41

44

20

20

. 6017

52
52
52
52
52

. 3983

. 1538

19
19
19
19
19

. 8469

. 4479

71

71
71
71
71

. 5521

40

40

24

21

6224

. 3776

1460

. 8540

4764

523b

39

tf

28

22

6432

. 35t8

. 1384

8bJ6

. 504b

. 4W2

38

32

32

23

. 6H39

. 3361

1307

. 8693

. 5332

466e

37

3tt?

36

24

. 6846

. 3154

1230

. 8770

5616

4384

36

24

40
44

48

25

26
27

. 7059

. 7259

7466

52
52
52
51
51

. 2948

2741

. 2534

1152

1075

. 01198

19
19
19
19
19

. 8848
. 8925
. 9002

. 5900

. 61*4

64bb

71
71
71
71
71

. 4100
. 3bl6
. 3532

35

34
33

20
16
12

52

28

. 7673

. 2327

. 0921

. 9079

. 6752

324ft

32

e

56

29

. 7879

. 2121

. 0843

. 9157

. 7036

. 29b4

31

4

30

. 8085

51
51
51
51
51

. 1015

. 0768

19
19
19
19
19

. 9234

. 7319

71
71
71
71
71

. 2681

39

54

4

31

. 82P1

. 1709

. 0688

. 9312

. 7603

. 23*7

29

5o

8

32

8497

1503

. 0611

9389

. 7880

. 2114

28

52

12

33

. 8703

. 1297

. 0533

. 9467

. 8170

1830

27

4e

16

34

. 8909

. 1091

. 0456

. 9544

. 8453

. 1547

26

44

20

35

. 9114

51
51
51
51

51

. 0886

. 0378

19
19
19
19
19

. 9022

. 8736

71
71
71
71
71

. 1264

25

40

24

36

. 9319

. 0681

. 0300

. 9700

. 9019

. 0981

24

3t>

28

37

. 0525

. 0475

. 02:3

. 9777

. 9392

069b

23

32

32

38

. 9730

. 0270

. 0145

. 9855

. 9585

. 0415

22

20

36

39

. 9935

. 0065

. 0067

. 9933

. 9868

. 0132

21

24

40

40

.7910140

51
51
51
51
51

.979860

.999989

19
19
19
19

19

.070011

.790151

71
71
71
71
71

.909849

20

20

44

41

. 0345

. 9655

. 9911

0089

. 0434

. 9566

19

16

48

42

. 0549

. 9451

. 9833

. 0'67

. 0716

. 9284

18

12

52

43

. 0754

. 9246

. 9755

. 0245

. 0999

. 9001

17

*

56

44

. 0958

. 9042

. 9877

. 0323

. 1281

. 8719

16

4

T

45

. 1162

51
51
51
51

51

. 8838

. 9599

19
20
19
19
20

. 0401

. 1563

70
70
70
70
70

. 8437

15

53

4

46

. 1366

. 8634

. 9521

. 0479

. 1846

. 8154

14

56

8

47

1570

. 8430

. 9442

. 0558

. 2128

7872

13

52

12

48

. 1774

. 8226

. 9364

. 0636

. 2410

. 75-0

12

48

16

49

. 1978

. 8022

. 9286

. 0714

. 2692

. 7308

11

44

20

50

. 2181

51
51
51
51
51

. 7819

9207

19
20
19
20
19

0793

. 2974

70
70
70
70
79

7026

10

40

24

51

. 2385

. 7615

. 9129

. 0871

. 3256

. 6744

9

36

28

52

. 2588

. 7412

9050

. 0950

. 3538

. 6462

8

32

32

53

. 9791

. 7209

. 8972

. 1028

. 3819

. 6181

7

28

36

54

. 2994

. 7006

. 8893

1107

. 4101

. 5899

6

24

40

55

. 3197

51

51
50
50
50

. 6803

. 8814

90
20
20
20
20

. 1186

. 4383

70
70
70
70
70

. 5617

5

20

44

56

. 3400

. 6600

. 8736

. 1964

. 4664

5336

4

16

48

57

. 3*503

. 6397

. 8657

1343

. 4946

5054

3

12

52

58

. 3805

. 6195

. 8578

1422

. 5227

4773

9

8

56

59

. 40U7

. 5993

R499

1501

. 5508

. 4492

1

4

T

60

60

9.7*4210

0.975790

9.9*8421

0.071579

9 795789

0.904211

59

m | •

'

L-Co*.

Difll
for
15"

L.S0C

L.8in.

Diff
for

15"

L.Cotec.

L.Cot

Diff
for
15"

L.Taitf.

'

m

$

Hour*.

Deg.

Dcff.

Houit.

orl'

orl'

orl*

m

IS

It©

r-

-4* 10

— 4«

1

4o

*

Digitized by VjOOQIC

TABUS II. — LOO. 1INE1, TANo'l, &C.

63

3*

33°

1

— 15" 1-

— 15'

1»«150

1*70

9»

Diff

Ditf.

Dilf-

1

Hours.

Deg.

L.8in.

fbr
15"
orl'

L. Cotec.

L. Cot.

for
15"
orl«

L.Sec

L. Tang.

fnr
15'
orl'

L, CoL

Dog.

Hours.

"J" 7 "

'

"

» o

9.7*4210

50
50
50

0.3757 JO

9.938421

SO
20
30
30
30

0.071579

9.795789

:m

0.304211

to

51

4

1

4412

. 5588

. 8342

. 1658

. 6070

:■■-

Tu
"...I

;&jo

51

5>

H

9

4614

. 5386

. 8263

. 1737

. 6351

. :ii 4'J

58

52

12

3

. 4816

. 5184

. 8184

. 1816

. 6632

, 33 8

57

48

16

4

. 5017

50
50

. 4983

. 8104

1896

. 6913

3087

56

44

•20

5

5319

50
50
50
50
50

4781

. 8025

20
30
30
20
30

. 1975

. 7194

TO

in

. 3*06

55.

40

•21

6

. 5420

. 4580

. 7946

. 2054

. 7474

, ^2b

54

36

28

7

. 5622

. 4378

. 7867

. S133

. 7755

To

. e*t5

53

32

32

8

. 582J

. 4177

. 7787

. 2213

. 8036

7U

. 1'JM

52

28

35

9

. 6024

. 3976

. 7708

. 2293

. 8316

70

. 11.84

51

24

40

10

6325

50
50
SO
SO
50

. 3775

. 7629

20
30
30
30
30

. 2371

. 8596

-u

1104

50

20

44

11

6425

. 3574

. 7549

. 3451

. 8877

. 1123

49

16

48

12

. 662**

. 3374

. 7469

3531

. 9J57

111
7,1

08 13

48

12

52

13

. 6837

. 3173

. 73:10

. 2610

. 9437

to

• 70

Don:)

47

8

56

14

. 7027

. 2973

. 7310

. 2690

. 9717

Q&3

46

4

9

15

. 7328

50
50

3773

. 7331

IS

30
30
30

. S769

. 9997

%

, 0003

45

51

4

16

7428

. 2572

. 7151

. 2849

.800277

7:i

4947*3

44

■>•»

«

17

. 762-*
. 782-*

. 2372

. 7071

. 2J29

. 0557

7 1

. W43

43

52

1-2

18

50
50
50

. 2173

. 6991

. 3009

. 0837

:•!

■ lift

42

48

16

19

. 80J7

. 1973

. 6911

. 3089

. 1116

70

6384

41

44

20

28
22
33

20
21
22
23
24

. 8327
. 8427
. 862 >
. 8825
. 9024

50
50
50
50
50

1773
. 1573

1374
. 1175
. 0976

. 6831
. 6752
. 6671
. 65)1
. 6511

30
30
30
20
30

. 3169
. 3248
. 3329
. 3409
. 3489

. 1396
. 1675
. 1955
. 2234
. 3513

TO
70
79

Tu

7='

; 8004
, *325
. *» 45

77tfc
. 7487

40
39
38
37
36

40

Mi
32
28
24

40
44

48
5*

25
23

27

23

. 9321

9422

. 9621

. 98

50
50
50

50
49

. 0777
. 0578
. 0379
. 0180

. 6431
. 6351
. 6270
. 6190

30
30
30
SO
30

. 3560
. 3649
. 3730

. a-iio

. 3793
. 3071
. 3351
. 3630

TO

7m
Tu

■ 7208

. !.<*>

M 19
fl S70

35
34
33
32

20
16
12
8
4

56

23

.730018

.309982

. 6110

3890

. 3908

Tu

. 0&62

31

10

4

-i

12
16

30
31
32
33
34

. 0216
. 0415
. 0M1
. 0311
. 1009

49
49
49
49
49

. 9784
. 9585
. 9387
. 9189
. 8991

. 6029
. 5149

. 5W
. 5788
. 5707

80
30
*Q
30
30

. 3971
. 4051
4132
. 4212
. 4393

. 4187

4466

4745

. 5023

. 5302

TO

70

To
70

70

. 5-13

5534

. 5255

. 4177

. J. . 18

30
29
28
27
36

50

56
52
48
44

20
24

2-i
32
35

35

36
37
38
39

. 1*>S
. 1401
. 1602
. 17 >9
. 1936

49
49
49
49
49

. 8794
. 8516
. 8318
. 8201
. 6004

. 5621
. 5545
. 54u5
. 5384
. 5303

30
SO
30
30
30

. 4374
. 4455
. 4535

. 4616
. 4697

. 5580
. 5851
. 6137
. 6415
. 6693

h'l

09

tVi

4110
441

33*5
. 3307

35
34
33
22
21

40
36
32
28
24

40
44

48
52
53

40
41
42
43
44

. 2113
. 31>0
. 35*7
. 37=>4
. 2380

49
49
49
49

49

. 7807
. 7610
. 7413
. 7216
. 7020

. 5222

514)

. 50i0

. 497>

. 4837

30
30
30
30
30

. 4778
. 485'J
. 4940
. 5021
. 5103

. 6971
. 7249
. 7527
7805
. 8083

69
69

. 1«r>9

. 2751

521771

. 21 ■«

30
19
18
17
16

20
16
12
8
4

11

4

8
12
16

45
46

47
48
49

. 3177
. 3173
. 3.511
. 3715
. 3961

49
49
40
49
49

. 6823
. 6627
. 6431
. 6215
. 6033

. 4816
. 4735
. 4IS5T
. 4572
. 4490

30

30
30
30
20

. 5184
. 52 i5
. 5347
. 5428
. 5510

. 8311
. 8638
. 8916
. 9193
. 0471

69
fl r >

6D

01

. 1K39

1*2

007

15

14
13
12
11

49

56
52
48
44

20
24

2"*
12
35

50
51
52
53
54

. 4157
. 4TVI
. 4549
. 4744
. 4333

49
49
40
4->
49

. 5843

. 5147
. 5451
. 5251
. 5031

. 4401
. 4328
. 4246
. 41<>i
. 4082

30
20
30

:o

30

' . 5511
. 5172
. 5754

. 583.i
. 531ri

. 9748
.8 10025
0101
. 0580
. 0857

419

r, 1
\V>

1. 1
69

0iS2
is rys
0607
B ito
IN 43

10
9

8
7

40
36
32

28
24

11

40
44
4«
52
5 »
•>0

•

55
56
57
58
51
60

. 5115
. 5110
. 5524
. 571 »
. 5HI
9 73-1103

49

4»
4"
49

. 4865
. 4671
. 447'!
. 42=51
. 408'«
0.391811

4001
. 3111
. 3817
. 3755

3671
9.931591

30
30
30
20
.*0

. 5919
. 6081
. 6163
. 6245
. 6327
0.076409

1134

. 1410
. 1687
. 1964
. 3241
9.8135H

m
1.

09

P9T>r,

KPO

t Kin

WW.
0*l47<t3

5

4
3
2
1

48

m

20
16
12
8
4

•

•

/

Diflr

for
15"

Diff

far

L. Cosec.

L. Cot.

Diff.
f»r

Hours. |Deg.

L. Cos.

L. Sec.

L. Sin.

15"

15"

L. Tang. t )^„
1 '

Hour*.

«4*

1

orl*

1

orr

or !•

i;

44-

V ■■

-4* i<

>«-4*

3»J3

3*

Digitized by VjOOQIC

84

TABLE II. — LOO. 8INB8, TANo's, Ac

ft*

330

V

— 15" 1"

— 15'

1»— 15©

1499

r

I

Diff

Dim

out

1

Hour*.

Deg.

L. Sin.

for
15"
orl*

L. Coiec

L.C<m.

for
15"
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L.8ee.

L.Taof

for
15"
orr

UCoc

D«fJ Hoti

«■

•

i

'

.' .

19

9.730101

48
48

48
48
48

0.9438J1

9.993591

90
20
20
20
20

9.07040

9.8 1251*

6fe
6b
69
89

89

0.1874C*

10

41

4

1

. 6303

. 8iKI7

. 350.

. 64tfi

. 27U

. 7*fc

it

8

2

. 649*

. 3502

. 3427

. 6573

. 3071

. 692-

5*

12

3

. 6695!

. 3308

. 3345

. eras

. 3347

. U&s

57

.-

ia

4

. 6881

. 3114

. 3203

. 6737

. 3623

. 6377

56

1

20

5

. 7080

48
46
48
48

48

. 2320

. 3181

21
20
21

21

. 6819

3899

89
0b

*

89

. 610J

55

24

6

. 7274

. 2720

. 30 *

. 6MB

. 4176

. 5824

54

id

7

. 740>

. 2532

. 301<

6984

. 4452

. 554>

53

„

32

8

. 7601

. 233.*

. 2j33

. 70J7

. 4728

. 5272

52

_«

36

9

. 7855

. 2145

. 2fc51

. 7149

. 5004

. 4MH

51

.4

40

10

. 804*

48

48
48
48
48

. ltt-2

. 27G8

20
21
20
21

21

. 7232

9280

09
89
09
80
89

. 478B

58

■

44

11

. 8 '41

. 17.-. '

. 2.i8i)

. 7314

. 5555

. 4445

48

48

12

. 8434

. 15

. &.03

. 7X*7

. 5*31

. 4h*

48

.

52

13

. 8f27

. 1373

. 2520

. 7480

. 6107

. 393

47

•

56

14

. 8820

. 11 tO

. sttto

. 7502

. 63B2

3618

4*

1 i

13

15

. 901?

48

48
48
48
48

. 0087

. 2355

. 7045

. 6C58

89

6*
69
89

. 3343

45

47

4

10

. fr(W.

. 07:14

. 2272

11

. 772f

. 6934

. 30a

44

s

8

17

. 934F

0602

. 21d:

XI

21
21
21

. 7811

. 720J

. 87 1

43

12

18

. P5"0

. 0410

. 210b

. 78.<4

. 74H

. 251i

42

10

19

. 9783

. 0217

. 2023

. 7977

. 77L0

. 8240

41

'*

20

20

. 9975

48

48
48
48
48

. 0025

. 1940

21
21
21
21
21

. 80C0

. 8035

69
89
89
89

1965

48

24

21

.740 »«7

.959833

. It57

. 8143

. 8310

. I6IO

39

2d

22

. 035

. 9641

1774

. 822*

. 8585

. 1415

39

-

32

23

. 0551

. 9449

. lttitl

. 8303

. »0

. 1140

37

30

24

. 0742

. 9258

. 1C07

. 83J3

. 9135

. 6815

36

r

40

25

. 0134

48
48

48
48
48

. 9066

. 1524

21
21
21
21
21

. 8476

. 9410

89

69
89
69
89

. oy©

35 ' •-•

44

20

. 1125

. 8H75

1141

. 8559

. 9t84

. 031.

* 1

48

27

. 13 IP

8684

. 1357

8043

. »*»

. 0041

33

52

23

150"

. 84f>2

. 1274

. 8726

.890234

.17V7i 4

at

50

29

. 1698

. 8302

. 11*0

. 8810

. 0508

. 9412

31

■ 4

14

30

. 1890

48

48
48

. 8110

. 1107

21
21
'1
21
21

8893

0783

89
89
08
68
68

. 9217

30

44

4

31

. 20=0

. 720

1023

. 8977

. 1057

. 8HC

29

8

32

. 2271

. 7729

. 0)3.

. f.Q61

. 1332

. 866f

89

12

33

. 24<*2

47

47

. 7538

. 0856

. 9144

1108

. 6394

27

10

34

. 2052

. 7348

. 0772

. 9226

. 1880

. 8120

26

*«

20

35

. 8342

47
4?
47

. 7158

. 0688

21
21
21
21
21

. 9312

. 2154

68
68
88
68
68

. 7*4f

85

24

36

. 3033

. 6W7

. 0J04

. 931)6

. 8429

. 7571

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. 0431.

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65

340

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Digitized by VjOOQIC

68

TABLE II.— LOO. 1INB1, TANo'l, &C.

8T°

for
15"
orl*

!•— 15" 1- — 15' 1»— 15°

14fto

Tifit
for
15"
orl*

Hours. Deg.

38

90

30

31

31

45
4 46

L.8in.

0.77J463

. 9630

. 97U6

9960

.780133

. 0300

0467

0834

. 0601

. 0368

. 1135
1301

. 1467
1634
1800

1966
. 3139
. 2398
. 3464

3796
3963
3127
33<i3
3457

Hoars.

3788
3953
4118

4447
4613
4776
4941
5105

5433

5597
5761

6089
6353
6416
6579
6743

6906
7069
7233
7394
. 7557

. 7780

. 7883

. 8045

. 8308

. 8370

. 8533
8694

. 8P57
0018

. 9180
9 78934*

Def.

L.COS.

L.Cotee.

43
42
42
43
42

42
42
42
42
42

42
43

43
43
41

41
41
41
41
41

41
41
41
41
41

41
41
41
41
41

41

41 l
4ll

41

41

41
41
41
41
41

41
41
41
41
41

41
41
41
41
41

41
41
41

40

0.3*0537

. 0370

. 0-202

0034

.1119*67

. 9700

. 9533

. 9366

. 9199

. 6032

8533
8366
8200

8034

7868
7702
7536
7370

7304
7038
6873
6708
6543

6377
6313
6047
5883
5717

5553
5388
5224
505?
4«95

4731
4567
4403
4339
4075

3911
3748
3584
3421
3357

3094
2P31
8768
3606
3443

8380

3117

1955

179

. 1630

- 1468
1306
1143

. 0983

0830

O.910658

L.COS.

Ditf.
for
15"
orl*

0.002349 a .

. 2J53 *J

. 2J58 **

. 3063 |J

1967 JJ

. 1873
. 1776
. 1681
1585
. 1490

. 1394
. 1298
. 1,02
1106
. 1010

. 0914

. 0818

. 0722

. 0636

. 05:9

. 0433
. 0337
. 6340
. 0144
0047

.809951
. 9854
. 9757
. 9661
. 9564

. 9467
9370
. 9373
. 9176
. 9079

. 8981
8884
.' 8787
. 8689
. 8593

•* 1»*»

DiflT.
for
15"

orl»

84
84
84
84
84

84
34
84
84
84

84
84
84
34
84

84

84
34
34
84

84
84
34
84
34

84
84
84
84
84

84

84
84
84
84

84

84
34
34
*4
8006 ^
7508 J*
7810 *J
. 7712 £
. 7614 §J

. 7516 «,

• 7418 J*
. 7320 1 g
. 7223 g

. 7123 *

. 7035! u

• <*** H

• 6729 g
. 6631 1 l\

9.806538 **

8495
8397

8301
8104

L.8cc.

L.8*c.

L.8in.

DifF
for
15"

orl*

097151

7747
7842
7937
8033

8128
8224
8319
8415
8510

8702
8798
8894

9086
9182
9378
9374
9471

9567
9663
97110
9856
9953

100049
0146
0243
0339
0436

0533
0630
0727
0824
0921

1019
1116
1313
1311
1408

1505
1603
1701
1799
1896

1994
3092
3190
8388
3386

9484
3582
3*80

8778
8877

L. Tang.

0.877114
. 737'
. 7640
. 7i03
. 816L

. 84%

. 86U1

. 8P53

. 921b

. 947b

. 9741

.880003

02t5

0528

. 07b0

. 1058
1314
1576
183*

. 8101

8625
3887
3148
3410

3672
3934
419*
4457

4719

5503
5765

6036

3975
3074
31 7i
3271
3369
0.103468

L. OMee.

6549
6810
7072
7333

7594
7855
8117
8378
8639

. 8900

. 9161

. 9432

. 9682

. 9943

.800304
. 04*5
. 0725
. 0981
. 1347

. 1507
. 17(*
. 202P
. 8381J
. 254!
0. 802810

L.Cot.

66

16

66

65
65
66
65

65
66
66
65
65

65
65
65
65
65

65
65
65
65
65

65
65
65
65
65

65
65
65
65

65
65
65
65

65
65
65
65
65

65
65
65
65
65

65
65
65
65
65

L Col

O.l 23n*

. *2~
23tO
20.

. 1634

. 1572
. lbO.
104;
. 07b4
. 0522

. 025b
.1199*

9735
94~i2
9210

Def. Hon

10
59
58
57
56

55
54
53

58
51

50
49
48
47

31 «o

5©
5* 1
4ff ,
«« .

40
3b

* ■
24

16

8946 45

668b 44

8434 43

8162 48

31

789b

7637
7375
7113
6852
65,0

5804
5543
5281

. 475U

. 4497

. 4235

3974

. 3713

. 3451

311*0

. 8928
2667

. 8406

. 3145

1883

1682

1361

1100
. 0839
. 0578

0318
. 0057

.109796
. 9535
. 9975
. 9014
8753

. 8493
8333
7971

. 7711

7451

0.107190

Wff
for
15"

orl*

L. Tttftf

41

99

36

2*|
24

20
16
12

«

4

5b
52
48
44

40
36
32

20

16
12

t»
4

56
52
48
44

l-«.

98

36
32
2H
24

20
16
19
H
4

Houi

V -. ¥ lo — 4"

~555 J-

Digitized by VjOOQIC

TABLE II.— LOG. SINES, TANG 9 *, &C.

09

a»

360

I 1

— 15" 1-

— 15'

1» — 150

1410

9»

Dili.

but:

Diff.

Hours.

Oe<.

L.8in.

for

15"
orl*

L. Cosec.

L. Co*.

for
15"
orl*

L.8ec

L. Tang

for
15"
orl*

Lb Cot.

Deg.

Hours.

m

3*

a

'

M

«

9.789342

40

4ti
4G
4(1
4(J

O.910658

9.896532

25
25
25
25
25

0.10346b

9.892810

62
62
62
62
62

O.1O7190

60

87

60

4

1

. 9533

. 04J7

. 6433

. 3567

. 3070

. 6930

59

56

8

2

. 9665

. 0335

. 6334

. 3.KU)

. 3331

. 6669

58

52

12

3

. 9827

. 0173

6236

3764

. 35J1

. 6409

57

48

16

4

9988

. 0012

6137

. 3863

. 3851

. ouy

56

44

SO

5

.1*9014')

40
40
40
40
41

.303851

. 6038

25
25
25
45
95

3962

. 4111

62
62
62
62
62

. 5889

55

40

24

6

. O310

. 9690

. 5939

. 4061

. 4371

. 569S

54

36

2d

7

. 0472

. 9528

. 5840

4160

. 4632

. 536*

53

32

32

8

. 0633

. 9367

. 5741

. 4253

. 4892

. 510*

52

,

28

3d

. 0793

. 9207

. 5641

. 435J

. 5152

. 48k

51

24

40

10

. 0954

4C
4C
4C
4C
4C

. 9046

. 5542

95
95
95
95
95

. 4458

. 5412

62
62
62
62

62

. 4588

50

30

44

11

1115

. 8885

. 5443

. 4557

. 5672

. 4328

49

16

48

12

. 1275

. 87.'5

5343

4657

. 5932

. 4068

48

12

52

13

. 1436

. 8564

. 5244

. 4756

. 6192

. 3808

47

8

56

14

1596

. 8404

. 5144

. 485b

. 6452

. 3548

46

4

33

15

. 1757

4C
40
40
40
40

. 8243

5045

95
95
95
95
95

. 4955

6712

62
62

et

62
62

. 3288

45

27

4

16

1917

. 8083

4945

. 5055

. 6972

. 3028

44

56

8

17

. 9077

. 7923

. 4846

. 5154

7231

, . 276ii

43

52

12
16

18
19

, 2 -'37

. 23J7

. 7763
. 7603

4746
. 4646

. 5254
. 5354

. 7491
. 7751

. 2509

. 224fl

42
41

48
44

20

90

. 2557

40

40
40
40
40

7443

4546

95
95
95
95
95

. 5454

. 8011

62
62
62
62
62

r . I98n

40

40

24

21

. 2716

. 7284

. 4446

5554

. 8270

. 173C

39

36

28

22

. 2?76

. 7124

. 4346

. 5654

. 8530

. 147C

38

32

32

23

. 3035

. 6965

4246

5754

. 8789

. 1211

37

28

36

24

. 3195

. 6805

4146

. 5854

. 9049

| . 0951

36

24

40

25

. 3354

40
40
40
40
40

. 6646

4046

95
95
95
95
95

. 5954

. 9308

62
62
62
62
62

- . 0692

35

20

44

26

3514

. 6486

. 3946

6054

. 9588

. 0433

34

m

48

27

. 3573

. 6327

. 3846

. 6154

. 9827

. 017?

33

12

52

28

. 3832

. 6168

3745

. 6255

.900087

.09391 a

32

8

56

29

3991

. 6009

. 3645

. 6355

0346

. 9654

31

4

34

30

. 4150

40
40
40
40
40

. 5350

. 3544

95
95

95
95

25

. 6456

. 0605

62

62
62
62
62

j . 9392

30

36

4

31

. 4308

. 56921

. 3444

. 6556

. 0864

. 9131

29

56

8

32

4467

5533

. 3343

. 6657

1124

. 887f

28

52

12

33

. 4626

5374

. 3243

. 6757

1383

. 8617

27

48

16

34

. 4784

. 5216

. 3142

. 6858

. 1642

. 8358

26

44

20

35

. 4942

39
39

39
39
39

. 5058

3041

95
95
95
95
95

. 6959

1901

62
62
62
62
62

. 8098

25

40

24

36

. 5101

. 4899

. 2940

. 7060

. 2161

. 7839

24

36

28

37

. 5259

. 4741

. 2839

. 7161

. 2420

. 7580

23

32

32

38

. 5417

. 4533

. 2738

. 7282

. 9679

. 7321

22

28

36

39

. 5575

. 4425

. 2637

. 7363

. 9938

. 7052

21

24

40

40

5733

39
39
39
39
39

. 4267

. 2536

95
95
95
95
95

. 7464

. 3197

62
62
65

65
65

. 6801

20

20

44

41

. 5891

. 4109

24.35

. 7565

3456

. 6544

19

16

48

42

. 6049

. 3951

. 2334

. 7666

3715

. 6285

18

12

52

43

. 6206

. 3794

. 2233

. 7767

. 3973

. 6027

17

8

55

44

. 6364

. 3636

. 2132

. 7868

. 4232

. 576?

16

4

35

45

. 6511

39

39
39
39
33

. 3479

2030

95
95
95
95

25

. 7970

. 4491

65
65
65
65

e5

. 5500

15

25

4

46

. 6679

. 3321

. 1921

. 8071

4750

. 5250

14

56

8

47

. 6836

. 3164

. 1827

. 8173

5003

. 4991

13

52

13

48

. 6W3

. 3007

1726

. 8274

. 5267

. 4733

12

48

16

49

. 7150

. 2850

. 1624

. 8376

. 5526

. 4474

11

44

■20

50

7307

39
33
39
33
39

. 2693

. 1523

95
25
95
95
25

. 8477

. 5784

64
65
65
6)
65

. 4216

10

22'

24

51

7464

. 2536

1421

. 857* 1

. 6043

. 3957

9

36 ,

23

52

7621

. 2373

. 1311

. 8681

. 6302

. 3698

8

32

32

53

7777

. 2223

. 1217

. 8783

. 6560

. 3440

7

28

36

54

. 7934

2066

1115

. 8885

. 6819

. 3181

6

24

40
44

55

51

. 8030
. 8217

39
39
39
39
39

1910
1753

1013
0311

25
25
25
25

i'5

. 8987
. 903>>

7077
7336

65

' 65
65
64
64

. 2923
9664

5
4

20
16

48 57

8103

1537

. <W>

. 9191

7534

2401

3

12

52 58

. R510'

1440

0707

. 9233

. 78W

. 2147

2

8

53 51

8716!

12*4

0S05

. 9395

. 8111

. 18H9

1

4

m

fiO

• 1

60

9 793872

0.90112*

9.890503

0.1034W

9.909369

0.091631

i

24

/

I

Diflf.
for
15"

Diff.

for
15"

Diff

for
15"

Hours* Deg.

L.Cos.

L. Sec

L.8in.

L. Cosec. I

L. Cot.

L. Tang.

Deg.

Hoars.

1

'

orl«

orl*

f

1

orl»

8»

12

8e

1' =

4» IO

-4"

ft

to a

»

Digitized by VjOOQIC

70

TABLE II.— LOO. SINKS, TANft'S, &C.

»»

390

1

mm 15" 1»

— 15 1

]* — ISO

140*

•»

Diff.

Diff.

Diff

j

Hoars.

D*.

L. Sin.

for

15"
or 1*

UCosec.

L.Co*

for
15"
orl*

L-Sec

L. Tang.

for

15"
or r

L.Cot.

»«*.

Boars. [

•

•

t

S3

•

36

9.793872

a.

39
39
39
39

o.aoiife

9.890503

38
35
38
85
88

0.109497

9.908369

65

64
64
64
84

0.O91631

60

69 ;

4

1

. 9028

. 0J72

. 0400

9600

. 8638

1372

59

y

8

3

. 9184

. 0816

. 03J8

. 9703

8886

J114

58

52

13

3

. 933U

. 0661

. 0195

. 9805

. 9144

. 025t

57

4r

18

4

. 9495

. 0505

. 0093

. 9907

. 9403

• 059b

58

44

30

5

. 9651

39
39
39
39
39

. 0349

.889991

85

86
88
88
86

110009

. 9560

M

84
64
84
64

. 0340

59

«!

24

6

. 9806

. 0194

. 9888

. 0113

. 9918

. 0983

54

* i

38

7

. 996V

. 003"

. 9785

. 0315

.910177

.08983:1

53

52

33

8

.800117

.199883

. 9683

. 0318

. 0435

. 9565

58

2* '.

38

9

. 0372

. 9738

. 9579

. 0431

. 0893

. 9307

51

24 '.

40

10

. 0437

39
39
39
39
39

. 9573

. 9476

88

86
86
88
88

. 0534

. 0951

64
64
64
64
64

. 9049

59

»l

44

11

. 0582

. 9418

9373

. 0637

. 1S0J

. 8791

49

It

48

19

. 0737

. 93ft3

. 9370

. 0730

1467

. 8533

48

i:

53

13

. 0893

. 9108

. 9167

. 0833

1735

. 6475

47

* '

56

14

. 1047

. 8953

. 9064

. 0936

. 1983

. 8917

46

4

37

15

. 1301

39
39
36
38
38

. 8799

. 8961

86
86
86
88
86

. 1039

. 8840

64
84
84
84
84

7760

45

8)3

0t

4

18

. 1356

. 8644

. 8858

. 1143

. 8498

. 7503

44

56

8

17

. 1511

. 8489

. 8755

1345

. 8756

. 7844

43

52

13

18

. 1665

. 8335

. 8651

1349

. 3014

. 0986

49

4rf

16

19

. 181!)

. 8181

. 8548

. 1453

. 3871

6739

41

44

30

30

. 1973

38

38
38
38
38

. 8037

8444

88
88
88
88
88

. 1558

. 3539

84
84
84
64
84

. 6471

49

40

34

21

. 3138

. 7873

\ 8341

1659

. 3787

. 6813

39

* 1

33

S3

. 3382

. 7718

. 8337

. 1763

. 4045

. 5955

38

'.t?!

33

33

. 3436

, 7564

. 8134

1866

. 4302

. 589r

37

*!

38

34

. 8590

. 7410

. 8030

. 1970

. 4580

. 5440

38

**

40

35

. 2743

38
38
38
38
38

. 7357

. 7936

88
86
88
86
86

. 8074

. 4817

84
84
84
84
84

. 5183

3$

-20 |

44

26

. 3897

. 7103

. 7832

. 8178

. 5075

. 4935

34

lo

48

27

. 3050

. 6950

. 7718

. 3383

5332

. 46t*

33

12 1

52

28

. 3804

. 6796

76M

. 3386

. 5590

. 4410

38

r

56

23*

. 3357

. 8843

. 7510

. 8490

. 5847

. 4153

31

4

38

30

. 3511

38

38
36
38
38

. 8489

. 7406

88

36
88
38
86

. 8594

. 6105

84
64
64
64
84

. 3895

30

aa

4

31

. 3664

. 6336

. 7303

. 3698

. 8362

. 3838

39

5t»

8

33

. 3817

. 6183

. 7198

. 8802

. 8819

. 3381

38

52

13

33

. 3970

6030

.. 7093

. 8907

. 8877

. 3133

37

4<-

16

34

. 4133

. 5877

. 6989

. 3011

. 7134

. 386t>

88

44

30

35

4376

38

38
38
3«
38

. 5724

. 6885

36
36
38
86
38

. 3115

. 7391

84
84
84
64
64

. 880"

35

40

34

38

. 4438

. 5573

. 6780

. 3330

7648

. 835"

34

3fi

3d

37

- 4581

. 5419

. 6676

. 3334

. 7905

. 8095

23

J?

33

38

. 4734

. 5366

. 6571

. 3439

. 8163

. 1837

23

>

36

39

. 4886

. 5114

. 6486

. 3534

. 8430

. 1580

31

24

40

40

. 5039

38

38
38
38
38

. 4961

. 6363

88
38
28
88

. 3638

. 8877

64
84
84

. 1333

80

30

44

41

5191

4809

. 6367

. 3743

. 8934

10W

19

1ft

48

49

. 5343

. 4657

. 6153

. 3848

. 9191

. 0800

18

12

52

43

. 5495

. 4505

. 6047

. 3953

. , 9448

64

. 0553

17

p

56

44

. 5647

. 4353

. 5943

38

. 4058

. 9705

84

. 0995

16

4

39

45

5799

38
38
38
38

38

. 4301

. 5837

88
88
88
36
86

4163

. 9963

84

84

. 0038

15

8)1

4

46

. 5951

. 4049

. 5733

. 4368

.990319

.OT9781

14

56

8

47

. 6103

. 3897

5637

. 4373

. 0476

84
64

. 9534

13

52

13

48

. 6354

. 374f

. 5531

4479

0733

. 9367

13

41*

16

49

. 6406

. 3594

. 5416

. 4584

. 0990

64

. 9010

11

44

30

50

. 6558

38
38
38
38
38

. 3443

. 5311

36
88
96
38
88

. 4689

1847

64

8753

10

40

34

51

. 6701

. 3391

. 5306

. 4794

1503

64

. 8497

9

3fi

38

53

. 6860

3140

. 5100

. 4900

1760

84

. 8940

8

*K>

33

53

. 7011

. 8981

. 4994

. 5006

. 8017

64

7983

7

■*-

36

54

. 7163

. 8837

. 4889

. 5111

. 8874

64

. 7736

6

40

55

. 7314

38
38
3»
38
38

. 8686

. 4783

88

38
38
26
88

. 5317

. 8531

64

. 7489

5

f

44

56

. 7465

. 3535

. 4678

. 5332

. 8787

64

. 733

4

IA 1

48

57

. 76l«

. 8384

. 4573

. 543 a

3044

64

. 695*

3

,: i

53

58

. 776P

. 3234

4466

. 5534

3300

84
64

6700

3

M

5*

53

. 7^ 7

. 20«3

. 4360

. 5640

. aw-

. 6443

1

4 |

39

60

60

9909068

0.191933

9.884354

0.115746

9.98)3814

0.078186

9)0

m

•

'

Diff.
for

15"

Diff.
for
15"

ntff.

for
15"

•

m

1

(Toon.

D«f.

L.CM.

L.B4C

L.S40.

L-Cotec

L.Cot.

L.T»if.

D«*

Horn 1

.

orl«

l«l'

orr

1

lA

1

W>

1'-

-4' lO

— 4*

w

to

8»

Digitized by VjOOQIC

TABU II.— LOO. SINES, TAJio'8, &C.

71

» 40©

V — 15" 1" — W 1» — 150

for
15"
or !•

139o

D*.

I* Sin.

40

41

49

4
«

u
16

20

24
*J
J2
36

40
44

46
52
56

4
8
13
ltf

•20
24
28
32
36

40
44

44
52
56

4
8

19

16

30
24

2-<
32
35

43

44

48

56

4

8
12
16

20
24
2-<
32
36

40
44
48
52
5i

1

s

3

4

5

6

7
8
9

il
IS
13
14

IS
10
17
18
IS

41
43*
43
44

45

46
47
48
40

50
51
53
53
54

O.8OJ0.8

. 8318

. 836u

. 8519

. 8669

. 8619

. 8J69

. 91 19

. 9909

. 9419

. 9569

. 9718

. 9868
410017

. 0167

. 0316

. 0465

. 0814

. 0763

. 0912

. 1001

. 1210

. 1358

. 1507

. 1658

. 1804

. 1952

. 3100

. 3348

. 83J6

. 3544

. 3092

. 3340

. 8188

. 3135

. 33*3

. 3430

. 3578

. 3725

. 3373

. 4019

. 4166

. 4313

. 4460

. 4007

. 4753

. 4900

. 504 *

. 5193

. 5333

. 54R5

. 5632
5778

. 5934

. 6070

. 6215

. 63ftl

. 6507

. 6652
6798
0.816943

Diff
for
15"
or l"

37
37
37
37
37

37
37
37
37
37

37
37
37
37

37

37
37
37
37
37

37
37
37
37
37

37
37
37
37
37

37
37
37

37
37

37
36
36
38
36

36
36
36
36
36

LGofM.

101332
1783
1631
1481
1331

1181
1031
0881
0731
0581

0431
Q3d3
0133
.189 P3
9433

9684
9535
9380
9337

9088

8790
8643
8493
8344

8196
8048
7900
7752
7604

7456
7308
7160
7019
6865

6717
6570
6432
6375
6134

5981
594

L.CO*.

0.884354

. 4148

. 4043

. 3936

. 3cJ39

3733

. 3d 17

. 3510

. 3404

. 33J7

. 3191

. 308

. 8978

. 3571

. 8764

. 8557

. 355(1

. 3443

. 83W

554

5347
5100
4954
4807
4661

4515
4368
4333

4076
3930

3785
3039
3493
3348
3103
183057

. 8131

. 8014

. 1907

. 1799

. 1092

. 1584

. 147

. 136!

. 12 I

. 11£

. 1045

. 0337

. 0**i

. 073*

. 0613

. 0505

. 03)7

. 03*»

. 0U0

. 0072

.873961

. 9855

. 974(3

. 963?

. 9533

. 9430

. 9311

. 9302

. 0033

. 8984

. 8875

. 8706

. 8657

. 8547

. 8438

8338

. 8919

. 8103

. 7999

. 7890
0.8 17780

for
15"
orl«

86
86
3d
37

86

86
87
37
87
87

87
87
87
87
87

87
87
87
87
37

87
87
87
37
37

87
87
37
37
87

87
37
37
37

87

87
37
37
37
87

87
87
87
87
87

87
87
87
87
87

87
37
87
87
87

87
87
87
87
87

L.Sec

0.115746

5552

5358
0064
6171

6377
6383
6490
6536
6703

6803
6916
7032
719<

7336

7343
7450
7557

7664
7771

7879
7986
8033
8301
833;

8416
8523
8631
8739
8847

0170
9878
933?

9405
9603
9711

190037
0145
0254
0363
0471

0580
0689
0798
0907
1016

1135
1334
1343
1453
1563

1673
1781
1891
3001
3110
1*3330

L. Tang.

0.093814

4070
. 4327
. 4583
. 4840

. 5096

. 5353

. 5J09

. 5865

. 6133

6378
. 6634
. 6890
. 7146
. 7403

. 76V

. 7915

. 8171

. 8427

. 8940

. 9196

. 9451

. 9708

. 9964

.030320
. 0475
. 0731
. 0387
. 1343

. 1499
1755

. 8010
8266

. 3522

. 3778
3033

. 3389
3545

. 3800

. 4056
4311

. 4567
. 4893
. 5078

. 5333

. 5589

. 5844

. 6100

. 6355

. 6610

. 6866

. 7131

7377

. 7887
. 8142
. 8398
. 8653
890*
0.039163

64
64
64
64
64

64

64
64
64

64
64
64
64
64

64
64
64
64
64

64
64
64
64
64

64
64
64
64
64

64
64
64
64
64

64
64
64
64

64
64
64
64
64

64
64
64
64
64

64
64
64
64
64

64
64
64
64
64

L.Col.

0.0*6186

. 5330

. 5673

. 5417

. 4160

. 5904

. 4648

. 4391

. 4135

. 3878

. 33»i»

. 3110

. 8 J 54

. 3537

. 8341

. 3085

. 1833

. 1573

. 1317

1060

. 0i04

. 0549

. 03*

. 0036

.069780

. 9535

. 9369

. 9013

. 8757

. 8501

. 8345

. 7930

. 7734

. 7478

7333 35
6967 34

»eg.

6711
6455
6*00

5044

5683
5433
5177
4933

4667
4411
4156
3300
3645

. 3134

. 8879

. 9633

. 8368

. 3113

1858

. 1603

. 1347

1092

0.080837

Hours

10

10

18

5i
4C
44

40
Jo
33
28
A

30
16
13

8

4

56
53
48
44

40
36
33
28
24

20
16
3
8
4

56
52
48
44

40

36
33
28
24

20

16

17

56
52
48
44

40

36
33
38
34

30
16
13

8

4

Hours. Deg.

I»Cm.

8i iW>

mm

for
15"
orl*

L.8M.

L.Sin.

lift
for
15"
or !•

L.Comc.

L. Cot,

Oiff

for
15"
orlM

L. Tang.

Deg. Hour*.

V—4* lo-4«

~T& »r

Digitized by VjOOQIC

72

TABLE n. — LOO. SINKS, TANo's, &C.

«o

1*— 15" !■ — 15' 1»— 150

1M«

jJirt".

fr£

TRI"

i

Itodm*

D**.

L. Sin.

for
15"
orl'

L. Cosec.

L.Cos.

for
15"
orl*

L.Sec

L.Tanf.

for 1

15"

orl*

L-Cot.

De* lb -•

n:

»

'

'

13

44

g

D.HliV.ltf

30
30
30

<> 183057

9.877780

27

27
27

O.132220

9.939163

64
64

64
64

o.ooMr.

66

-i

1

70*?

2912

. 7670

2330

. 9418

05d

59 ■ '

I

3

, 7233

. 2767

. 7500

2440

. 9673

032:

50

12

3

* 7*ra

, 2622

. 7450

2550

992b

0072

57

H.1

4

. 7534

30
30

2476

. 7340

27
27

. 2660

.940184

.059eib

56 ' *

to

M

5

7
8

, 7009

795fl
i BIOS

30
:io

30
30

3-

2331

2187

. 2042

1897

. 7230

7120

. 7010

. 6899

27

27
28
28
28

. 2770
2880

. 2990
3101

. 0439
0693

. 0048
1204

64
64
64

64
64

. 9561
. 9307
. 9052

. 87tt>

55 *

54

53

51

;j-j

fl

. hji;

1753

. 6789

. 3211

1458

8542

51

40

10

- K4 >3

30
•M
30
30
30

1608

. 6678

28
28
28
28
28

3322

. 1714

64

. e28l

31

11
48

11

. S5.1A

1464
1319

. 6508
. 6458

. 3432
. 3542

. 1968
2323

. 8JO*
. 7777

49
48

->±

n

. 8825

1175

. 6347

3653

2478

75*

47 ,

50

14

8009

i 1031

6236

. 3764

2733

. 72*7

46 1

f">

15

. 0113

30
30
33
30
30

. 0887

. 6125

28
28
28
28
28

. 3875

3988

6*

64
64
64
64

. 7013

45 1*

■1

H

13

16
17

, 9257

. NftJ

. P515

. 0743

05>9

. 0455

6014
. 5904
. 5793

. 3986
4096
4207

3243

3497

. 3752

6757
. 65&i
. 624fi

44
43
42

u

10

. Bdeu

0311

. 5682

. 4318

4007

5393

41

-JO

20

. 9833

36
30
30
30
30

. 0167

. 5571

28
28
28
28
28

. 4429

. 4262

64
64
64
64
64

5738

• 1 ':

88

21
33

007(1

.eaorso

. 0024
.179880

. 5459
. 5348

. 4541

. 4652

. 4517
4772

5483
. S23r

* ! -

32

33
34

. Q2KI
0406

9737
. 9594

. 5237
. 5125

. 4763
. 4875

. 502h
. 5281

4*74
. 471.

«: ::

40
11

35
30

, 0L2WO
, 0503

30

S

30
30

. 9450
. 9307

. 5014
. 4903

28
28
28
28
28

. 4986
. 5097

. 5536
. 5790

64
64

64

64

4464

4210

35 ' •*
34 '

!■■»

37

. 0836

9164

. 4791

. 5209

6045

. 3.55

33

5*

jW

. OttO

. 9021

. 4680

. 5320

. 6299

3701

»

50

23

i urn

. 8878

. 4568

. 5432

. 6554

344U

31

'

46

i)

30

. 12P5

30
30
30
30
36

8735

. 4456

28
28
28
28
28

5544

6809

64
64
64
64
64

. 3191

9 "\

1

31

1407

8593

4344

. 565G

7063

81»37

29 1 '

-

33

1550

8450

. 4232

. 5768

. 7318

. 26*9

*

U

XI

. 18ft

8307

. 4121

5879

. 7572

. 24**

27 1 r

In

:«4

- 1635

8165

. 4009

. 5991

. 7826

. 2174

"1 "

J£l

35

]G7fl

35

35
35
33

35

. 8022

. 3897

28
28
28
28
28

. 6103

. 8081

64
64
64

Si

. 1919

25 ! •

-.:■;

3tf

* 2130

7880

. 3784

6216

. 8336

1664

24 >

^

37

. Ml

t 7738

. 3672

6328

. 8590

1410

23

*

30

3d
3tf

* 3404
9546

7596
. 7454

. 3560
3447

6440
. 6553

. 8844
. 9099

1156
0001

22
21

10

40

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. 3335

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63
63
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41

mm

. 7170

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6777

. 9807

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35
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. 2885

. 7115

. 0371

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35

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28
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i 6462

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63
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. 424£

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35

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m

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. 5614

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. 8019

. 2405

63
63
63

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53
54

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35
35

. 5473
; 5332

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1755

81 32
8245

2659
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. 7341
. 7087

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m

. 4*08
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35

. 5192
5051

1641
1528

28

28
28
29
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. 8359
. 8472

. 3167
. 3421

63
63
63
63
63

. 6833
. 657*

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57
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59

. 5*230
. 5370

35
35
35

. 4910

. 4770

4630

1415
1301
1187

. P5 J 5
. 8699
. 8813

. 3675
. 4183

6325
6071
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3
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9.871074

0.1248926

9 954437

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nor

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for
15"

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L.Sln.

for
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for
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nul *"

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Digitized by VjOOQIC

TABLE II.— LOO. SINES, TANG 1 8, <fec.

73

5»

490

1

• — 15" 1"

-15'

1»-150

1370

9*

Dirt:

»>,<:

I)-,l1

mirm.

Dot.

L Bin-

for
15"
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L. C«cc

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for
15"
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15"
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L- CM.

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1

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33
35
35

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38
28
38
28 ,
33

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0.954437

63
U3
ra

03

0.045563

141

11

60

4

1

. 5651

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. 0960

9040

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♦ 5399

59

56

8

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. 5701

* 4209

. 0846

. 9154

. 4045

. 5ik55

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52

1*

3

i 5031

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. 5100

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57

48

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. 3928

. 0618

. 9399

. 5454

4546

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44

90

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35
35
35
35
35

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38

38 '
20
28
38

. 9496

. 5707

63
63
63
63
63

4393

55

40

44

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3640

, 0390

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, 4039

54

36

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7

+ 0401

, ism

. 0276

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0215

. 37S.J

53

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8

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3370

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53

m

36

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35

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35
35

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50

20

44

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9474

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35
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, 9360

39
29
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. 89*6

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11

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, 3255

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, 2116

9130

0870

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48

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, 1B38

. 8900

1100

i 9262

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. 0738

11

44

90

30

, 83oi

35
35
35
35
34

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. B785

39

30
29

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39

. 1215

+ 9516

63

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40

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34

31

. 8430

. P.j'.i

. 8670

1330

. 9769

63
63

, 0231

19

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28

33

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. 8555

1445

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38

32

32

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63

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34

34
34
34
34

. 1007

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29
39
29

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► 1701

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63
63
63
63
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. 4335

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5311

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, 8394

6004

. 3906

5603

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. 1398

16

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91

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30
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52
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5770

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♦ 610*

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63

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34
34
34

34

. 7985

4 5651

4347

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13

12

48

. 3153

. 7^48

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63
f3

. 31*4

12

16

49

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. 7712

. 54 r9

. 45*1

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11

20

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. 2425

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31

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29
29
29
29
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. 2833

34
34

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5050

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63

. 3117

7

36

54

Ml

7031

4-13

, 5 167

. ^136

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. 1864

6

40

55

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mg

. 4716

29
99
29
29
20

p <52*4

, 8380

63
63
63
63
63

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5

26
16
13

44

56

1941

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6751

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k 8643

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4

48

57

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74

TABLE II.— LOG. SINES, TANG's, &C.

m

430

]

• — 15" 1-— 15' 1» — 150

19«o

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. 3538

30
30
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63
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63

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n

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34
34
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63
63
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63

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for

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l~ T*n f

*

Flour*

Hef.

D, t

itoi«

|l

|1

,*s

V -

4* 1$

- r

■

P

m

Digitized by VjOOQIC

TABLX II. LOO. SINES, TANG V 8, &C.

flft

440

1

• — 15" 1"

-15'

1» — 150

I350 -9*

thtt 1

L.COMC.

L.OM.

Uiff.

bit

warm.

Deff.

L.8io.

for
15"

for
15"
orl*

L.8r*.

L.Tang.

for
15"
or V

L.Cot.

Dog.

Hon

■

•

o

$

i

*

«

9.841771

33
33
33

0.1 6*339

9.856934

30
30
30
30
31

0.143066

9.984837

63
63
63
63
63

0.015163

60

8

4

1

. 1903

80U8

. 6813

. 3188

. sono

. 4910

59

8

3

3033

. 7907

. 6690

. 3310

. 5343

. 4657

58

1*2

3

. 3164

. 7836

. 6568

. 3433

. 5596

. 4404

57

16

4

. 3994

33

33

. 7706

.. 6446

. 3554

. 5848

. 4153

56

30

5

. 9434

33
39

7576

6333

31
31
31
31
31

3677

. 6101

63
63

8

63

. 3899

55

34

6

. 3555

. 7445

. 6301

. 3799

. 6354

. 3648

54

-28

7

. 3885

. 7315

. 6078

. 3993

. 6607

3393

53

33

8

. 9816

39

. 71?4

. 5956

. 4044

. 6860

. 3140

58

36

. 3946

39

39

. 7054

. 5834

. 4166

. 7113

. 3888

51

40

10

3076

33
33
33
33
33

. 6934

. 5711

31
31
31
31
31

. 4390

. 7365

63
63
63

. 3635

50

44

11

. 3906

. 6794

. 5588

. 4413

. 7618

3383

49

4S

13

. 3336

. G664

. 5465

. 4535

. 7871

3139

48

5*3

13

. 3465

6535

. 5343

. 4658

. 8133

63

1877

47

56

14

. 3595

. 6405

. 5319

. 4781

. 8376

63

1634

46

ST

15

. 3735

39

. 6375

. 5096

31
31
31
31
31

. 4904

. 8639

63
63
63
63

. 1371

45

8

4

16

. 3855

. 6145

. 4973

. 5037

8683

. 1118

44

8

17

. 3084

32
39
33
39

. 6016

. 4850

. 5150

. 9134

0666

43

13

18

. 4114

. 5886

. 4737

5373

. 9387

0613

48

16

19

. 4343

. 5757

. 4603

. 5397

. 9640

63

0360

41

30

30

4373

5637

. 4480

31
31
31
31
31

. 5530

. 9893

63
63

. 0107

40

34

31

. 4503

33
33

. 5496

. 4357

. 5643

.990145

.009855

30

38

33

. 4631

. 5369

4333

5767

0308

63

9603

38

33

S3

4760

39
33
39

. 5340

. 4109

. 5891

. 0651

63

9349

37

3ft

34

. 4880

. 5111

. 3988

. 6014

. 0903

63

9097

36

40

SS

. 5018

33

. 4989

. 3863

31
31
31
31
31

. 6138

. 1156

63

. 8844

35

44

36

. 5147

. 4853

. 3738

6309

. 1400

63

8591

34

48

37

. 5376

38

4734

. 3614

. 6386

1663

63
63
63

8338

33

53

38

. 5404

39

. 4596

. 3490

. 6510

. 1914

8086

38

56

90

. 5533

39
39

. 4467

. 3366

. 6834

. 8167

7833

31

58

30

5069

. 4338

. 3343

31
31
31
31
31

. 6758

. 8430

63

. 7580

30

8

4

31

. 5790

39

. 4310

. 3118

. 6883

. 3673

63

7338

89

8

33

. 5919

33

. 4081

. 3994

. 7006

. 3935

63

7075

38

13

33

. 6047

39

. 3953

. 3869

. 7131

. 3178

63

6833

37

16

34

. 6175

39

33

. 3835

. 3745

. 7355

. 3430

63

. 6570

86

30

35

6303

. 3697

. 3630

31
31
31
31
31

7380

3663

63

. 6317

35

34

36

6433

33

3568

. 3496

. 7504

. 3936

63

6064

34

38

37

. 6560

39

3440

. 9371

7639

. 4189

63

5811

83

33

38

6686

33

3319

. 3347

. 7753

. 4441

63

5550

38

36

39

.' 6816

33
33

. 3184

. 3133

7878

. 4694

63

5300

31

40
44

48
53
56

40
41
49
43
44

. 6044
. 7071
. 7199
7337
. 7454

33
33
33
33
39

. 3056
. 3939
. 3801
. 3673
. 3546

. 1997

. 1873

. 1747

1633

1497

31
31
31
31
31

8003
. 8138
. 8353

8378
. 8503

. 4947
. 5199
. 5453
. 5705

. 5957

63
63
63
63
63

: SR

. 4548
. 4395
. 4043

30
19
18
17
16

59

45

7583

. 3418

. 1373

31
31
31
31
31

8638

. 6810

63

. 3790

15

1

4
8
13

46

47
48

. 7709

. 7836

7964

33
33
33

. 3391
. 3164
. 3036

. 1346
. 1131
. 0906

. 8754
. 8879
. 0004

. 6463
. 6715
. 6968

63
S

3537
. 3385
. 3033

14
13
13

16

40

. 8001

33
33

. 1900

. 0870

. 9130

. 7331

63

3779

11

30

50

8318

1783

. 0745

31
31
31
31
31

. 9955

. 7473

63

. 3537

10

34

51

. 8345

33

1655

. 0619

. 9381

. 7736

63

9374

9

38
33

53
53

. 8473
8599

33

33

. 1538
1401

. 0493
. 0368

. 9507
. 9633

. 7979
. 8331

63
63

9031

. 1769

8

7
6

36

54

. 8736

33
39

. 1374

. 0343

. 9758

8484

63

1516

59

40
44

48
53
56
60

55
56

57
58

50
60

. 8853
. 8979
. 9106
9333
. 9358
9.849485

33
33
33
33
33

. 1147
1031

. 0894
0768

. 0643
0.150515

. 0116
.849990

. 9864

9737

. 9611

9.840485

31
31

31
31
31

. 9884

.150010

. 0138

0363

0389

O.150515

8737
. 8989
. 9343
. 9495
. 9747
0.000000

63
63
63
63
63

. 1363
. 1011
. 0758
. 0505
. 0353
0.000000

5

4
3

9
1

O

m

a

t

LCcm.

Diff.
for
15"
orl*

L.80C

L.8in.

Diff
for
15"
orl»

L.OOMC

L.COC

DUE
for
15"
orl*

L. Tang.

*

a

D*

Soon.

D*

Hoi

" 8*

1&

13

1'-

■ 4» I©

-4"

"* 4

50 —

8\

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE III.

PROPORTIONAL LOGARITHMS.

Digitized by VjOOQIC

TABLE HI.— PROPORTIONAL LOGARITHMS.

79

II

o° c

OO v

OO 2'

OO 3'

OO 4'

OO 5'

OO 6'

oo r

OO 8'

it

9

9.9553

1.9549

1.7789

1.6539

1.5563

1.4771

1.4103

1.3522

1

4.0334

.3481

.9506

.7757

.6514

.5549

.4759

.409)

.3513

1

3

3.7334

5410

.9471

.7734

£406

£534

.4747

.408)

.3504

2

3

.5563

5341

.9435

.7710

.6478

£590

.4735

.4071

.3495

3

4

.4314

5979

.9400

.7686

.6460

£508

.4793

.4061

£486

4

5

3345

5905

5

6

.9553

.9139

.9331

.7639

.6495

£477

.4699

.4040

£468

6

7

.1883

5073

.9996

.7616

.6407

£463

.4688

.4030

.3459

7

8

.1303

5009

.9969

.7593

.6390

£449

.4676

.4020

.3450

8

9

.0793

.1946

.9998

.7570

.6379

£435

.4664

.4010

.3441

9

10

.0334

.1883

.9195

.7547

.6355

£491

.4659

.4000

.3439

10

11

9.9990

.1823

.9169

.7594

.6338

£407

.4640

£989

.3493

11

13

.9549

.1761

.9198

.7501

.6390

£393

.4699

£979

.3415

19

13

.9195

.1701

.9098

.7479

.6303

£379

.4617

£969

£406

13

14

.8873

.1649

.9063

.7456

.6286

£365

.4606

£959

£397

14

15

.8573

.1584

.9031

.7434

.6969

£351

.4594

£949

.3388

15

16

.8993

.1596

8999

.7419

.6953

£337

.4589

£939

.3379

16

17

.8030

.1469

wtn

.7390

.6935

£394

.4571

£999

£371

17

18

.7789

.1413

.8935

.7368

.6918

£310

.4559

.3919

.3369

18

19

.7547

.1358

.8904

.7346

.6901

£998

.4548

.3910

.3353

19

90

.7394

.1303

.8873

.7394

.6185

.5283

.4536

.3900

.3345

90

SI

.7119

.1949

.8849

.7309

.6168

.5969

.4595

.3890

.3336

91

93

1010

.1196

.8811

.7981

.6151

£956

.4514

.3880

.3397

99

93

.6717

.1143

.8781

.7959

.6135

£949

.4509

£870

.3319

93

94

.6533

.1091

.8751

.7938

.6118

£999

.4491

£860

£310

94

SS .6355

.1040

.879!

.7917

.6103

.5915

.4480

£851

£301

95

90 .6185

.0989

.8691

.7196

.6085

£909

.4468

.3841

£993

96.

97 .6091

.0939

.8661

.7175

.6069

£189

.4457

£831

, .3984

97

98

.5863

.0889

.8639

.7154

.6053

£174

.4446

.3821

3976

98

99

.5710

.0840

.8602

.7133

.6037

£163

.4435

£813

£967

99

30

.5563

.0799

.8573

.7119

.6021

£149

.4494

.3802

£959

30

31

.5491

.0744

.8544

.7091

.6005

£136

.4412

.3792

.3950

31

39

.5983

.0696

.8516

.7071

.5989

£193

.4401

.3783

£949

39

33

.5149

.0649

.8487

.7050

£973

£1)0

.4390

£773

£933

33

34

.5019

.0603

.8459

.7030

£957

£097

.4379

3764

£995

34

35

.4894

.0557

.8431

.7010

.5041

£084

.4368

£754

£916

35

30

.4771

.0519

£403

.6990

.5925

.5071

.4357

£745

.3908

36

37

.4659

.0467

.8375

.6970

.5809

£058

.4346

.3735

£199

37

38

.4536

.0493

£348

.6950

£894

£045

.4335

£796

£191

38

39

.4494

.0378

.8390

.6930

£878

£039

.4325

.3716

£183

39

40

.4314

.0334

.8293

.0910

.5863

£019

.4314

.3707

£174

40

41

.4906

.0291

£966

.6890

£847

.5006

.4303

.3697

.3166

41

43

.4103

.0948

.8239

.6871

£833

.4994

.4992

£688

£158

49

43

.4000

.0906

.8212

.6851

£816

.4981

.4981

.3678

£149

43

44

.3900

.0164

£186

.6839

£801

.4968

.4970

£669

£141

44

45

.3809

.0199

.8159

.6819

.5786

.4956

.49T0

.3060

£133

45

46

.3707

.0081

.8133

.6793

£771

.4943

.4249

£650

£134

46

47

.3613

.0040

.8107

.6774

£755

.4931

.4238

.3641

£116

47

48

.3599

.0000

■8081

.6755

£740

.4918

.4998

.369

£108

48

49

.3439

1.9960

£055

.6736

£735

.4906

.4917

£100

49

50

.3345

.9990

£030

£717

.5710

.4894

.4906

£613

£091

50

51

.3259

.9681

.8004

.6698

.5695

.4881

.4196

£604

£083

51

59

.3174

.9842

.7979

.6679

£680

.4869

.4)85

.3595

.3075

59

53

.1091

.9803

.7954

.6661

.5666

.4856

.4175

.3586

.3067

53

54

.3010

.9765

.7939

.6649

£651

4844

.4164

.3576

£059

54

55

.9931

.9797

.7904

.6694

£636

.4839

.4154

.3567

£051

55

56

.9H53

.9690

.7879

.66J05

£691

.4890

.4143

£558

£043

56

57

.9775

.9659

.7855

.6587

.5607

.4808

.4133

£549

£034

57

58

5700

.9615

* .7830

.6568

£599

.4795

.4)99

.3540

.3096

58

59

9.3896

1.9579

.7806

.6550

£578

.4783

.4119

£531

.3018

59

t
i
\

OO

' OO 1

' OO 9

' OO 3

' OO 4

' OO 5

' OO 6

' OO 7

0° »

•

Digitized by VjOOQIC

80

TABLE III.-— HUMPORTIONAL LOGARITHMS.

II

0O 9'

OO 10'

OO ii'

OO 19'

OO 13'

OO 14'

OO 15*

00 16»

"1

oo ir

it

1.3010

1.9353

1JU39

1.1761

1.1413

1.1091

1.0799

1.0519

1.0948

1

. 09

. 45

. 39

. 55

. 08

. 86

. 87

. 07

. 44

1

3

.9994

. 38

. 96

. 49

. 09

. 81

. 83

. 09

. 40

2

3

. 86

. 31

. 19

. 43

.1397

. 76

. 77

.0498

. 35

3

4

. 78

5

. 70

. 17

. 06

. 31

. 86

. 66

. 68

. 89

. 77

5

6

. 69

. 10

.9099

. 95

. 80

. 61

. 63

. 84

. 93

6

7

. 54

. 09

. 93

. 19

. 74

. 55

. 58

. 80

. 19

7

8

. 46

.3495

. 86

. 13

. 69

. 50

. 53

. 75

. 14

8

9

. 39

. 88

. 89

. 07

. 63

. 45

. 49

. 71

. 10

9

10

. 31

. 81

. 73

. 01

. 58

. 40

. 44

. 67

. 06

10

11

. 93

. 74

. 67

.1895

. 59

. 35

. 39

. 69

. 09

11

IS

. 15

. 67

. 61

. 89

. 47

. 30

. 34

. 58

.0197

19

13

. 07

. 60

. 54

. 83

. 49

. 95

. 30

. 53

. 93

13

14

.9899

. 53

. 48

. 77

. 36

. 90

. 95

. 49

. 8*

14

15

. 91

. 45

. 41

. 71

. 31

. 15

. 90

. 44

. 85

15

16

. 88

. 38

. 35

. 65

. 95

. 09

. 15

. 40

. 81

16

17

. 76

. 31

. 98

. 60

. 90

. 04

. 11

. 35

. 76

17

18

. 68

. 94

. 99

. 54

. 14

.0099

. 06

. 31

. 79

H

19

. 60

. 17

. 16

. 48

. 09

. 94

. 01

. 96

. 68

19

90

. 59

. 10

. 09

. 49

. 03

. 89

.0896

. 93

. 64

90

91

. 45

. 03

. 03

. 36

.1998

. 84

. 99

. 18

. 60

91

99

. 37

.93396

.1996

. 30

. 99

. 79

. 67

. 13

. 56

S3

S3

. 99

. 89

. 90

. 94

. 87

. 74

. 89

. 09

. 51

93

94

. 91

. as

. 84

. 19

. 89

. 69

. 78

. 04

. 47

94

95

. 14

. 75

. 77

. 13

. 76

. 64

. 73

. 00

. 43

95

96

. 08

. 68

. 71

. 07

. 71

. 59

. 68

.0395

. 39

96

97

.9798

. 69

. 65

. 01

. 68

. 54

. 63

. 91

. 35

27

98

* 91

. 55

. 58

.1595

. 60

. 49

. 59

. 87

. 31

98

99

. 83

. 48

. 59

. 89

. 55

. 44

. 54

. 89

. 96

99

30

. 75

. 41

. 46

. 84

. 49

. 39

. 49

. 78

. 93

30

31

. 68

. 34

. 39

. 78

. 44

. 34

. 45

. 74

. 18

31

33

. 60

. 97

. 33

. 79

. 39

. 99

. 40

. 69

. 14

33

33

. 53

. 90

. 97

. 66

. 33

. 94

. 35

. 65

. 10

33

34

. 45

. 13

. 91

. 61

. 98

. 19

. 31

. 60

. 06

34

35

. 38

. 07

. 14

. 55

. 93

. 14

. 96

. 58

. 03

35

36

. 30

. 00

. 08

. 49

. 17

. 09

. 91

. 59

.0038

36

37

. 99

.99393

. 09

. 43

. 19

. 04

. 17

. 47

. 93

37

38

. 15

. 86

.1896

. 38

. 07

.0899

. 19

. 43

. 89

38

39

. 07

. 79

. 89

. 39

. 01

. 94

. 08

. 39

. 85

39

40

. 00

. 79

. 83

. 96

.1196

. 89

. 03

. 34

. 81

40

41

.93899

. 66

. 77

. 90

. 91

. 84

.0598

. 30

. 77

41

49

. 85

. 59

. 71

. 15

. 86

. 80

. 94

. 96

. 73

49

43

. 78

. 59

. 65

. 09

. 80

. 75

. 89

. 91

. 69

43

44

. 70

. 45

. 59

. 03

. 75

. 70

. 85

. 17

. 65

44

45

. 63

. 39

. 59

.1498

. 70

. 65

. 80

. 13

. 61

45

46

. 55

. 39

. 46

. 99

. 64

. 60

. 75

. 08

. 57

46

47

. 48

. 95

. 40

. 86

. 59

. 55

. 71

. 04

. 53

47

48

. 40

. 18

. 34

. 81

. 54

. 50

. 66

. 00

. 49

48

49

. 33

. 19

. 98

. 75

. 49

. 45

. 69

.0995

. 44

49

50

. 96

. 05

. 99

. 69

. 43

. 40

. 57

. 91

. 40

50

51

. 18

.9198

. 16

. 64

. 38

. 35

. 59

. 87

. 36

51

59

. 11

. 99

. 09

. 58

. 33

. 31

. 48

. 89

. 33

53

53

. 04

. 85

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. 98

. 96

. 43

. 78

. 98

53

54

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. 78

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. 47

. 93

. 91

. 39

. 74

. 94

54

55

. 89

. 79

. 91

. 41

. 17

. 16

. 34

. 70

. SO

55

58

. 89

. 65

. 85

. 36

. 19

. 11

. 30

. 65

. 16

56

57

. 74

. 59

. 79

. 30

. 07

. 06

. 95

. 61

. 19

57

58

. 67

. 59

. 73

. 94

. 09

. 01

. ~n

. 57

08

58

59

. 69

. 45

. 67

. 19

.1097

.0797

. 16

. 59

. 04

59

ii

■

QO V

OO 10'

OO 11'

OO 19'

OO 13'

00 14'

OO 15'

OO 16'

OO 17'

•

Digitized by VjOOQIC

TABLB IH.— PBOFOBXIOlf At LOOABITBItt.

81

"

QO 18»

DOW

0O90'

0O91'

DOS*

DO 23'

DO 34'

DO 25'

0O86'

0O ST

90 88'

»' 89'

a

10000

9765

9549

9331

9198

8935

8751

8973

8408

8939

8981

7999

1

9906

61

39

97

35

32

'48

70

00

36

79

96

1

3

09

58

35

94

39

89

45

68

8397

34

76

94

3

3

88

54

32

90

19

98

43

65

95

31

73

91

3

4

84

50

98

17

15

S3

39

63

99

88

71

19

4

5

80

46

94

6

76

49

91

10

09

17

33

56

86

83

68

14

6

7

79

39

17

06

06

13

30

53

84

80

63

11

7

e

68

35

14

03

09

10

37

50

81

18

61

09

6

9

64

31

10

00

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07

34

47

78

15

68

66

9

10

60

97

06

95006

96

04

31

44

75

13

55

64

10

11

56

93

03

03

99

01

18

48

79

10

63

01

11

IS

59

90

94V90

60

80

8696

15

39

70

07

50

7890

19

13

48

16

06

86

86

95

13

36

67

04

48

96

13

14

44

19

99

83

83

93

09

33

64

03

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94

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40

08

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79

79

88

06

30

61

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36

05

85

76

76

85

03

37

50

96

40

89

16

17

39

01

81

72

73

89

00

34

56

94

37

67

17

18

98

9607

78

60

70

79

8697

33

53

91

35

64

18

19

94

03

74

66

66

76

94

10

50

88

33

89

19

90

90

90

71

69

63

73

91

16

48

86

30

79

80

91

16

86

67

59

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70

88

13

45

83

97

77

21

99

19

89

64

55

57

67

85

10

49

81

35

74

32

93

08

78

60

52

53

64

83

07

39

78

93

79

93

94

05

75

56

49

60

61

79

04

37

75

80

69

34

95

01

71

53

45

47

57

76

08

54

73

17

67

35

96

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67

40

49

44

54

73

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31

70

14

64

96

97

63

64

46

39

41

51

70

96

38

67

12

C2

97

98

80

60

49

35

37

48

67

93

36

65

09

59

98

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85

56

39

39

34

45

64

90

33

69

07

57

99

30

81

52

35

98

31

43

61

87

90

SO

04

55

30

31

77

49

39

95

38

39

58

84

18

57

03

fi

31

39

73

45

98

92

94

36

55

89

15

54

79*9

33

33

69

41

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18

91

33

59

79

13

59

97

47

33

34

65

38

91

15

18

30

40

76

09

49

94

45

34

39

61

34

18

13

15

37

46

73

07

46

93

49

35

38

58

30

14

08

19

84

43

70

04

44

89

40

36

37

54

96

11

05

08

31

40

67

01

41

f 7

37

37

36

50

33

07

01

05

17

37

65

89J88

38

84

35

38

39

46

19

04

9108

09

14

35

69

96

36

81

39

39

40

49

15

00

05

8999

11

39

59

93

33

79

30

40

41

38

13

9307

91

06

08

99

58

90

31

76

88

41

49

34

08

93

88

99

05

86

' 53

88

88

74

35

49

43

30

04

90

85

80

03

83

51

85

85

71

83

43

44

97

01

66

61

86

6999

80

48

89

83

69

80

44

45

93

9997

83

78

83

96

17

45

79

80

68

18

45

48

19

93

79

75

80

93

14

49

77

17

64

15

46

47

15

90

76

71

77

90

11

39

74

15

61

13

47

48

11

86

79

68

73

87

08

37

S

19

59

10

48

49

07

89

69

65

70

84

05

34

10

56

08

49

50

03

79

65

69

67

81

09

31

66

07

54

06

50

51

00

75

69

58

64

78

8500

38

63

04

51

03

51

*•

9996

71

58

65

61

75

97

85

61

09

40

01

59

53

99

68

55

59

58

79

04

83

58

8990

46

7798

53

1 M

98

64

51

48

54

69

91

80

55

97

44

96

54

55

84

61

48

45

51

68

88

17

53

94

41

94

55

58

80

57

44

49

48

63

85

14

50

91

39

91

56

57

77

53

41

38

45

60

88

11

47

80

36

89

57

58

73

50

37

35

49

57

79

09

44

86

34

86

58

59

69

46

34

33

39

54

76

06

49

64

31

84

50

it

00 18'

0O 19

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0O21'

i'"

0O98'

0O33'

00 34'

0O25'

0O36'

©0 27'

0O28'

0O89'

ii

Digitized by VjOOQIC

ft.

TABLE III.— -FB0FOJKX1ONAL LOOA1UTHM8.

It

©0 30'

0O31'

©0 38'

0© 33'

0O34'

OP 35'

0036*

0O37'

OO 38'

OO 30'

OO 40*

OO 41'

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7788

7039

7501

7368

7938

7113

6990

6871

6755

6643

6538

6485

1

79

37

7499

65

36

, 10

88

69

53

40

30

83

1

8

. 77

! 34

97

63

34

* 08

86

67

51

38

89

81

3

3

; 74

38

94

61

32

06

84

65

49

37

87

80

3

4

1 78

30

98

59

89

04

88

63

47

35

35

18

4

5

69

5

6

67

85

88

54

85

00

78

59

43

31

31

14

6

7

65

83

65

53

83

7098

76

57

48

89

19

13

7

6

68

80

83

50

81

96

74

55

40

87

18

11

8

60

18

61

48

19

93

73

53

38

85

16

09

10

57

16

79

46

17

91

70

51

36

84

14

07

10

11

55

13

76

44

15

80

68

49

34

83

18

06

11

IS

53

11

74

41

13

87

66

47

33

80

10

04

13

13

50

09

78

39

10

85

64

45

30

18

09

08

13

14

48

07

70

37

06

83

63

43

88

16

07

00

14

15

45

04

67

39

06

81

60

48

86

14

05

6308

15

16

43

03

65

33

04

79

58

40

85

IS

03

07

16

17

41

00

63

39

03

77

56

38

83

11

01

95

17

18

38

7597

61

88

00

75

54

36

81

09

00

93

18

19

36

09

58

86

7196

73

53

34

19

07

6498

01

19

' 90

34

03

56

84

06

71

50

38

17

05

06

00

80

31

31

00

54

88

93

69

48

30

15

03

04

68

81

!»

89

88

53

80

01

67

46

88

13

01

OS

66

88

83

8

86

50

17

80

65

44

86

11

00

01

84

93

84

63

47

15

87

63

48

84

09

6598

80

83

84

85

88

81

45

13

85

61

40

88

08

06

67

81

89

86

10

79

43

U

63

50

38

80

06

94

89

70

86

87

17

77

41

09

81

57

36

18

04

93

84

77

87

88

14

74

38

07

70

55

34

16

03

90

88

76

88

80

18

78

36

04

77

53

33

14

00

89

80

74

89

30

10

70

34

08

75

50

30

13

6698

87

78

78

30

31

07

67

38

00

78

48

88

11

06

85

76

71

31

38

09

69

89

7»98

70

46

86

09

94

83

75

69

38

33

03

63

87

06

68

44

84

07

08

81

73

67

33

34

00

69

85

04

66

43

83

05

91

79

71

65

34

39

7608

58

83

01

64

40

80

03

89

78

68

64

39

36

06

56

81

89

68

38

18

01

87

76

67

68

36

37

03

54

18

87

60

36

16

6799

65

74

66

60

37

38

01

51

16

65

58

34

14

97

83

73

64

58

38

30

68

49

14

83

56

38

18

OS

81

70

68

57

30

40

88

47

18

81

54

30

10

03

79

68

60

59

40

41

64

44

09

79

58

88

08

01

77

67

50

53

41

48

81

48

07

76

49

86

08

80

76

65

57

51

48

43

70

40

05

74

47

84

04

87

74

63

55

50

43

44

77

38

03

78

45

83

OS

85

78

61

53

48

44

45

74

39

01

70

43

80

00

84

70

50

51

46

45

46

78

33

7898

68

41

18

6808

88

68

58

50

44

46

47

70

31

06

66

39

16

06

60

66

56

48

43

47

48

67

88

04

64

37

14

04

78

04

54

46

41

48

40

69

86

08

61

35

13

08

76

63

53

44

30

40

50

63

84

00

50

33

10

90

74

61

50

43

38

50

51

60

88

87

57

31

08

88

78

50

48

41

36

51

58

58

10

85

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06

86

70

57

47

39

34

58

53

55

17

83

53

87

04

84

68

55

45

37

as

53

54

53

15

81

51

84

03

83

66

53

43

39

31

54

59

51

13

79

49

83

00

81

64

51

41

34

89

55

56

48

10

76

46

80

6998

79

63

50

39

38

87

56

57

46

08

74

44

18

96

77

61

48

38

30

89

57

58

44

06

78

48

16

04

75

50

46

36

88

84

58

50

41

03

70

40

14

83

73

57

44

34

87

83

50

tt

ft© 30*

0°31'

90 38'

Oo 33'

OO34'

0O39'

OO 36'

0©37'

OO 38'

OO 30'

OO 40*

OO 41'

•

Digitized by VjOOQIC

TABLE III.— PROPORTIONAL LOGARITHMS.

8ft

II

OO 49'

03 43'

0O 44'

00 45'

OO 46'

0©47'

OO 48'

OO 49'

(P58'

OO 51'

0O59'

0O53'

ii

6320

6416

6118

6021

5995

5833

5740

5651

5503

5477

5393

5310

1

19

16

17

19

94

30

39

49

63

76

91

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1

9

17

15

15

17

93

39

37

48

60

74

90

07

9

3

15

13

13

16

30

97

36

46

59

73

89

06

3

4

13

11

19

14

19

96

34

45

57

71

87

05

4

5

19

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6

10

08

08

16

54

7

08

08

07

03

14

91

30

40

53

67

83

00

7

8

06

05

05

08

13

19

98

39

51

66

89

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8

9

05

03

03

06

11

18

97

37

50

64

80

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9

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09

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36

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79

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10

11

01

00

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03

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15

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35

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61

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01

06

13

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33

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19

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39

44

59

75

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13

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14

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09

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41

56

79

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15

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00

07

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40

54

70

88

16

17

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06

15

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38

53

69

87

17

18

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04

13

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37

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68

85

18

19

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86

87

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03

19

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36

50

66

84

19

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85

85

89

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01

10

91

34

49

65

83

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33

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81

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07

18

31

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68

80

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81

79

61

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89

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08

17

30

45

61

79

93

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79

78

79

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88

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04

15

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43

59

77

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78

76

77

81

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93

03

14

27

49

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76

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76

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76

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01

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71

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76

81

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10

93

37

54

79

98

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71

69

71

74

80

87

97

08

91

36

53

71

99

30

69

68

69

73

78

86

95

07

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35

51

69

30

31

67

66

67

71

77

1 84

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05

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68

31

39

65

65

68

69

75

83

93

04

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39

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66

39

33

64

63

64

68

74

81

91

09

16

30

47

65

33

34

69

61

63

68

79

80

89

01

14

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46

64

34'

35

60

60

61

65

70

78

88

5599

13

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44

69

35

36

59

58

59

63

09

77

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11

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43

61

36

37

57

56

58

61

67

75

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96

10

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41

60

37

38

55

55

56

60

66

74

83

95

08

93

40

58

38

39

54

53

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58

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79

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07

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39

57

39

40

59

51

53

57

63

71

80

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08

91

37

50

40

41

50

50

51

55

61

69

79

91

04

19

36

54

41

49

48

48

50

54

60

68

77

89

03

18

35

53

49

43

47

46

48

59

58

68

76

88

01

16

33

59

43

44

45

45

46

50

56

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Digitized by VjOOQIC

TABLE III.— PROPOftTIONAL LOGARITHMS.

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3P5'

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30 11'

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80 15*

•

Digitized by VjOOQIC

TABLE III.— PROPORTIONAL LOOARTTHMf .

91

n

90 16'

90 17*

90 18'

SO 18'

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02

TABLE III.— PROPORTIONAL LOOARITHXS.

•
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90 98'

90 99'

90 30'

90 31'

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20 33'

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Digitized by VjOOQIC

TABLft III.— MtOFORTIONAL LOOARITHM8.

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SO 41'

30 43'

SO 43'

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30 45'

80 46'

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•

Digitized by VjOOQIC

TABLE HI.— PROPORTIONAL LOGARITHMS.

•
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80 51'

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80 54'

80 55'

80 56'

80 57'

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91

66

48

18

16

17

£

41

16

90

65

40

15

91

66

48

J7

17

18

41

15

90

65

40

15

90

66

41

17

18

19

66

40

15

89

64

39

14

90

65

41

17

19

SO

65

40

14

89

64

39

14

89

65

40

16

89

SI

65

39

14

89

63

39

14

89

64

40

16

81

22

64

39

13

88

63

38

13

89

64

40

15

88

83

64

38

13

88

63

38

13

88

64

39

15

83

84

64

38

13

87

68

37

19

88

63

39

15

84

85

63

38

19

87

68

37

18

87

63

38

14

85

86

63

37

18

87

61

36

18

87

68

38

14

86

87

68

37

11

86

61

36

11

87

08

38

13

87

88

68

36

11

86

61

36

11

86

68

37

13

88

88

61

36

11

65

60

35

10

86

61

37

18

89

30

61

35

10

65

60

35

10

85

61

36

18

39

31

61

35

10

84

59

34

10

85

60

36

18

31

38

60

35

09

84

59

34

09

84

60

36

11

39

33

60

34

09

84

58

34

09

84

60

35

11

33

34

56

34

68

83

58

33

08

84

59

35

10

34

35

59

33

08

83

58

33

08

83

59

34

10

16

36

58

33

08

88

57

38

67

63

58

34

10

36

37

58

33

07

88

57

39

07

89

58

34

69

37

38

58

38

07

81

56

31

67

88

57

33

69

39

30

57

39

08

81

56

31

66

88

57

33

68

39

40

57

31

66

81

56

31

06

81

57

38

68

40

41

56

31

OS

80

55

30

OS

81

56

39

68

41

48

56

30

05

89

55

30

05

80

56

31

67

48

43

55

30

05

79

54

99

05

80

55

31

07

43

44

55

30

04

79

54

99

04

80

55

31

66

44

45

55

89

04

79

53

89

04

79

55

30

66

45

46

54

89

03

78

53

88

03

79

54

30

66

46

47

54

88

03

78

53

88

03

78

54

89

65

47

48

53

88

09

77

58

87

03

78

53

89

65

46

49

53

87

OS

77

58

87

68

77

53

89

64

40

50

58

87

68

76

51

86

68

77

53

98

64

59

51

58

97

01

76

51

86

01

77

59

88

04

51

58

58

86

01

76

61

86

01

76

58

87

03

59

53

51

86

00

75

50

85

00

76

61

87

03

53

54

51

85

60

75

50

85

60

75

51

87

69

54

55

56

85

60

74

49

84

00

75

61

96

69

55

56

50

84

0199

74

49

84

0099

75

50

86

09

59

57

50

84

99

74

48

84

99

74

59

85

•1

57

58

49

84

98

73

48

83

98

74

49

85

01

59

59

49

83

98

73

48

83

98

73

49

86

69

59

it

$

P49'

*> sty

10 51'

10 58'

¥>&

10 54'

10 55'

¥>W

*>57M

ro5B'

10 59*

it

•

Digitized by VjOOQIC

TABLE IV.

COURSES, DISTANCE, DEPARTURE, AND
DIFFERENCE OF LATITUDE.

Digitized by VjOOQIC

96 Distance 1 to 50 miles.

TABLE IT.

§

C. i Pt.

c *

PL

C. f Pt.

C. 1 Pt.

cupt

ai v Pt.

CUR.

C9Pta.

Jj

t £

a

1

•

■

s

T ■* *•* T

££

» Z £ fc* it £ *'

SB ad

ri %

fc ori

oo* SB

SB 00

80 SB

SB 00

oo sb

SB 00

n as

K a*

a SB

SB ad

B SB, SB 00 CZ;

d-lat.

dep.

d. lat. dep.

d. lat. dep.

d.lat.

dep.

d-lat.

dep.

LUt

dep.

Hat.

dep. tin. *»j

1

1

L

.1

L

0.1

1.

0.3

L

0.3

L

as

a9

13 19 ft 4

s

3

0.1

3.

0.3

3.

0.3

8.

0.4

1.9

0.5

L9

0.6

1.9

0.7. L6 ft-

3

3

0.1

3.

0.3

3.

0.4

8.9

0.6

3.9

0.7

3.9

0.9

3.8

L | 9.8 1 i

4

4

0.8

4.

0.4

4.

0.6

3.9

0.8

3.9

1.

3.8

L3

3.8

13, 17 li

5

5

0.3

5.

0.5

4.9

0.7

4.9

L

4.9

L2

4.8

1.5

4.7

1.7

4.6 1>

6

6

0.3

6.

0.6

5.9

0.9

5.9

1.2

5.8

1.5

5.7

1.7

5.6

9.

5.5 13

7

7

0.3

7.

0.7

6.9

L

6.9

1.4

6.8

1.7

6.7

9.

6.6

9.4

65 I"

8

6

0.4

a

0.8

7.9

1.3

7.8

1.6

7.8

1.9

7.7

9.3

7.5

9.7

74 3;

9

9

0.4

9.

0.9

8.9

1.3

8.8

1.8

8.7

8.3

8.6

9.6

15

1

8.3 li

10

10

0.5

10.

1.

9.9

1.5

9.8

8.

9.7

3.4

9.6

9.9

9.4

3.4

9.3 1?

11

11

0.5

10.9

1.1

10.0

1.6

10.8

8.1

10.7

8-7

10.5

3.3

10.4

3.7

H3 4 5

13

13

0.6

11.9

1.3

11.9

1.8

11.8

8.3

11.6

8.9

11.5

3.5

11.3

4.

1L1 4 4

13

13

0.6

13.9

1.3

13.9

1.9

13.8

3.5

13.6

3.3

18.4

as

IS. 9

4.4

IS. i

14

14

0.7

13.9

1.4

13.8

8.1

13.7

8.7

13.6

3.4

13.4

4.1

13.9

4.7

119 it

15

15

0.7

14.9

1.5

14.8

8.3

14.7

9.9

14.6

3.6

14.4

4.4

14.1

5.1 13.9 17

18

18

0.8

15.9

1.6

15.8

8.3

15.7

3.1

15.5

3.9

15.3

4.6

15.1

& 4] 14.8 41

17

17

0.8

16.9

1.7

16.8

8.5

16.7

3.3

16.5

4.1

16.3

4.9

16.

IT 117 45

18

18

0.9

17.9

1.8

17.8

3.6

17.7

3.5

17.5

4.4

17.8

5.3

16.9

6-1 16.6 4*

19

19

0.9

18.9

1.9

18.8

8.8

18.6

17

18.4

4.6

1&3

5.5

17.9

6-4

H.6 T 2

. 90

30

L

19.9

3.

19.8

8.9

19.6

3.9

19.4

4.9

19.1

5.8

18.8

6.7

U5 ::

31

81

1.

30.9

8.1

30-8

3.1

30.6

4.1

30.4

5.1

30.1

6.1

19.8

7.1

19.4 & |

33

33

1.1

31.9

3.3

31.8

3.3

31.6

4.3

31.3

5.3

31.1

6.4

80.7

7.4

96.3 84>

83

C

1.1

38.9

3.3

83.8

14

33.6

4.5

38.3

5.6

88.

6.7

91.7

7.7

tL« #*

34

1.3

33.9

3.4

33.7

3.5

83.5

4.7

83.3

5.8

33.

7.

83.6

8.1

99.9 tX

35

35

1.3

34.9

8.5

84.7

3.7

84.5

4.9

84.3

6.1

83.9

7.3

83.5

14

93.1 %*

36

36

1.3

85.9

3.6

35.7

3.8

35.5

5.1

35.3

6.3

84.9

7.5

84.5

as

94. % *

37

37

1.3

38.9

8.6

86.7

4.

36.5

5.3

86.3

6.6

35.8

7.8

85.4

9.1

94.9 111

88

38

1.4

37.9

3.7

37.7

4.1

87.5

5.5

87.8

6.8

86.8

8.1

86.4

9.4

919 wr

39

39

1.4

38.9

3.8

38.7

4.3

88.4

5.7

38.1

7.

37.8

8.4

97.3

9.8

96.6 H t

30

30

1.5

89.9

8.9

39-7

4.4

89.4

5.9

89.1

7.3

38.7

8.7

98.9

10.1

97.7 US

31

31

1.5

30.8

3.

30.7

4,5

30.4

6.

30.1

7.5

89.7

9.

99.3

16.4

98.6 It*

33

33

1.6

31.8

3.1

31.7

4.7

31.4

6.9

31.

7.8

30.6

9.3

30.1

10.6

99,6 lit

33

33

1.6

33.8

3.3

33.6

4.8

33.4

6.4

33.

&

31.6

9.6

31.1

11.1

30.5 114

34

34

1.7

33.8

3.3

33.6

5.

33.3

8.6

33.

8.3

33.5

9.9

39.

11.5

3L4 a

35

35

1.7

34.8

3.4

34.6

5.1

34.3

6.8

34.

8.5

33.5

10.3

33.

11.6

IS Ut

36

36

1.8

35.8

3.5

35.6

5.3

35.3

7.

34.9

8.7

34.4

10.5

33.9

19.1

33.3 a*

37

37

1.8

36.8

3.6

36.6

5.4

36.3

7.8

35.9

9.

35.4

10.7

34.8

18.5

34 9 141

38

38

1.9

37.8

3.7

37.6

5.6

37.3

7.4

36.9

9.3

36.4

11.

35.8

18.8

311 Hi

39

39

1.9

38.8

3.8

38.6

5.7

38.3

7.6

37.8

9.5

37.3

11.3

36.7

13.1

31 14 »

40

40

3.

39.8

3.9

39.6

5.9

39.3

7.8

38.8

9.7

3a 3

11.6

37.7

13.5

37. UJ

41

41

8.

40.8

4.

40.8

6.

40.3

8.

39.8

10.

39.3

11.9

38.6

13.8

37. » UT

43

41.9

8.1

41.8

4.1

41.5

6.3

41.3

8.3

40.7

10.3

40.3

13.3

39.5

14.1

38.6 j41

43

48.9

3.1

43.8

4.9

43.5

6.3

43.3

8.4

41.7

10.4

41.1

18.5

40.5

14.5

39-7 H>

44

43.9

3.3

43.8

4.3

43.5

6.5

43.8

8.6

43.7

10.7

48.1

13.8

41.4

14.8

417 M-

45

44.9

3.8

44.8

4.4

44.5

6.6

44.1

18

43.7

10.9

43.1

13.1

49.4

15.9

4L6 17*

46

45.9

3.3

45.8

4.5

45.5

8.7

45.1

9.

44.8

11.3

44.

13.4

43.3

15.3

49.4 17*

47

46.9

3.3

46.8

4.6

46.5

6.9

46.1

9.3

45.6

11.4

45.

13.6

44.3

15.9

414 If*

48

47.9

8.4

47.8

4.7

47.5

7.

47.1

9.4

46.6

11.7

45.9

13.9

45.9

16.8

44.3 1*4

49

48.9

8.4

48.8

4.8

48.5

7.8

48.1

9.6

47.5

11.9

46.9

14.9

46.1

16.5

413 I*»

50

49.9

8.5

49.8

4.9

49.5

7.3

49.

9.8

48.5

13.1

47.8

14.5

47.1

16.8

418 91

g

dep.

d.Ut

dep.

d.lat

dep.

Hat.

dep.

d-lat.

dep.

d.lat.

dep.

d.lat.

dep. d.lat.

dep. d.Ut

3 3j P P

M

P w

M

W H

33

W W

3 3 P» P

s*

P> P

33

pa N

351 ?•«

as S»

j» as

*- *•
Z p

as p

w as

cd as

as »
as 5°

m a*

S3

P> as

as ?
a* P

» as

ip -a.

3? ?8{e

*

C. 7| PU.

C.71Fte.

C.7}Pt*.

C. 7Pti.

C.6IPU.

C.HTU-

C.6*Pta.

C.6Pta ,

Digitized by VjOOQIC

Distance 1 to 50 miles.

TABLS IV.

97

1

1

C.SlPt*.

C.9|PU.

0.9}Pte.

C. SPta.

C.3}Pta.

C.3|Pta.

C.3JPti.

C.4PU.

c

* ri **

*p4

fc*

.

■* **
£*

* ad

25 rf

-s

« OB

•4m •*

* ad

•44 T*

2

* ad m as

* m a* 2s

* *

ad 2:

W H

fc*

* H

& »t

» w

**.

c4 rf

**

»H ** .2

ft to ad 85

85 00 00 25

=5 09

m SB

25 nd

OQ 25

25 CD

ad 25

25 ad

OQ 25

25 ad

m 25

250d 00*25*

1

d. lat. dep.

d. lat. dep.

d. lat.

dep.

d. lat.

dep.

d.lat.

dep

d . lat.

dep.

<Llat.

dep.

d. lat. dep.

0.9 0.4

0.9 0.5

0.9

0.5

0.8

0.6

0.8

0.6

0.8

0.6

0.7

0.7

0.7 1

1.8 0.9

1.8 0.9

1.7

1.

1.7

1.1

1.6

1.9

1.5

1.3

1.5

1.3

1.4 3

«.7 1.3

9.6 1.4

9.6

1.5

9.5

1.7

9.4

1.8

9.3

J.9

9.3

3.

3.1 3

3.8 1.7

3.5 1.9

3.4

9.1

3.4

9.9

3.9

9.4

3.1

9.5

3.

8.7

8 8 4

4.5 8.1

4.4 9.4

4.3

9.6

4.3

9.8

4.

3.

3.9

3.9

3.7

3.3

3.5 5

5.4 8.6

5.3 9.8

5.1

3.1

5.

3.3

4.8

3.6

4.6

3.8

4.4

4.

4.3 6

0.3 3.

6.9 3.3

6.

3.6

5.8

3.9

5.6

4.3

5.4

4.4

5.3

4.7

4.9 7

7.8 3.4

7.1 3.8

6.9

4.1

6.7

4.4

6.4

4.8

6.3

5.1

5.9

5.4

5.7 8

&1 3.8

7.9 4.9

7.7

4.6

7.5

5.

7.9

5.4

7.

5.7

6.7

6.1

6.4 9

0. 4.3

8.8 4.7

8,6

5.1

8.3

5.6

8.

6.

7.7

6.3

7.4

6.7

7.1

10
7.8 11

9.9 4.7

9.7 5.9

9.4

5.7

9.1

6.1

8.8

6.6

8.5

7.

8.3

7.4

10.8 5.1

10.6 5.7

10.3

6.9

10.

6.7

9.6

7.1

9.3

7.5

8.9

8.1

a5 19

11.7 5.6

11.5 6.1

11.9

6.7

10.8

7.9

10.4

7.7

10.

8.3

9.6

a7

9.3 13

19.7 6.

19.3 6.6

19.

7.9

11.6

7.8

11.9

8.3

10.8

8.9

10.4

9.4

9.9 14

13.0 6.4

13.9 7.1

19.9

7.7

19.5

83

19.

8.9

11.6

9.5

11. 1

10.1

10.6 15

14.5 6.8

14.1 7.5

13.7

8.9

13.3

&9

19.9

9.5

19.4

10.9

lt.9

10.7

11.3 16

15.4 7.3

15. &

14.6

8.8

14.1

9.4

13.7

10.1

13.1

10.8

19.6

11 4

• 18. 17

16.3 7.7

15.9 8.5

15.4

9.3

15.

10.

14.5

10.7

ft

11.4

13.3

18.1

18.7 18

17.9 &1

16.8 9.

16.3

9.8

15.8

10.6

15.3

11.3

19.1

14.1

19.8

13.4 19

18.1 8.6

17.6 9.4

17.3

10.3

16.6

11.1

16.1

11.9

15.5

187

14.8

13.4

14.1 80

19. 9.

185 9.9

18.

10.8

17.5

11.7

16.9

19.5

16.3

13.3

15.6

14.1

14.8 21

19. 9 9. 4

19.4 40.4

18 9

11.3

18 3

!9.9

17.7

13.1

17.

14.

16.3

14.8

15.6 oo

90 8 9.8

90.3 10.8

19.7

11.8

19.1

19.8

18.5

13.7

17.8

14.6

17.

15.4

16.3 23

91.7 10.3

91.9 11.3

90.6

19.3

90.

13.3

19.3

14.3

18.6

15.3

17.8

16 1

17. M

98.6 10.7

99. 11.8

91.4

19.9

90.6

13.9

90.1

14.9

19.3

15.9

18.5

16.8

17.7 S

93.5 11.1

99.9 19.3

99.3

13.4

91.6

14.4

90.9

15.5

90.1

16.5

19.3

17.5

ia4 9Q

94.4 11.5

93.8 19.7

93.9

13.9

99.4

15.

91.7

16.1

90.9

17.1

90.

iai

19.1 £

95.3 IS.

94.7 139

94.

14.4

93.3

15.6

29.5

16.7

91.6

17.8

90.7

18.8

19.8 as

96.9 19.4

95.6 13.7

94.9

14.9

94.1

16.1

93.3

17.3

99.4

18.4

91.5

19.5

90.5 no

97.1 19.8

96.4 14.1

95.7

15.4

94.9

16.7

94.1

17.9

93.3

19.

99.9

90.1

91.9 J

9a 13.3

97.3 14.6

96.6

15.9

95.8

17.9

94.9

18.5

94.

19.7

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90.8

91.9 31

99.6 2

23 ' 3 33
94. S

94.7 g

98.9 13.7

989 15.1

97.4

16.5

96.6

17.8

95.7

19.1

94.7

80.3

93.7

91.5

99.8 14.1

99.1 15.6

98.3

17.

97.4

18 3

96.5

19.7

95.5

90.9

94.5

93.9

30.7 14.5

30. 16.

99.9

17.5

983

18 9

97.3

90.3

36.3

91.6

95.9

99.8

31 6 15.

30.9 16.5

30.

18.

99.1

19.4

98.1

90.8

97.1

99.9

95.9

93.5

39.5 15.4

31.7 17.

30.9

18.5

99.9

90.

98.9

91.4

97.8

99.8

96.7

34.3

95.5 «n
96.9 J?
86.9 g

33.4 15.8

39.6 17.4

31.7

19.

30.8

90.6

99.7

99.

986

93.5

97.4

84.8

34.4 16.9

33.5 17.9

39.6

19.5

31.6

21 1

10.5

99.6

99.4

94.1

98.9

955

35.3 16.7

34.4 18.4

33.5

90.1

39.4

91.7

31.3

93.3

30.1

94.7

289

96.9

W 'S 39

36.3 17.1

35.3 18.9

34.3

90.6

33.3

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39.1

93.8

30.9

95.4

99.6

96.9

98.3 ^o

37.1 17.5

36.9 19.3

35.9

91.1

34.1

99.8

33.9

94.4

31.7

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30.4

97.5

99. 41

38. 18.

37. 19.8

36.

91.6

34.9

93.3

33.7

95.

39.5

96.6

31.1

98.9

99.7 J5

38 9 18 4

37.9 90.3

36.9

99.1

35.8

93.9

34.5

95.6

33.3

97.3

31.9

98.9

30.4 J3

39.8 18 8

38.8 90.7

37.7

99-6

36.6

94.4

35.3

96.9

34.

97.9

39.6

99.5

31.1 2

40.7 19.9

39.7 31.9

38.6

93.1

37.4

95.

36.1

96.8

34.8

98.5

33.3

30.3

31.8 S

41.6 19.7

40.6 91.7

39.5

93.6

38.9

956

36.9

97.4

35.6

99.9

34.1

30.9

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49.5 90.1

41.5 99.9

40.3

94.9

39.1

96.1

37.8

98.

36.3

99.8

34.8

31.6

S-* 47

4a 4 90.5

49.3 99 6

41.9

94.7

39.9

96.7

38.6

98.6

37.1

305

35.6

39.9

*•* %

44.3 91.

43.9 93.1

49.

95.8

40.7

97.9

39.4

90.9

37.9

31.1

30.3

39.9

34.6 £

45.9 91.4

44.1 93.6

49.9

95.7

41.6

97.8

40.9

99.8

38.7

31.7

37.

33.6

35.4 g

dep. d. tat.

dep. d. lat.

dep.

d. lat.

dep.

d. lat.

dep. d. lat.

dep.

d. lat.

dep.

d. lat.

dep. d. lat. ~

^ 1 « 2

as » ob a>

25 »

» 58

*, »

» 5K

as dd

00 »2

as m

oo a*

as pa

oo as

as* a 2 ?

k * * *

S 3 * *

S*

P 9 in

*.<

w w

"s?

P W

55

P ft

**

P w

'$$ ^P I

k*
**

5*

* »

55

9

**"*

MM •*•

**

C. 5| Pta.

G 5* Pta.

can*.

C.5PU.

C. 44PU.

C.4JPt».

C.4*Pte.

0.4PU.

Digitized by VjOOQIC

98 Distance 51 to 100 miles.

TABLK1V.

i

C. *Pt.

C t Pt.

C. t Pt.

C. lPt.

o.urt

a it Pt. | a if pt.

carta.

p4«

•4* •+•

* « E£

1

i

■^2

•4m -*

tt

as to

DO *

fc flD

flD 85

as od

z s

as «

od as

as od

«: as

as «d a; as

a: m

ob » i ai at m as

d.lat.

dep.

d.lat.

dep.

d. lat.

dep.

d. lat. dep.

d-lat.

dep.

d. lat. dep.

d.Ut.

dep. d. lat. dep.

%

50.0

8.5

50.8

5.

50.4

75

50.

0.0

40.5

13.4

48.8 14.8

4a

178

47.1 19.5

51.0

8.6

51.7

5.1

51.4

7.6

51.

10.1

50.4

13.6

408 151

40.

175

40. I9L9

53

52.0

8.6

58.7

5.8

58.4

7.8

58.

10.3

51.4

13.0

50.7 154

49.0

17.9

49. 90.3

54

53.0

8.6

53.7

5.3

53.4

7.0

53.

105

53.4

13.1

51.7 15.7

50.8

18.9

49.9 90.7

55

54.0

8.7

54.7

5.4

54.4

8.1

53.0

10.7

53.4

13.4

59,6 16.

51.8

18.5

5tv8 SI.

56

55.0

8.7

55.7

5.5

55.4

8.3

54.0

10.0

54.3

13.6

53.6 16.3

59,7

18.9

51.7 9X4

57

56.0

8.8

56.7

5.6

56.4

8.4

55.0

11. 1

55.3

13.8

54.5 16.5

53.7

19.9

59X7 91.8

53

57.0

8.8

57. 7

5.7

57.4

85

56.0

11.3

56.3

14.1

55.5 16.8

54.6

10.5

5X8 99,3

59

58.9

8.0

58.7

5.8

58.4

8.7

57.0

11.5

57.3

14.3

58.5 17.1

55.6

19.9

54.5 99,6

60

50.0

8.0

50.7

5.0

50.4

8.8

588

11.7

58.3

14.8

57.4 17.4

56.5

90,8

55,4 S3.

61

60.0

3.

60.7

6.

60.3

0.

50.8

11.0

50.8

14.8

58,4 17.7

57.4

90.8

89,4 93.3

68

61.9

3.

61.7

6.1

61.3

0.1

60.8

13.1

60.1

15.1

50.3 1&

58.4

90.9

57.3 93.7

63

68.0

3.1

68.7

6.8

68.3

0.8

81.8

19,3

61.1

15.3

60.3 18.3

59,3

91.9

5&8 94.1

64

63.0

3.1

68.7

6.3

63.3

0.4

63.8

13.5

68.1

15.6

61.8 18.6

60.3

31.6

58.1 94.5

65

64.0

3.8

64.7

6.4

64.3

8.5

63.8

13.7

63.1

15.8

69,3 18.0

6L8

31.9

•9,1 94.0 ,

66

65.0

3.8

65.7

6.5

65.3

0.7

64.7

18.0

64.

16.

63.3 19.3

69,1

98.3

6L 95.3

67

66.0

V*

66.7

6.6

66.3

0.8

65.7

13.1

65.

16.3

64.1 10.4

63.1

39,6

61.0 95.6

68

67.0

67.7

6.7

67.3

10.

66.7

13.3

66.

16.5

65.1 10.7

64.

98.9

69,8 96.

60

68.0

3.4

68.7

6.8

68.3

10.1

67.7

13.5

66,0

16.8

66. 30.

65.

93.9

63.7 96.4

70

60.0

3.5

60.7

6.0

60.3

ia3

68.7

13.7

67.0

17.

67. 30.3

65,9

93.6

64.7 9BL8

71

70.0

3.5

70.7

7.

70.2

ia4

60.6

13.9

68.0

17.3

67.0 98.6

86.8

93.9

65.6 97.3

73

71.0

3.6

71.7

7.1

71.8

10.6

70.6

14.

60.8

17.5

68.0 90.9

67.6

34.3

89>5 97.6

73

72.0

3.6

78.6

7.8

73.3

10.7

71.6

14.3

70.8

17.7

60.0 81.3

68,7

34.6

67.4 97.0 '

74

73.0

3.7

73.6

7.3

73.3

iao

73.6

14.4

71.8

18.

70.8 31.5

69.7

94.9

89,4 9H3

75

74.0

3.7

74.6

7.4

74.8

11.

73.6

14.6

73.8

iao

71.8 31.8

70.6

85.3

60.3 98.7 j

76

75.0

3.8

75.6

7.4

75.3

11.3

74.5

14.8

73.7

1&5

73.7 89,1

71.6

85.6

70,8 99.1 '

77

76.0

3.8

76.6

7.5

76.3

11.3

75.5

15.

74.7

18.7

73.7 89,4

79,5

85.9

71.1 99.5 '

73

77.0

3.0

77.6

7.8

77.8

11.4

76.5

15.3

75.7

10

74.6 89,6

73.4

96.3

79,1 99,8 '

70

78.0

3.0

78.6

7.7

78.1

11.6

77.5

15.4

76.6

10.8

75.6 89,9

74.4

96.6

73. 30.3 '

80

70.0

4.

78.6

7.8

70.1

U.7

78.5

15.6

77.6

10.4

76.6 83,3

75.3

97.

73.9 39,6 j

81

80.0

4.

80.6

7.0

80.1

U.0

70.4

15.8

78.6

10.7

77.5 83.5

76.3

37.3

74.8 SL 1

83

81.0

4.1

81.6

8.

81.1

13.

80.4

16.

79.5

10.0

78.5 33.8

77.3

87.6

75,8 31.4

83

88.0

4.1

88.6

8.1

88.1

13.8

81.4

16.3

80.5

80 3

79.4 841

78.1

98

76,7 31.8 i

84

83.0

4.8

83.6

8.3

83.1

13.3

88.4

16.4

81.5

304

80.4 94.4

70.1

98.3

77.6 39,1

85

84.0

4.8

84.6

8.3

84.1

13.5

83.4

16.6

89,5

80.7

8L3 84.7

80.

98.6

78.5 39,5,

88

65.0

4.3

85.6

8.4

85.1

13.6

84.3

16,8

83.4

80.0

89,3 35.

81.

90.

79.5 39.9!

87

88.0

4.3

86.6

&5

86.1

18.8

85.3

17.

84.4

31.1

83.3 35.3

81.0

30.3

80.4 3X3.

86

87.0

4.4

87.6

&6

87.

13.0

88.3

17.3

85.4

81.4

84.3 85.5

89,9

39.6

81.3 33.7

80

88.0

4.4

88.6

8.7

88.

13.1

87.3

17.4

86.3

81.6

85.3 35.8

83.8

30,

68.9 34.1

00

80.0

4.5

80.6

&8

80.

13.3

88.3

17.6

87.3

81.0

86.1 86.1

84.7

30.3

83.1 34.4

01

00.0

4.5

00.6

8.9

00.

13.4

80.3

17.8

88.3

83.1

87.1 96,4

85.7

30.7

84.1 34.8

OS

01.0

4.6

01.6

0.

01.

13.5

00.3

17.8

80.3

83.4

88. 36.7

86.6

31.

85, 35.9

03

08.0

4.6

08.6

0.1

03.

13.6

01.3

18.1

00.8

89,6

89. 97.

87.6

31.3

85,9 35.6

04

03.0

4.7

03.5

0.3

03.

13.8

08.8

183

01.3

89,8

00. 87.3

885

31.7

88,8 36.

05

04.0

4.7

04.5

0.3

04.

13.0

03.3

18.5

08.8

33.1

00.0 87.6

89.4

39,

87.8 36,4

06

05.0

4.8

05.5

0.4

05.

14.1

04.3

18.7

03.1

83.3

01.0 37.0

90.4

39,3

88.7 36.7

07

06.0

4.8

06.5

0.5

06.

14.8

05.1

18.0

04.1

33.6

08.8 88.3

01.3

39,7

89.6 37.1

08

07.0

4.0

07.5

0.6

06.0

14.4

06.1

10.1

03.1

83.8

03.8 88.4

09,3

33L

00.5 37.5

go

08.0

4.0

08.5

0.7

07.0

14.5

07.1

10.3

06.

84.1

04.7 88.7

03.3

33.4

01.5 37.9

100

00.0

4.0

00.5

0.8

08.0

14.7

08.1

10.5

07.

34.3

05.7 89.

94.3

33.7

09,4 39.3

5

dap

d. lat.

dep.

d. lat.

dep.

1. lat.

dep.

d. lat.

dep.

d. lat.

dep. d. lat.

dep.

d. lat.

dep. d. lat.

M

P P

•s-s

p p

•8?

p p

33

P P

3*

p p

S 3 P P

33

P P

33 P*

s

* p

p p

P 91

P P

* P

P as

P P

XX

p as

xx

as P
P *

XX

p ?

P as

xx **
p£ *S

as P

p P

P as

p p

3< **

C 7| Pta.

C71PU.

C.7iPta.

C. 7Pte.

C.6JPta.

0. 6iPU-

C.6}Pta.

C6Pta.

Digitized by VjOOQIC

Distance 51 to 100 miles.

TABLE IT.

C.SiPM.

CSft Pta.

C. 8| Ptt.

C. 3PU.

C.3JPta.

C.3JPU.

C.3|Fti.

C.4FU.

it

«2

25 QQ

. 2*

is

aS

as oq

2 CD

m at

* B

m fe

U ftj

M

» z

fc*

ta pa

**

ri ri

**

•«!4 £*

i

B

2 «G

OD 2

* m

m 2

as *

QD 2

2 OD

as 2

2 W

m 2

as a

to 2

as m

00 2

2od oq2

m
*

s

d-tat.

dep

d.lat.

dep.

d.lat.

dep.

d. lat.

dep.

d.lat.

dep.

d.lat.

dep.

d. lat.

dep.

d.lat dep.

46-1

31.8

45.

34.

43.7

86.3

43.4

38.3

41.1

30.4

39.4

33.4

37.8

34.3

36.1

51

47.

83.3

45.9

34.5

44.6

36.7

43.3

38.9

41.8

31.

40.8

33.

38.5

34.9

36.8

53

47.9

33.7

46.7

35

45.5

37.8

.44.1

89.4

43.6

31.6

41.

33.6

39.3

35.6

37.5

53

48.8

33.1

47.6

35.5

46.3

87.8

44.9

30.

43.4

38.3

41.7

34.3

40.

36.3

3as

54

49.7

83.5

48.5

25.9

47.3

88.3

45.7

306

44.3

39.8

48.5

34.9

40.8

36.9

38.9

55

50.6

33.9

49.4

86.4

48.

89.8

46.6

31.1

45.

33.4

43.3

35.6

41.5

37.6

39.6

56

51.5

34.4

50.3

36.9

48.9

89.3

47.4

31.7

45.8

34.

44.1

36.3

48.8

38.3

40.8

57

53.4

34.8

51.8

37.3

49.7

89.8

48.8

38,8

46.6

34.6

44.8

36.8

43.

39.

41.

58

53.3

35.3

53.

87.8

59.6

30.3

49.1

38.6

47 4

35.1

45.6

37.4

43.7

39.6

41.7

59

54.2

35.7

53.9

88.3

51.5

30.8

49.9

33.3

48.3

35.7

46.4

38.1

44.5

40.3

43.4

60

55.1

96.1

53.8

98.8

53.3

31.4

59.7

33.9

49.

36.3

47.3

38.7

45.3

41.

43.1

61

56.

96.5

54.7

89.9

53.3

31.9

51.6

34.4

49.8

36.9

47.9

39.3

45.9

41.6

43.8

69

57.

36.9

55.6

89.7

54.

39.4

53,4

35.

50.6

37.5

4&7

40.

46.7

48.3

44.5

63

57.9

87.4

50.4

39.3

54.9

39,9

53.3

35.6

51.4

38.1

49.5

40.6

47.4

43.

45.3

64

5&8

87.8

57.3

30.6

55.8

33.4

54.

36.1

53,3

38.7

50.3

41.3

48.3

43.7

46.

65

59.7

88.8

59.8

31.1

56.6

33.9

54.9

36.7

53.

39.3

51.

41.9

48.9

443

46.7

66

60.6

38.6

59.1

81.6

57.5

34.4

55.7

37.3

53.8

39.9

51.8

48.5

49.6

45.

47.4

67

61.5

39.1

60.

33.1

58.3

35.

56.5

37.8

54.6

40.5

53.6

43.1

50.4

45.7

48.1

68

63.4

89.5

60.9

33.5

59.9

35.5

57.4

38.3

55.4

41.1

53.3

43.8

51.1

46.3

48.8

£9

J 63.3

89.9

61.7

33.

60.

36.

5a 2

38.9

56.8

41.7

54 1

44.4

5L9

47.

49.5

70

64.3

30.4

63.6

33.5

60.9

36.5

59.

39.4

57.

43.3

54.9

45.

53.6

47.7

50.9

7"

65.1

30.8

63.5

33.9

61.8

37.

59.9

40.

57.8

48.9

55.7

45.7

53.3

48.4

50.9

79

C6.

31.8

64.4

34.4

63.6

37.5

69.7

40.6

58.6

435

56.4

46.3

54.1

49.

» 51.6

73

66.9

31.6

65.3

34.9

63.5

38.

61.5

41.1

59.4

44.1

57.3

46.9

54.8

49.7

588

74

67.8

33.1

66.1

35.4

64.3

88.6

68.4

41.7 60.3

44.7

58.

47.6

55.6

50.4

53.

75

SS-I

33,5

67.

35.8

65.8

39.1

63.3

43:8

61.

45.3

5R7

4a 2

56.3

51.

53.7

76

• 69.6

38.9

67.9

36.3

66.

39.6

64.

48.8

61.8

45.9

59.5

4a 8

57.1

51.7

54.4

77

70.5

33.3

68.8

36.8

66.9

40.1

64.9

43L3

69.7

46.5

60.3

49.5

57.8

58.4

55.3

78

71.4

33.8

69.7

37.9

67.8

40.6

65.7

43.9

63.5

47.1

61.1

50.1

58.6

53.1

55.9

79

73.3

34.3

70.6

37.7

68.6

41.1

66.5

444

64.3

47.7

61.8

50.8

59.3

53.7

56.6

80

73.fi

346

71.4

»a 2

69.5

41.6

67.8

45.

65.1

48.3

63.6

61.4

60.

54.4

57.3

81

74.1

35.1

73.3

38.7

70.3

48.8

68.8

45.6

65.9

48.8

63.4

58.

60.8

55.1

58.

89

75.

35.5

73.3

39.1

71.8

48.7

69.

46.1

66.7

49.4

64.8

58.7

61.5

55.7

58.7

83

75.9

35.9

74.1

39.6

73.

43.3

69.8

46.7

67.5

50.

64.9

53.3

69.8

56.4

59.4

84

76.8

36.3

75.

40.1

73.9

43.7

70.7

47.9

68.3

so. 6

65.7

54.9

63.

57.1

60.1

85

77.7

36.8

75.8

40.5

73.6

44.8

71.5

47.8

69.1

51.8

66.5

54.6

63.7

57.8

60.8

86

78.6

37.9

76.7

41.

74.6

44.7

73.3

48.3

69.9

61.8

67.3

55.9

64.5

58.4

61.5

87

79.6

37.6

77.6

41.5

75.5

45.8

73.8

48.9

70.7

53.4

68.

55.8

65.8

59.1

68.8

86

80.5

38.1

78.5

43.

76.3

45.8

74.

49.4

71.5

53.

6a 8

56.5

65.9

59.8

63.9

89

81.4

385

79.4

48.4

77.8

46.3

74.8

50.

78.3

53.6

69.6

57.1

66.7

60.4

63.6

99

83.3

38.9

80.3

43.9

78.

46.8

75.7

50.6

73.1

54.3

ras

57.7

67.4

61.1

64.3

91

83.3

39.3

81.1

43.4

78.9

47.3

76.5

Sll

739

54-8

71.1

58.4

68.9

6k 8

65.1

98

84.1

30.8

83.

43.6

79.8

47.8

77.3

61.7

74.7

554

71.9

59.

68.9

68.5

65.8

93

85.

40.3

83.9

44.3

80.6

4a3

78.8

58.3

75.9

56,

73.7

59.6

69.6

63.1

66.5

94

85.9

40.6

83.8

44.8

81.5

48.8

79.

53,6

76.3

56.6

73.4

60.3

70.4

63.8

67.3

95

86.8

41.

84.7
85.5

45.3

88.3

49.4

79.8

53.3

77.1

57.8

74.3

60.9

71.1

64.5

67.9

96

87.7

41.5

45.7

83.8

s:

80.7

53.9

77.9

57.8

75.

61.5

71.9

65.1

68.6

97

88.6

41.9

86.4

46.8

84.1

81.5

54.4

78.7

58.4

75.8

68.3

73.6

65 8

69.3

98

89.5

48.3

87.3

46.7

84.9

50.9

83.3

55.

79.5

59.

76.6

08.6

73.4

66.5

70.

99

90.4

43.8

88.9

47.1

65.8

61.4

63.1

55.6

80.8

59.6

77.3

63.4

74.1

67.8

70.7

100

dep.

d. lat.

dep.

d. lat.

dep.

d. lat.

dep.

d. lat.

dep.

d. lat.

dep.

d. lat.

dep.

d. lat.

dep. d. lat.

f

* .*

C6 2

2 J»

cc 2

2 OD

po >

■z »

<» 2

2 OD

QD 2

* P

OD 2

2 S»

P 2

2p» a 2

* *

p r

**

P p

**

gr

k*

P p

*H

*'*

M

P p

ft

P p

j$? ^P

P B

4*. «*•

3$

P*

Si

I 9

S*
**

p fco

*• **

**

f"r

5*

P P

P P

L.-

P P

■»• *•

P p

**

•S*

*«

1 C. 5i Pt».

C.JJPu.

C.5*PU.

C.5TU.

C. 4|Pti.

C.4|Fta.

C.4*PU.

C.4PU.

Digitized by VjOOQIC

distance 101 to 150 miles.

TABLE IV.

!.**.

C. *Pt.

C. f Pt.

C. lPt.

CliPt

C. it Pt.

CHPt.

C.SPta.

Si

p4 ri

tt

w ri
**

**
**

03 flD S5

as n

rf fc

fc m

2 £

X CO

m B

fc n

m 86

as a*

at fc

fc ei

ad acjac m

m X

lat. dep.

d. lat.

dep.

d. let.

dep.

d. let. dep.

d.let.

dep.

d. lat. dep.

d.lat.

dep. 4. lat.

dep-

).9 5.

100.5

9.9

99.9

14.8

99.1

19.7

98.

84.5

96.7

89.3

95.1

84.

93.3

3B.7

1.9 5.

101.5

10.

100.9

15.

100.

19.9

98.9

84.8

97.6

89.6

96.

34.4

94.9

39,

1.9 5.1

103.5

10.1

101.9

151

101.

30.1

99.9

85.

98.6

89.9

97.

34.7

95.9

39.4

1. 9 5. 1

103.5

10.9

108.9

15.3

108.

80.3

100.9

85.3

99.5

30.9

97.9

35.

96.1

39.8

1.9 5.8

104.5

10.3

103.9

15.4

103.

90.5

101.9

85.5

100.5

30.5

98.9

35.4

97.

40.3

J.9 5.3

105.5

10.4

104.9

15.6

104.

80.7

108.8

85.8

101.4

39.8

99.8

35.7

97.9

40. C

1.9 5.3

108.5

10.5

105.8

15 7

104.9

80.9

103.8

88.

108.4

31.1

100.7

38.

98.9

40. t

r.9 5.3

107.5

10.6

108.8

15.8

105.9

91.1

104.8

86.3

103.3

31.4

101.7

36.4

99.8

4X3

J.9 5.3

106.5

10.7

107.8

18.

108.9

31.3

105.7

36.5

104.3

31.6

108.6

36.7

109.7

41.7

>.9 5.4

(09.5

10.8

108.8

16.1

107.9

31.5

108.7

86.7

105.3

31.9

103.6

37.1

10L6

49.1

1.9 5.4

110.5

10.9

109.8

16.3

106.9

81.7

107.7

97.

106.3

38.9

104.5

87.4

KB. 6

49,5

1.9 5.5

111.5

11.

110.8

16.4

109.8

31.9

108.8

97.9

107.8

38.5

105.5

37.7

103.5

49.9

1.9 5.5

113.5

U.1

111.8

18.6

110.8

83.

109 6

97.5

108.1

38.8

108.4

38.1

104.4

419

1.9 5.6

113.5

11. 3

118.8

16.7

1U.8

33.3

110.6

97.7

109.1

33.1

107.3

38.4

105.3

416

1.9 5.8

114.4

11.3

113.8

16.9

118.8

88.4

111.6

87.9

110.

38.4

108.8

38.7

106.9

44.

i.9 5.7

115.4

11.4

114.7

17.

113.8

83.6

118.5

98.9

111.

33.7

109.3

39.1

107.8

44.4

1.9 5.7

116.4

11.5

115.7

17.9

114.8

88.8

113.5

88.4

118.

34.

110.9

39.4

108.1

44.0

r.9 5.8

117.4

11.6

li6.7

17.3

115.7

83.

114.5

38.7

113.9

34.3

111.1

39.8

109.

45.9

J. 9 5.8

118.4

11.7

117.7

17.5

118.7

83.8

115.4

88.9

113.9

34.5

118.

40.1

109.9

45.5

>.9 5.9

119.4

1L.8

118.7

17.6

117.7

83.4

116.4

99.9

114.8

34.8

113.

40.4

110.9

45.9

1.9 5.9

190.4

11.9

119.7

17.8

118.7

33.6

117.4

89.4

115.8

35.1

113.9

40.8

111.8

40.3

1.9 6.

191.4

13.

190.7

17.9

119.7

83.8

lift 3

89.6

116.7

35.4

114.9

41.1

118. 7

46.7

5.9 6.

133.4

13.1

181.7

18.

180.6

84

119.3

89.9

117.7

35.7

115.8

41.4

1116

47.1

L9 61

133.4

18.3

188.7

18.9

181.6

84.3

180.3

aai

118.7

36.

116.8

4i.8

114.6

47.5

1.8 6.1

134.4

18.3

183.6

18.3

188.6

84.4

131.3

80.4

119.6

36.3

117.7

48.1

115.5

47.8

18 6.9

135.4

18.4

184.6

18.5

183.6

84.6

133.3

30.6

180.6

36.6

118.6

48.4

116.4

48.9

L8 6.3

138.4

18.4

135.6

1&6

184.6

94.8

133.8

30.9

131.5

36.9

119.6

48.8

117.3

40.6

r.8 6.3

137.4

18.5

136.6

18.8

185.5

85.

134.8

31.1

188.5

37.9

190.5

48.1

118.3

49.

18 6.3

138.4

18.6

187.6

18.9

188.5

85.3

185.1

31.3

183.4

37.4

181.5

415

119.9

49.4

1.8 6.4

139.4

18.7

188.6

19.1

187.5

95.4

188.1

31.6

184.4

37.7

188.4

418

189,1

49-7

>.8 6.4

130.4

13.8

189.6

19.9

138.5

85.6

187.1

31.8

185.4

SB.

183.3

44.1

181.

5Sil

1.8 6.5

131.4

13.9

130.6

19.4

189.5

85.8

198.

39.1

186.3

3B.3

184.3

44.5

188.

50.5

(.8 6.5

133.4

13.

131.6

19.5

130.4

85.9

189.

33.3

187.3

S&6

185.8

44.8

188.9

50.9

18 6.6

133.4

13.1

138.5

19.7

131.4

88.1

130.

38.6

isaa

38.9

186.3

45.1

183.8

5L3

1.8 6.8

134.8

13.2

133.5

19.8

133.4

86.3

131.

38.8

189.9

89.8

187.1

45.5

184.7

31.7

i.8 6.7

135.3

13.3

134.5

90.

133.4

96.5

131.9

33.

130.1

30.5

198.

45.8

185. 6

59.

L8 6.7

136.3

13.4

135.5

90.1

134.4

86.7

138.9

33.3

131.1

89.8

189.

46.3

190.6

59.4

r. 8 6. 8

137.3

13.5

136.5

80.3

135.3

88.9

133.9

33.5

138.1

40.1

189.9

46.5

187.5

39,8

L8 6.8

138.3

13.6

137.5

80.4

138.3

87.1

134.8

33.8

133.

40.3

130.9

46.8

188.4

53.9

L8 6.9

139.3

13.7

138.5

80.5

137.8

87.3

135.6

34.

134.

40.6

131.8

47.3

189.8

53.6

1.8 6.9

140.3

13.8

139.5

80.7

13& 3

87.5

136.8

34.3

134.9

46.9

138.8

47.5

139.8

54-

1.8 7.

141.3

13.9

140.5

80.8

139.3

87.7

137.7

34.5

135.9

419

133.7

47.8

131.9

54.3

18 7.

143.3

14.

141.6

81.

140.3

97.9

138.7

34.7

136.8

41.5

134.6

48.8

138.1

54.7

18 7.1

143.3

14.1

148.4

91.1

141.3

98.1

139.7

35.

137.8

41.8

135.6

48.5

183.

55.1

1.8 7.1

144.3

14.9

143.4

91.3

148.8

98.3

140.7

35.3

13R 8

43.1

136.5

48.8

134.

35.5

i.8 7.9

145.3

14.3

144.4

81.4

143.3

98.5

141.6

85.5

139.7

48.4

137. S

49.9

134.9

55.9

.8 7.3

146.3

14.4

145.4

91.6

144.9

98.7

143.6

85.7

140.7

48.7

138.4

49.5

135.8

59.3

.8 7.3

147.3

14.5

148.4

81.7

145.8

98.9

143.6

86.

141.6

43.

139.3

49.9

186.7

56.6

.8 7.3

148 3

14.6

147.4

31.9

146.1

99.1

144.5

86.8

148.6

48.3

140.3

50.9

137.7

57.

.8 7.4

149.3

14.7

148.4

98.

147.1

99.3

145.5

36.4

1415

43.5

141.9

59.5

13B.6

37.4

►. d. let

dep.

d. let.

dep.

d. tat.

dep.

d.let.

dep.

d.let.

dep.

d.let.

dep.

diet.

dep.

cut

3 n p

•s*

►* P

M

pi

M

p] W

33

P P

3*

V» »B

**

P *■

•5?

P "

XX

XX

as »

XX

» *

a as

* as

XX

as «

as ■

XX

a a;

as •

* ?
P p

7|Pte.

C.7|PUk

C.7*Pta.

• 7Pt«.

C.6|PU.

CC|PU-

COtPta.

CSPta.

LjO(

8 Distance 101 to 150 miles.

TABLE IT.

C.SiFU.

0.3* Pta.

C.9|PU,

C.3Pta.

C.SJPte.

C. 3| Pta.

C.3}Pta.

C.dPte.

g4 u

35 *

CD *•

ii

•4m **

35 CD

"*• -4*

35 4

CO 35

* «

cd z

< m

GO* 35

^ «

**

**

M

W H

i»

fid rf

Be*

<*<* **

55 OB

oi as

% OQ

co 35

35 *5

QD 35

25 cd

00 35

25 CO

OB 35

35 CO

CO 35

35 CO

CO 35

35 a5 OB35

(Mat

dep.

d.lat.

dep.

d.lat.

dep

d. lat.

dep.

d. lat.

dep.

d . lat. dep.

d.lat.

dep.

d. lat. dep.

91.3

43.9

89.1

47.6

86.6

51.9

84.

56.1

81.1

60.3

78.1

64.1

74.8

67.8

71.4

OS. 9

43.6

90.

48.1

87.5

53,4

84.8

56.7

81.9

60.8

78.8

64.7

75.6

68.5

73.1

93.1

44.

90.8

48.6

88.3

53.

85.6

57.9

83.7

61.4

79.6

65.3

76.3

69.3

73.8

94.

44.5

91.7

49.

89.3

53.5

86.5

57.8

83.5

63.

80.4

68.

77.1

69.8

73.5

94.9

44.9

93.6

49.5

90.1

54.

87.3

58.3

84.3

08.5

81.9

66.6

77.8

70.5

74.3

95.8

45.3

93.5

50.

00.9

54.5

88.1

589

85.1

63.1

81.9

67.3

78.5

71.8

75.

96.7

45.7

94.4

50.4

91.8

55.

80.

59.4

85.9

63.7

83.7

67.9

70.3

71.9

75.7

97.6

46.9

95.9

50.9

93.6

55.5

89.8

60.

88.7

64.3

83.5

68.5

80.

73.5

76.4

98.5

46.6

96.1

51.4

93.5

56.

90.6

60.6

87.5

64.9

84.3

69.1

80.8

73,3

77.1

99.4

47.

97.

51.9

94.4

56.6

91.5

61.1

88.4

65.5

85.

69.7

81.5

73.9

77.8

100.3

47.5

97.9

59.3

95.8

57.1

93.3

61.7

80.9

66.1

85.8

70.4

83.3

74.5

78.5

101 9

47.9

98.8

53.8

96.1

57.6

93.1

63.8

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68.7

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75.8

79.8

103.9

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99.7

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67.3

87.4

71.7

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67.9

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73.3

84.5

76.6

80.6

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54.9

98.6

59.1

95.6

63.9

92.4

68.3

88.9

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85.3

77.8

81.3

104.9

49.6

103.3

54.7

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59.6

96.5

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73.6

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55.3

100.4

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69.7

90.4

74.8

86.7

78.6

83.7

106.7

50.5

104.1

55.6

101.8

60.7

98.1

65.5

94.8

70.3

91.3

74.9

67.4

79.8

83.4

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50.9

104.9

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61.3

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51.3

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56.6

103.9

61.7

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66.7

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71.5

93.8

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88.9

80.6

84.9

100.4

51.7

106.7

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103.8

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100.6

67.3

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73.1

93.5

76.8

89.7

81.3

85.6

110.3

53.9

107.6

57.5

104.6

63.7

101.4

67.8

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78.7

94.3

77.4

90.4

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58.5

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119.3

58.9

107.9

64.3

103.9

69.4

100.4

74.5

96.6

79.3

93.6

83.9

88.4

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53.9

111.1

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104.8

70.

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75.1

97.4

70.9

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84.6

89.1

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54.3

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108.9

65 3

105.6

70.5

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75.7

98.3

80.6

94.1

85.3

89.8

115.7

54.7

113.9

60.3

109.8

65.8

108.4

71.1

103.8

76.3

98.9

81.9

94.8

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116.6

55.9

113.8

60.8

110.6

66.3

107.3

71.7

103.6

76.9

99.7

81.8

95.6

86.6

91.3

117.5

55.6

114.6

61.3

111.5

66.6

108.1

73.3

104.4

77.4

100.5

83.5

96.3

87.3

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61.8

113.4

67.3

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73.8

105.8

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101.3

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97.1

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93.6

119.3

56.4

116.4

63.8

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109.8

73.3

106.

78.6

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83.7

97.8

88.6

93.3

190.9

56.9

117.3

63.7

114.1

68.4

110.6

73.9

106.8

79.3

103.8

84.4

98.5

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191.1

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63.8

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80.4

104.4

85.6

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95.5

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64.1

116.7

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86.3

100.8

91.3

96.3

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58.6

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70.4

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76.7

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106.7

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97.6

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190.9

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60.7

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109.8

90.1

105.8

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100.4

199.3

61.1

199.1

67.4

133.7

73.5

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79.4

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85.8

110.5

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113.6

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130.5

60.8

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180.5

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153

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79.9

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79.7

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98.3

114.8

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109.6

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137.6

73.5

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80.9

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86.7

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92.9

130.6

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115.6

104.6

110.3

156
157
158
159
160

141.9 67.1

138.5

74.

134.7

80.7

130.5

87.9

126.1

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191.4

99.6

116.3

105.4

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142.8 67.6

139.3

74.5

135.5

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106.1

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143.7 68.

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81.7

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83.8

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114.6

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116.

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161
163
163
164
165

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168
169
170

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152.8 72.3

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171
172
173
174
175

155.5 73.5
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139.7

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176
177
173
179
180

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160.9 78.1
161.8 76.5
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157.9

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84.4
84.9

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153.5

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99.5

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148.8

149.7

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131.9
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13a4

163.6 77.4
164.5 77.8
165.4 78.2
166.3 78.7
167.2 79.1

159.6
160.5
161.4
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86.7
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182
183
184
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169.9 80.4
170.9 80.8
171.8 81.9

164.

164.9

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88.9
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132.3
132.9
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180
187
188
189
190

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173.6 82.1
174.5 82.5
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159.6
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193
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179.9 85.1
180.8 85 5

179.9
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92.9
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93.8
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101.8
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900.8 9.9 boo. 19.7

108.8 29.5

197.1 30.9

195.

48.8

198.3 58.3

189.3 67.7

185.7 7S.9

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801.8 9.9

801. 10. 8

190.8 29.6

198.1 39.4

195.9

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193.3 58.6

190.2 66.1

186.6 773

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308.8 10.

908. 19.9

300.8 89.8

190.1 39.6

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194.3 58.9

191.1 68.4

187.5 77.7

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803.8 10.

803. 80.

801.8 29.9

800.1 39.8

197.9

49.6

195.9 59.9

192.1 68.7

186.5 7b. I |

205

804.8 10.1

804. 80.1

208.8 30.1

801.1 40.

198.9

49.8

196.9 50.5

193. 69.1

189.4 78.5

906

905.8 10.1

905. 90.3

203.8 30.2

208. 40.2

199.8

50.1

197.1 50.8

194. 62.4

190,3 7&9

807

808.8 10.8

206. 80.3

204.8 30.4

203. 40.4

900.8

50.3

198.1 60.1

194.9 69.7

191.3 79.2

908

807.7 10.8

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805.7 30.5

304. 40.6

201.8

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199. 60.4

195.8 70.1

192.3 79.6;

900

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808. 90.5

806.7 30.7

805. 40.8

303,7

50.8

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196.8 70.4

193.1 80. ,

810

900.7 10.3

909. 20.6

807.7 30.8

806. 41.

903.7

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801. 61.

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911

910.7 10.4

910. SO. 7

808.7 31.

206.9 41.2

804.7

51.3

901.9 61.3

19a7 71.1

194.9 80.7

918

311.7 10.4

811. 90.8

800.7 31.1

207.0 41.4

205.6

51.5

303.0 61.5

199.6 71.4

195.9 81.1

813

818.7 10.5

813. 20.9

810.7 31.3

208.9 41.6

808.6

51.8

903.8 61.8 1200.5 71.8

196.8 81 5

814

213.7 10.5

213. 81.

311.7 31.4

909.9 41.7

907.6

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901.5 72.1

197.7 81.9

815

214.7 ia5

314. 81. 1

818.7 31.5

210.0 41.9

808.6

52.2

805.7 68.4

903.4 79-4

198.6 82- J

816

915.7 10.6

315. 81.3

213.7 31.7

911.8 42.1

909.5

53.5

806.7 62.7I203.4 72.8

199.6 68,7'

817

316.7 10.6

916. 81.3

314.7 31.8

212.8 42.3

210.5

53.7

907.7 63.

904.3 73.1

900.5 83.

818

217.7 10.7

317. 81. 4

315.6 38.

213.8 42.5

211.5

53.

908.6 63.31205.3 73,4

201.4 8X4

819

218.7 10.7

317.0 81.5

916.6 33.1

214.8 42.7

213.4

53.3

800.6 63.6

996.2 73.8

208.3 83.*

880

319.7 10.8

818.9 21.6

817.6 33,3

215.8 42.9

213.4

53.5

210.5 63.0

207.1 74.1

803.3 84.2

981

880.7 10.8

219.9 21.7

aiae 32.4

916.8 4a 1

814.4

53.7

811.5 64.3

908.1 74.5

804.8 84.4

898

881.7 10.8

220.9 81.8

319.6 33.6

217.7 43.3

315.3

53.9

812.4 64.4

290. 74.6

805.1 rS. •

883

883.7 10.9

881.0 31.9

990.6 33.7

218.7 43.5

316.3

54.3

313.4 64.7

210. 75.1

208. £&3

894

883.7 11.

983.0 83.

831.6 33.9

219.7 43.7

217.3

54.4

814.4 65.

210.9 75.5

908.9 65 7

885

384.7 11.

323.9 28.1

823.6 33.

220.7 43.9

218.3

54.7

315.3 65.3

211.8 75.8

907.9 86.1.

896

885.7 11.1

8349 28.2

883.6 33.3

831.7 44-1

210.2

54.9

816.3 65.6

212,6 78.1

808.8 80.5 |

927

286.7 11.1

285.9 88.8

884.5 33.3

838.6 44.3

280.3

55.2

8173 65.9

813.7 76.5

990.7 8K.9 1

988

887.7 11.3

396.9 22.3

835.5 33.5

883.6 44.5

831.8

55.4
55.6

818.3 66.8

214.7 76.8

9J6L6 87 3

829

898.7 11.8

887.9 88.4

886.5 33.6

834.6 44.7

323.1

819.1 66.5

815: 6 77.1

211.6 87.6

830

289.7 11.3

298.9 28.5

837.5 33.7

825.6 44.9

233.1

55.9

830.1 66.8

816.6 77.S

833.5 83. ,

831

230,7 11.3

990.9 28.6

296.5 33.9

826.6 45.1

224.1

56.1

821.1 67.1

817.5 77.8

813.4 8B.4

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831.7 11.4

830.9 98.7

930.5 34.

287.5 45.3

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56.4

829, 67.3

318.4 78 3 *M.3 88? ;

833

838.7 11.4

831.9 99.8

230.5 34.2

398.5 45.5

286.

56.6

883. 67.6

810.4 78.5

215.3 80. i 1

834

833.7 11.5

838.9 88.9

931.5 34.3

930.5 45.7

237.

56.0

293.9 67.9

820.3 78.6

816.2 895

835

834.7 11.5

933.9 83.

938.5 34.5

830.5 45.8

238.

57.1

234.9 68.9

881.3 79.2

217.1 89.9,

936

835.7 11.6

934.9 23.1

833.4 34.6

831.5 46.

298.9

57.3

235.8 68.5

8839 79.5

818. 90.3 1

837

838.7 11.6

935.9 23.2

834.4 34.8

338.4 46.8

299.9

57.6

886.8 68.8

393.1 79.8

«9l 90.7

838

837.7 11.7

936.9 23.3

835.4 34.9

833.4 46.4

230.9

57.8

887.8 69.1

884.1 80.3

219.9 91.1 ,

830

838.7 11.7

237.8 23.4

836.4 35.1

834.4 46.6

231.8

58.1

898.7 69.4

295. 80.5

880.8 91.5

840

830.7 11.8

238.8 23.5

837.4 35.2

835.4 46.8

833.8

5a3

829.7 69.7

286. 80.0

881.7 9L5

941

940.7 11.8

230.8 23.6

238.4 35.3

236.4 47.

933.8

58.6

230.6 70.

296.0 81.2

232.7 98.2

948

841.7 11.9

240.8 23.7

230.4 35.5

237.4 47.2

934.7

58.8

231.6 70.2

227.0 61.5

283.6 92.4 |

943

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841.8 23; 8

94a 4 35.7

338.3 47.4

935.7

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998.8 81.9

824.5 93.

944

243.7 18.

843.8 23.9

241.4 35.8

239.3 47.6

936.7

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233.5 70.8

929.7 82 2

985.4 93.4

945

844.7 18.

243.8 24.

242.3 35.9

84a 3 47.8

937.7

50.5

234.5 71.1

230.7 62.5

826.4 93.*

846

245.7 18.1

244.8 24.1

943.3 36.1

241.8 48.

836.6

59.8

235.4 71.4

231.6 88.9

287.9 941

247

346.7 19.1

245.8 94.2

944.3 36.2

242.3 48.2

839.6

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832.6 83.2

239,2 94.5

248

847.7 18.3

246.8 24.3

245.3 36.4

243.3 48.4

840.6

60.3

237.3 72.

233.5 83.5

289.1 94.9

249

248.7 13.3

247.8 24.4

246.3 36.5

244.2 48.6

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60.5

938.3 72.3

234.4 83.9

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849.7 18.3

248.8 24.5

247.3 36.7

245.2 48.8

843.5

60.7

230.2 79.6

835.4 84.3

231. 95.7

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dep

d.lat dep.

HI. 7

85.9

177.3

94.8

173.4

10X3

167.1

111.7

161.4

119.7155.4 137.5

14a 9

135.

143.1

1<*2,6

88.4

178.1

95.9

17X3

10X8

168.

113.3

163.4

120.3 156.1 138.1

149.7

135.7

142.8

202

1<*.5

86.8

179.

95.7

174.1

104.4

16a 8

113.8

16X1

130.9 156.9 128.8

150.4

136.3

14X5

303

1*4.4

87. S

179.9

96.9

175.

104.9

169.6

11X3

163.9

131.5 157.7 189.4

151.3

137.

144.3

304

185.3

87.6

180.8

96.6

175.8

105.4

170.5

11X9

164.7

233 11158.5 130.1

151.9

137.7

145.

305

'h6,3

68.1

181.7

97.1

176.7

105.9

171.3

114.4

165.5

133, 7; 159. 3 130.7

152.6

13a 3

145.7

806

807

137.1

88.5

189.6

97.7

177.5

106.4

173.1

115.

166.3

19X3 160. 131.3

153.4

139.

146.4

l**.

88.9

183.4

98.1

178.4

106.9

179.9

115.5

167.1

133. 9| 160. 8 133.

154.1

139.7

147.1

308

1*3.9

89.4

184.3

98.5

179.3

107.4

173.8

116.1

167.9

124.5

161.6 133,6

154.9

140.4

147.8

309

13J.8

89.8

185.9

99.

180.1

10a

174.6

116.7

16a 7

125.1

163.3 13X3

155.6

141.

14a 5

310

19a 7

90.9

186.1

99.5

181.

108.5

175.4

117.3

169.5

125.7

16X1 13X9

156.3

141.7

149.9

311
313
913
814

*15

MI. 6

90.6

187.

99.9

181.8

209.

176.3

117.8

170.3

136.3

16X9 134.5

157.1

142,4

149.9

193.5

91.1

187.8

100.4

183.7

109.5

177.1

118.3

171.1

136.9

164.7 135.1

157.8

143.

150.6

193.5

91.5

188.7

100.9

183.6

no.

177.9

118.9

171.9

137.5

165.4 135.8

158.6

14X7

151.3

,194.4

91.9

189.6

101.4

184.4

110.5

17a 8

119.4

178,7

12a 1

166.9 136.4

159.3

144.4

153.

1
195.3

93.4

190.5

101.8

185.3

111.

179.6

ISO.

173.5

12a 7

167. 137.

160.

145.1

153.7

916
317
3i8
819
820

il96.8

99,8

191.4

109,3

86.1

111.6

180.4

130.5

174.3

139.3

167.7 137.7

160.8

145.7

15X4

197.1

93.9

199.3

103.8

187.

112.1

131.3

131.1

175.1

139.9

168.5 138 3

161.5

146.4

154.1

ua

93.6

193.1

103.3

187.8

113.6

182.1

131.7

175.9

130.5

169.3 13a 9

163.3

147.1

154.9

U98.0

94.1

194.

103.7

188.7

11X1

183.9

122.8

176.7

131.1

170.1 139.6

16X

147.7

155.6

'l99.8

94.5

194.9

104.3

189.6

11X6

18X8

133.8

177.5

131.6

170.8 140.9

16X8

148 4

156.3

931
332
333
334
235

|200.7

94.9

195.8

104.7

190.4

114.1

184.6

13X3

178.3

133,8

171.6 140.8

164.5

149.1

157.

1201.6

95 3

196.7

105.1

191.3

114.6

185.4

13X9

179.1

139,8

173.4 141.5

165.8

149.fi

157.7

'202.5
20X4

95.8
96.9

197.6
198.4

105.6
108.1

133,1
19X

115.3
115.7

186.3
187.1

it'

179.9
180.7

13X4
134.

17X3 143,1
17X9 143.7

166.
166.7

150.4
151.1

158.4
159.1

201.3

96.6

199.3

106.5

19X8

116.9

187.9

135.5

181.5

134.6

174.7 14X4

167.5

151.8

159.8

336
327
338

3!9
830

-295. 3

97.1

900.3

107.

194.7

116.7

188.7
189.6

136.1

182.3

135.3

175.5 144.

168.3

153.4

1G0.5

205. 1

97.5

901.1

107.5

195.6

117.9

136.7

18X1

135.8

176.2 144.6

168.9

15X 1

161.9

207.

97.9

J03.

107.9

196.4

117.7

190.4

137.8

18X9

136.4

177. 145. 3

169.7

153.8

161.9

■ 2J7.9

98.3

202.8

108.4

197.3

118. 2

191.9

137.8

184.7

137.

177.8 145.9

170.4

154.5

162.6

23*. 8

98 8

303.7

108.9

loa 1

lias

193,1

19a 3

185.5

137.6

17a 6 146.5

171.9

155.1

16X3

831
833
333
834
235

J 0.7

•in. 6

•211.5

J213.4

99.9
90.6

100.

100.5

204.6
305.5
306.4
307.3

103.4
103.8
110.3
110.8

199.
199.9
300.7
301.6

119.3
119.8
130.3
130.8

193.9
19X7
194.6
195.4

138.9
1394
130.
1305

186.3
187.1

isa

183.8

138.2
138.8
133.4
140.

179.3 147.3
180.1 147.8
180.9 14a 4
181.7 149.1

171.9
172.6
173.4
174.1

155.8
153.5
157.1
157.8

164.
164.8
165.5
166.3

313.3
•214.9
215. 1
214.1
217.

100.9
101.3
101.8
102.2
102.6

308.1

303.

203.9

210.8

311.7

111.9
111.7
112.9
112.7
113.1

303.4

20X3

2041

205.

305.9

131.3
131.8
132.4
132.9
13X4

196.9
197.1
197.9

19a 7

199.6

131.1
131.7
132.3
13-2.8
13X3

189.6

190.4

191.3

193.

193,8

140.6
141.2
141.8
143.4
14X

183.4 149.7
18X9 150.4
134. 151.
184.7 151.6

185.5 153.3

174.9
175.6
176 3
177.1
177.8

158.5
15J.2
151.8
1G0.5
161.3

166.9

167.6

168.3

169.

169.7

336
837
333
339
340

1
'217. 9

213.8
'211.7
: 2X1.6

22L5

101

1015

103.9

104.3

104.8

313.5
31X4
314.3
915.3
316.1

11X6

114.1

114.5

115.

115.5

306.7
207.6
303.4
203.3
210. 1

13X9
134.4
134.9
125.4
136.

300.4

201.9

203.

202.9

20X7

13X9
134 4
135.
135.5
136.1

19X6

194.4

195.3

136.

196.8

14X6
144.3
144.8
145.4
145.9

186.3 153.9
137.1 15X5
187.8 154.2
188.6 154.8

189.4 155.4

17a 6
179.3
130.1
130.8

181.5

161.8
163.5
163 3
16X9
164.5

170.4
171.1
171.8
172.5
173.8

341
343
343
344
345

ItW.4
'22X3
224.2
225.1
22 5.

nv 9

115.6
101.
10X5
100.9

917.
217.8
318.7
919.6
220 5

116.

116.4

116.9

117.4

117.8

211.

211.9

312.7

21X6

214.4

136.5

137.

127.5

133.

133.5

204.5
305.4

306.9

•207

307.9

136.7
137.2
1378
133.3

13a 9

197.6
19a 4
199.3
200.
200.8

148.5
147.1
147.7
148.3

14a 9

190.8 156.1

190.9 15& 7
191.7 157.3
D3.5 159.
193.3 153.6

139.3

183.
133.8
134.5
135.3

165.9
165.9
166.5
167.3
1879

173.9

174.7
175.4
178.1
176.8

846
347
348
24')
950

c

dep.

d-lat.

top.

d.let.

dap

d. lat.

dep

d. lat.

dep d. lat.

dtp. d. lat.

dep.

d. lat.

dep. d. lat.

75 •»

y z

2 »

« Z

2 «

» Z

z »

0D Jg

Z »

m a»

2 .» P 8 2

2 30

JD 'X

2?» »2

< .<

p* i«

<:»

.« W

k*

« K

'**

w *W

;«*

P* pi

$$ r *

'**

zt

^ *'*

9

s

z*
**

s?

kz
**

Z*
**

P* w

**

P «

i* ^

w ?

' *» <w

« P»

•»- ■*•

p* p

•+• H-

p »

C.SI

•s*

M

Pti.

C.5|Pu.

C 5* PU.

C 5 Ptt.

C. 4| Ptt.

C. 4| Ptt.

C. 4* PU.

C.4Pt«.

Digitized by VjOOQIC

106 Distance 251 to 300 miles.

TABLS xv.

1

C. *P£.

C. « PL

C. | PL

O. 1 Pt.

CI* PL

aupt.

0.11 PL

CSPta.

•4m H"

$*

*t

•

8
i

e

Ml

is

**

«w J**

« «

ti

£*
**

E4 (4

* *

55 £

» r fc

as OB OQ fc

as 2

2 5

SB «

m SB

fc ari

«* SB

SB m

a SB

SB a*

00 SB

fc OB

m SB

d. 1st. dep.

d. laL dep.

d. lat.

dep.

d. lat. dep.

d.lat

dep.

d. lat. dep.

d.lat.

dep.

d.laL

«*P-

950.7

13.3

840.8 84.6

848.3

36.8

846.3

49.

843.5

61.

340.3

78.9

336.3

84.6

931.9

96,1

85-4

851.7

13.4

850.8 34.7 1849.3

37.

847.3

49.3

844.4

61.3

841.1

73.8

837.3

84.9

839.8

96,4

85

858.7

18.4

851.8 84.8

350.3

37.1

348.1

49.4

345.4

61.5

843.1

73.4

338.9

85.3

833.7

90.6

85*

853.7

18.5

853.8 84.9

351.3

37.3

849.1

49.6

846.4

61.7

843.1

73.7

939.9

85.6

934.7

97.1

855

854.7

18.5

853.8 85.

858.8

37.4

850.1

49.7

847.4

68.

344.

74.

940.1

85.0

935.0

97.0

850

855.7

18.6

854.8 85.1

853.9

37.0

851.1

49.9

346. 3

68.3

845.

74.3

341.

80.8

939.5

98.

857

856.7

18.6

355.8 85.3

854.8

37.7

853:1

50.1

849.3

63.5

845.0

74.6

348.

86.6

937.4

98.1

850

857.7

18.7

856.8 85.3

855.8

37.0

853.

50.3

850.3

68.7

846.9

74.9

948.0

86.9

838.4

98.7

35U

858.7

12.7

857.6 85.4

856.3

38.

854.

50.5

351.3

62.9

847.8

75.9

843.0

87.3

930.3

99.1

8b0

359.7

13.8

858.7 85.5

357.8

38.1

855.

50.7

858.8

63.8

84a 8

75.5

844.8

87.6

840.3

90.5

881

860.7

13.8

359.7 85.6

358.8

38.3

858.

50.0

853.8

63.4

849.8

75.8

945.7

87.9

941.1

90.9

963

961.7

13.9

SH0.7 857

859.8

38.4

857.

51.1

354.1

63.7

850.7

76.1

346.7

86.3

848.1

100.3

863

868.7

13.9

361.7 85.8

860.3

38.6

857.0

51.3

355.1

63.9

851.7

76.3

847.6

88.6

943.

K8.I

854

863.7

13.

868.7 85.9

361.1

38.7

858.0

51.5

356.1

64.1

858.6

76.6

348.0

88.9

943.9

101.

865

864.7

13.

863.7 36.

368.1

38.0

'250.0

61.7

857.

64.4

853.6

76.9

849.5

89.3

344.8

ML 4

886

865.7

13.1

864.7 36.1

863.1

39.

860.9

51.9

858.

64.0

854.5

77.9

350.5

80.6

345.8

ML8

887

8H6.7

13.1

365.7 88.8

804.1

39.3

961.9

53.1

859.

64.0

855.5

77.5

851.4

89.0

346,7

M8.I

968

867.7

13.3

366.7 86.3

8H5.1

39.3

8A8.9

53.3

859.9

65.1

356.5

77.8

353. 3

00.3

947.8

109.1

ft*

Si>8.7

13.3

867.7 36.4

366.1

39.5

363.8

58.5

860.9

65.4

857.4

78.1

353.3

90.6

348.5

M8.9

870

419.7

13.3

368.7 86.5

367.1

39.6

364.8

58.7

361.9

65.6

358.4

78.4

354.3

01.

849.4

MAI

871

870.7

13.3

369.7 36.6

868.1

39.8

365.8

58.9

368.6

65.8

359.3

78.7

855.3

01.3

858,4

193.7

878

871 7

13.3

970.7 36.7

369.1

39.0

306.8

53.1

363.8

66.1

360.3

79.

850.1

01.6

851.3

104.1

873

878.7

13.4

971.7 86.8

370.

40.1

867.8

53.3

364.8

66.3

361.3

79.3

357.

99.

359.8

1013

874

/73.7

13.4

378.7 38.9

371.

40.3

368.7

53.5

365.8

66.6

363.3

79.5

358.

09.3

353.1

104.9

875

874.7

13.5

873.7 87.

378.

40.4

369.7

53.6

366.7

66.8

863.8

79.6

358.9

98t6

354.1

Mtt.9

878

375.7

13.5

374.7 87.1

873.

40.5

870.7

53.8

867.7

67.1

964.1

60.1

350.9

93.

955.

105.1

877

876.7

13.6

375.7 87.3

874.

40.6

871.7

54.

868.7

67.3

865.1

60.4

360.6

03.3

855,9

M6,

878

877.7

13.6

876.7 87.8

875.

40.8

378.7

54.3

369.6

67.5

866.

80.7

361.7

08.7

836,8

106.4

879

878.7

13.7

877.7 87.3

876.

40.9

873.6

54.4

370.6

67.8

867.

81.

368.7

04.

957.8

106.0

880

879.7

13.7

878.7 87.4

877.

41.1

374.6

54.6

371.6

68.

867.9

81.3

863.6

04.3

958.7

107. S

891

880.7

13 8

379.6 87.5

878.

41.8

875.6

54.8

373.5

68.3

368.9

81.0

364.6

04.7

950.0

187. 5

998

881.7

13.8

380.6 87.6

878.0

41.4

376.6

55.

373.5

68.5

360.0

81.9

365.5

05.

3C0.5

M7.9

883

888.7

13.9

381.6 87.7

879.9

41.5

877.6

55.3

874.5

68.8

870.6

88.3

366.5

85.3

961.5

108.3

884

883.7

13.9

288.6 87.8

880.9

41.7

878.5

55.4

375.5

60.1

871.8

88.4

367.4

05.7

969.4

M8.7

885

884.7

14.

383.6 87.9

881.9

41.8

870.5

55.6

870.4

69.3

878.7

88.7

368.3

96.

963.3

109.1

896

885.7

14.

384.6 88.

888.9

43.

380.5

55.8

377.4

69.5

873.7

83.

369.3

96.4

364.3

190.4

887

886.7

14.1

985.6 88.1

883.9

48.1

881.5

56.

878.4

09.7

874.0

83.3

370.3

06.7

865.9

189.1

888

887.7

14.1

386.6 38.3 384.9

43.3

3835

56. 3

379.3

70.

875.6

83.6

871.3

97.

966.1

118. 1

889

8887

14.3

387.6 38.3

885.9

48.4

8*3.4

56.4

380.3

70.3

876.6

83.9

373.1

07. 4*97.

110.4

890

889.7

14.3

888.6 38.4

386.9

48.6

884.4

56.6

881.3

70.5

877.5

84.9

373.

07.7

|867.9

11 h

891

890.6

14.3

389.6 88.5

887.0

48.7

885.4

56.8

888.8

70.7

878.5

84.5

374.

98.

368.8

UL4

898

891.6

14.3

390.6 38.6

388.8

48.8

886.4

57.

883.8

71.

370.4

84.8

374.0

98.4

860.8

111. 7

893

398.6

14.4

891.6 88.7

380.8

43.

387.4

57.8

384.8

71.8

360.4

85.1

375.9

08.7

970.7

119.1

894

893.6

14.4

893.6 88.8

390.8

43.1

888. 4

57.4

885.3

71.4

381.3

85.3

876.8

00.

971.0

119.5

895

J04.6

14.5

893.8 88.9

891.8

43.3

380.3

57.6

886.1

71.7

388.3

85.6

877.8

90.4

979.5

119.9

896

895.6

14.5

894.6 39.

7)3.8

43.4

890.3

57.7

887.1

71.0

383.3

85.0

3787

90.7

373.3

HX3

897

898.6

14.6

895.6 89.1

393.8

43.6

391.3

57.9

886.1

78.9

384.9

66.9

379.0

100.1874.4

113.7

9«8

•197.6

14.6

896.6 89.9

894.8

43.7

393.3

58.1

889.1

78.4

385.3

66.5

380.8

.100.4375.3

114.

8W

39*. 6

14.7

817.8 39.3 ,895.8

43.0

893.3

58.3

890.

78.7

366.1

86.8

381.5

100.7876.9

114.4

300

899.6

14.7

398.6 39.4 896.8

44.

894.3

58.5

891.

78.0

387.1

67.1

369.5

101.1

877.8

1148

•

5

dep.

d.lat.

dep. d. lat.

dep.

Hat.

dep.

d. lat.

dep.

d. lat.

dep.

d. lat.

dep. d. laL

dep.

d.laL

3*

W W

^^H

33

w p

3 3

W (Q

33

w w

33

ffl p

<*

P» P

33

n GB

5?

pa as

» a?

as j»

m a*

as P"

33

* 9"

S3

4M 4te

as a
33

P a

Pp

as »

» a:

aj*

»5

;5

P%

C. 71 Pta.

C7IJU.

C.7*Pte,

C. 7 Pta.

CO* Pta.

C6tPt«-

C.6±Ptt.

COPU.

Digitized by VjOOQIC

Distance 251 to 300 miles. table nr.

107

c.atFu.

H M

d-lat.

ad as

H
% Of

.9

397.8
17
.6
15

931.4
L3
233.9
334.1

L9
L8

837.7
1.7

839. «

949.9
841.4
348.3
343.9
344.1

0.8* Fta.

ad 2
» as

107.3 991.4 1193 815.3 199.

107.7

108.9

108.

11.4
1.9
LI

119.:

119.
994.9 190.!

r.9|Pti.

Mi **

ad m as

d. lat. dep. d. lat. dep.

d. lat. dep.

908.7
118.81916.1 199.6 909.5

3 917.

7 917.
8819

100.5 985.8
169.9395.7
119.3 997.5

110.7

llt.9999.3 199.6993. 133.7916.9 144.4 908.8

111
119.
119.

930.9
931.1
4 831.9 194.

183.5 994.

134.9

134.7

993.9

1.7
995.6 135.9|918.7

1I33|3397 m9|«7i3

118.9 999.8 194.4 998.

119.7

114,

114.

115.
115.

934.6 195.

3 835.5 __.

0930.4 195.

937.9 _I

4838.1 197.

1949.5 lift

859.4 1I&4

851.3 118.9

858.9 119.3

853.1 119.7

854.
854.9

855.8
856.7
857.6

5

4
3
861.3

J 363. 2

mi

■864.
1864.9

1965.8
'969.7

967.6
5
4

870.3
871.8

199.1
190.
191.
191.4
1319 331

189.3
198.7
183.1

183.6
194.

898.4

190.
191.1
181.
199.1

7|819.6
4
3
9

6 991.3

4 898.9

195.9 999.
196.3989.
196.8 830.

9

1.7

3|831.6

839.4.

845. 115.9)839. 197.7 __ ..

345.9 116.3 939.9 198.9 833.3

846.8 116.79498 1897

847.7 117.3 941.6 189.

849.11 117.6 949.5 199.61835.

4 130.

944.3 130.

945.3 131.

946.1 131.5

946.9 138.

3 941.
4948.7 138.9 941.
4949.
9943.
3944.5

947.8 139.:
7

949.6 133.'

5 133.1

3 134.:

3946.9

135.8 947.

136.9 947.

130.1
130.6
13L1

131.6
139.1
139.6
1338

C.3FU.

*«d

35 ad

d. lat.

310.4
911.9
818.

8199
813 7
314.5
815.4

139.4
140.

301.6

4

US. 7
136.8

136.8
137.3
137.8
138.3
138.8

139.

139.8

834.3 140.4

140.9

9 141.4

3835.3 150.5 917.7 161.4

8 151.1918.5 168.

837. 151.7 219.3 169.6

897.8 158.9390.1 163.9

1936.

6 837.

7 141.

6 148.4

4 143.9

3 143.4
9

940.9 143.9333.8

144.5
9 145.
7 145.5
6 146.

146.5

8945.

3 147.
147.5
148.1

148.6

859.3 134.
1 135.
854.

854.9

855.8 136.71348.7 I49.I941.1 161. 1

917.
917.8

919.5
3

891.3 147.6

983.7
894.5

150.

C. 3| Fta.

•it
Ob fc

dep. d . lat. dep. d. lat dep.

140.5 903.9
141. 1 904.

141.7 904.8

143.3905.6

149.8 906.4
143.3 307.8

143.9 906.

145.

145.51910.4
146.1911.8
146.7 919
147.3 3198

149.5
150.1
159 7
151.3
151.9

158.5
153.1
153.7
154.3
154.9

155.5

156.1
156.7
157.3
157.9

913.7

148.3 314.5
148.9 915.3

149.4 316.1

156.5
159.
159.
160.

916.9 160. 8 908.

C.3|Pte.

as ad ad as

194.

194.8

195.6

196.3

197.1

159.3
159.9
160.5
161.1
161.8

197.9 169.4

198.7 168.

199.4 163.7

900.2 164.3

901. 164.9

801.8
.5

.3

304.1
304.8

165.6
166.3
166.8
167.5
166.1

305.6 1697

4 169.4

8 170.

9 170.7
7 171.

1906.
6 907.
9907.

9 339.5 153.

839.3 153.

331.1 154.

338. 155.

155.

3 331.
9989.2

4 983.3

.7 164.

5 165.

3 165.6

834.1 166.3

9 166.8

5394.9

833.6 156.

834.5 156.

333.3 157.8

336.1 157.

837. 158.

1895.7
7339*

7 167.

5 168.

837.3 168.6

1 169.

31998.9 169.

194.4 956.6 U7.9

194.8 357.5 137.6

185.3358.4 138.
185.7 959.3

180.1360.8 139.

196.6 361.

187.

197.4

197.

861.9 140.
8
7

953.9
954.7 1597

1593 346.

399.8 140.5 355.6 153.3 347.

1 164.4837.7 179,3
946.9 165. 838.6 179.9

.8 198.8

.1 199.5
.8

1993^364.6 1414)957.3 154.3940.4 166.7941 178.7831.9 198.3)89913 9015

6363.7 140.9356.5 153.7948.

' 3 "

1351.
138.6 353.3
1853.

349.6 149.6
.5 150.1
3 150.
151.1
151

349
3498
6 343.6
344.5 163,3
345.3 163.9

837.8 158.

338.6 159.4

339.5 160.

340.3 1695

r.8 165.5939.4
L6 166.1940.3

5 1719 900.8 18

9193 1796101.5 18_ .

311 173.31093 183.3

C.SfFte.

as «d

00 OB 2

186.7
187.5
188.8
188.9

169.7
190.4
1919
1919
1996

193.4
194.1
194.9
195.6
196.4

197.1
197.8
198.6
199.3
1

3 100.

1696
169.9
169.9
170.6
171.9

171.9
173.6
173,3
173.9
174.6

175.3
175.9
176.6
177.3
179

1796

179.3

189

1896

181.3

C.4FH.

fcod odas

d.lat dep.

311.8 1798 393. 184.
338.7 15981880.9 163.88196 174.5983.8 184.7

4 3194 175.1

814.1 175.

314.9 179

815.7 177.

3194 177.

4 817.1

3 178.3

319 179

3198 179.

3319.5 180.9(910.
3

5 909.

8 330.3 1898 311.

1617 8397 1793

179 9

174.5

996.1 175.1

3399 175.7

1693834.5
1698835.3

177.5
1791

904.5 185.4
7905.9 186.

4909

906.7
6 307.5

9399.7 170.48311 18148119 1991

830.5 171 891.9 18919197 1997

831.3 171.68998 1897813.4 1994

1 173.9933.4 1893 314.1 194.1

3399 1798394.3 184. 314.9 194.8

334.9 184.6^15.
835.7
5
837.3 186.
899 187.

185.83194
185.9 917.

5 917.

1319

8 819.1

7

187.4
189

9099 1897

9 189.4

7 190.1

4 190.7

9 1914

9398 187.1
1894

831.1 189.7)891.

6 1994

1991

1 1996

.8 197.4

6 1991

177.5
1799
1799
179.6
1898

181

181.7

1894

1891

1898

184.6

185.3

189

1897

187.4

1891
1898
189.5
1999
1999

1916

1993

193.

1997

194.5

195.3
195.9
1)9 6
197.3
109

198.7
199.4
800.1
800.8
901.5

9093
3099
3096
904.4
905.1

905.8
966.5
807.9
807.9
9996

809.3

810.

910.7

8114

8191

|951

853
853
854

855

357

870

971
873

973
374
375

376
877
878

984

967

968

dep. d.laL

d.lat.

dep. d. lat.

dep. d. lat.

dep. d lat.

dep. d. lat. dep. d. lat.

dep. d. lat.

ae • « as

3* n

as « » ae

*** +

3* *

*• *" P* fed •*"

as 9» » ae

55 M

at 9i si as

^

as p a ae

1* J*

as •

o> as

as «

m as

H

«P

**

* P

C.5|rte.

C5«Pta.

C.5*Fta.

C.5Pte.

C. 4|Pto. C.4iPta.

C.4*Pti. C.4FU.

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE V.

DISTANCE, DIFFERENCE OF LATITUDE,
AND DEPARTURE.

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLK V.

Otwit 1°.
i, DUE Latitude and Departure.

Ill

dirt.

d.lal

dep.

dirt.

d.lal

dtp

dirt.

d.lal

dep.

dirt.

d.lal

dep,

dirt.

d.lai dep.

1

1.

0.

61

01.

1.1

19)1

191.

9.1

181

181.

34

841

941. 44

9

9.

0.

09

09.

1.1

9

199.

9.1

3

189.

34

9

8«*. 44

3

3.

0.1

03

03.

1.1

3

193.

9.1

3

isa

34

3

243. 44

4

4.

0.1

04

04.

1.1

4

194.

94

4

184.

as

4

814. 44

5

5.

0.1

05

05.

1.1

5

195.

9.9

5

Ma

as

5

94a 44

0*

0.1

08

08.

14

198.

94

1*8.

34

940. 4-3

7

7.

0.1

87

07.

14

7

197.

94

7

187.

3.3

7

947. 4.3

8

8

0.1

08

08.

14

8

198.

94

8

188.

34

8

948. 44

•

9.

04

09

09.

14

9

199.

94

9

189.

34

9

949. 44

10

10.

04

TO

70.

14

180

139.

94

100

130.

34

850

950. 4.4

11

11.

04

71

71.

14

1

131.

94

1

191.

34

1

951. 44

19

19.

04

73

79.

14

9

139.

94

9

199.

a4

9

859. 44

13

13.

04

73

73.

1.3

3

133.

94

3

19a

34

3

95a 4.4

14

14.

04

74

74.

14

4

134.

94

4

194.

a4

\ 4

954. 4.4

15

15.

04

75

75.

14

5

135.

9.4

5

195.

ad

5

SSa 44

10

10.

0.3

70

70.

14

138.

9.4

198.

a4

958. 44

17

17.

0.3

77

77.

14

7

137.

9.4

7

197.

3.4

7

957. ,4.5

18

18.

0.3

78

78.

1.4

8

138.

9.4

8

198.

34

8

958. 44

19

19.

03

79

79.

1.4

9

139.

9.4

9

199.

34

9

959. 4.5

90

90.

04

80

80.

1.4

140

140.

9.4

aoo

900.

34

860

980. 44

91

%L

0.4

81

81.

1.4

1

141.

94

1

901.

34

1

901. 44

99

99.

0.4

89

89.

1.4

9

149.

94

3

909.

34

9

909. 44

93

93.

0.4

83

83.

1.4

3

143.

94

3

9oa

as

3

98a 4.0

94

94.

0.4

84

84.

14

4

144.

94

4

804.

ao

4

984. 44

95

95.

0.4

85

85.

14

5

145.

94

5

90S.

ao

5

98a 4.6

99

98.

04

88

86.

14

140.

94

908.

ao

90a 4.0

97

97.

04

87

87.

1.5

7

147.

94

7

907.

ao

7

907. 4.7

98

98.

04

88

88.

14

8

148.

94

8

908.

ao

8

988. 4.7

90

99.

04

89

89.

1.0

9

149.

94

9

909.

ao

9

909. 4.7

80

30.

04

00

90.

14

150

150.

94

910

910.

2.7

870

970. 4.7

31

3L

04

91

91.

1.0

1

15L

94

1

911.

a7

1

971. 4.7

39

39.

0.0

99

99.

1.0

9

159.

9.7

9

919.

a7

9

979 4.7

33

33.

0.0

93

93.

1.0

3

153.

9.7

3

9ia

a7

3

97a 44

34

34.

0.0

94

94.

14

4

154.

9.7

4

914.

a7

4

974. 44

35

35.

04

95

95.

1.7

5

155.

9.7

5

915.

34

5

975. 44

38

38.

0.0

08

98.

1.7

158.

9.7

916.

34

8

970. 44

37

37.

0.0

97

97.

1.7

7

157.

9.7

7

817.

as

7

977. 4.8

SB

38.

0.7

98

98.

1.7

8

158.

9.8

8

918.

34

8

978. 44

30

39.

0.7

99

99.

1.7

9

159.

94

9

919.

as

9

979. 4.9

40

40.

0.7

100

100.

1.7

160

100.

94

980

990.

34

880

980. 4.0

41

41.

0.7

1

101.

14

1

101.

94

1

991.

ao

1

981. 4.9

49

49.

0.7

9

109.

14

9

109.

94

9

939.

ao

9

989. 44

43

43.

04

3

103.

14

3

103.

94

3

99a

ao

3

98a 4.9

44

44.

04

4

104.

14

4

104.

9.9

4

994.

ao

4

984. a

45

45.

0.8

5

105.

14

5

105.

9.9

5

99a

34

5

98a a

40

40.

0.8

108.

14

100.

9.9

936.

a»

98a a

47

47.

0.8

7

107.

14

7

107.

9.9

7

997.

4.

7

987. a

48

48.

04

8

108.

14

8

108.

9.9

8

998.

4.

8

98a a

49

49.

04

9

109.

14

9

109.

94

9

999.

4.

9

>989. a
990. ai

50

50.

04

110

110.

1.9

190

170.

3.

830

930.

4.

800

51

51.

09

1

111.

1.9

1

171.

3.

1

931.

4.

1

891. 5.1

59

59.

04

9

119.

9.

9

179.

3.

9

939.

4.

9

99a 5.1

53

53.

04

3

113.

9.

3

173.

3.

3

93a

4.1

3

99a 5.1

54

54.

04

4

114.

9.

4

174.

3.

4

934.

4.1

4

904. 5.1

55

55.

1.

5

115.

9.

5

175.

ai

5

93a

4.1

5

99a 5.1

58

50.

1.

118.

9.

178.

3.1

93a

4.1

998. 54

57

57.

1.

7

117.

9.

7

177.

ai

7

937.

4.1

7

997. 54

58

58.

1.

8

na

9.1

8

178.

at

8

938.

44

8

986. 54

59

59.

L

9

119.

9.1

9

179.

ai

9

939.

44

9

999. 54

"

00.

1.

190

190.

9.1

180

180.

3.1

940

940.

44

300

300. 54

dJrt.1

dep.

d. lat.

dirt.| dep.

d. lat

dirt.

dep.

d. lat.

dirt.

dep.

d.lat.

dirt.

dep. d.lat

Distance, Departure aad DUE Latitude.
C«VN 80O.

Digitized by VjOOQIC

112

TABLK V.

Dirtaaee, DUE Latitude and Departure.

dirt.

d. 1st.

dep.

dirt.

d. lat.

dap.

dirt.

d.lat. dep.

dirt.

d. lat dep.

dirt.

d. lat dap.

1

1.

0.

61

61.

3.1

iai

120.9 4.2

181

180.9 6.3

941

910.9 a4

S

9.

0.1

69

69.

9.9

9

191.9 4.3

9

181.9 6.4

9

941.9 &4

a

3.

0.1

63

63.

241

3

193.9 4.3

3

182.9 6.4

3

9424) &5

4

4.

0.1

64

64.

9.9

4

193.9 4.3

4

183.9 6.4

4

243.9 &5

5

5.

0.2

65

65.

9.3

5

194.9 4.4

6

184.9 6.5

5

9444) &6

6

8.

041

66

66.

9.3

6

125.9 4.4

6

1859 645

6

245.9 8L6

7

7.

0.3

67

67.

23

7

126.9 4.4

7

188.9 6.5

7

9464) a6

8

8.

0.3

68

68.

9.4

8

127.9 4.5

8

187.9 6.6

8

9474) &7

9

9.

0.3

69

69.

9.4

9

198.9 4.5

9

1884) 641

9

94&8 a7

10

10.

0.3

70

70.

9.4

180

199.9 15

100

18941 6.6

930

94943 a7

11

11.

0.4

71

71.

25

1

130.9 4.6

1

190.9 0.7

1

9508 as

19

19.

0.4

79

79.

9.5

9

131.9 4.6

9

191.9 07

9

9514) as

13

13.

0.5

73

73.

24J

3

132.9 4.6

3

1929 6.7

3

95243 a8

14

14.

0.5

74

74.

26

4

133.9 4.7

4

193.9 6.8

4

25343 84)

15

15.

0.5

75

75.

9.6

6

134.9 4.7

6

194.9 643

5

25443 6L9

ie

10.

0.8

76

78.

2.7

6

135.9 4.7

195.9 68

95543 841

17

2

0.6

77

77.

2.7

7

138.9 4.8

7

198.9 6.9

7

25843 a

18

0.6

78

78.

2.7

8

137.9 4.8

8

197.9 6.9

8

25743 9.

19

19.

0.7

79

79.

2.8

9

138.9 4.9

9

198.9 6.9

9

25843 9.

510

90.

0.7

80

80.

2.6

140

139.9 4J>

3300

199.9 7.

960

259.8 9.1

91

91.

0.7

81

81.

243

1

140.9 4.9

1

900.9 7.

1

28043 9.1

93

99.

0.8

89

89.

2.9

9

141.9 5.

2

901.9 7.

9

981.8 9.1

93

93.

0.8

83

89.9

2.9

3

149.9 5.

3

9029 7.1

3

9898 941

94

94.

043

84

83.9

2.9

4

143.9 5.

4

903.9 7.1

4

38343 941

95

95.

0.9

85

84.9

3.

5

144.9 5.1

5

904.9 7J

5

98443 941

98

98.

0.9

86

85.9

3.

6

145.9 5.1

6

905.9 741

6

98543 941

97

97.

0.9

87

86.9

3.

7

140.9 5.1

7

900.9 741

7

28843 941

98

98.

1.

88

87.9

3.1

8

147.9 5.3

8

907.9 741

8

28743 9.4

99

99.

L

89

88.9

3.1

9

148.9 5.9

9

908.9 741

9

98843 9.4

30

30.

1.

00

89.9

3.1

150

149.9 5.9

910

909.9 7.3

9310

98943 9.4

31

31.

1.1

91

90.9

3.3

1

150.9 5.3

1

910.9 7.4

1

87043 941

39

39.

1.1

93

91.9

3.3

9

151.9 5.3

9

911.9 7.4

9

37143 9.5

33

33.

1.9

93

93.9

3.3

3

159.9 5.3

3

212.9 7.4

3

279.8 941

34

34.

1.2

94

93.9

3.3

4

153.9 5.4

4

213.9 7.5

4

273.8 941

35

35.

1.9

95

94.9

3.3

5

154.9 5.4

5

214.9 7.5

5

27443 941

38

38.

1.3

96

95.9

3.4

6

155.9 5.4

6

315.9 75

275.8 941

37

37.

14)

97

98.9

3.4

7

158.9 5.5

7

310.9 7.6

7

27643 9.7

38

38.

1.3

98

97.9

3.4

8

157.9 5.5

8

317.9 7.6

6

27743 9.7

39

39.

1.4

99

964)

&5

9

158.9 5.5

9

918.9 7.6

9

97843 9.7

40

40.

1.4

too

99.9

35

160

159.9 5.0

990

819.9 7.7

9380

97943 94)

41

41.

1.4

1

100.9

3.5

1

160.9 5.6

1

990.9 7.7

1

98943 94)

49

49.

1.5

9

101.9

3.6

9

161.9 5.7

9

23141 7.7

9

28143 94)

43

43.

1.5

3

109.9

3.6

3

169.9 5.7

3

922.9 7.8

3

38243 94)

44

44.

1.5

4

103.9

3.6

4

163.9 5.7

4

223.9 74)

4

28343 941

45

45.

1.8

5

1049

3.7

5

164.9 5.8

5

994.9 7.9

5

28443 941

40

48.

1.6

105.9

3.7

6

165.9 5.8

8

225.9 7.9

285.810.

47

47.

1.6

7

108.9

3.7

7

168.9 5.8-

7

238.9 7.9

7

28843 10.

48

48.

1.7

8

107.9

3.8

8

167.9 5 9

8

227.9 8.

8

257.8 10.1

49

49.

1.7

1084)

3.8

9

168.9 5.9

9

228.9 8.

9

238.8 J0.1

50

50.

1.7

110

109.9

3.8

110

169.9 5.9

330

229.9 a

900

23943101

51

51.

1.8

1

110.9

3.9

1

170.9 6.

1

230.9 8.1

1

230.8 1041

59

52.

1.8

9

1» 1.9

3.9

9

171.9 6.

9

331.9 8.1

9

39143 1041

53

53.

18

3

119.9

3.9

3

173.9 6.

3

332.9 8.1

3

392.8 10.9

54

54.

1.9

4

11X9

4.

4

173.9 6.1

4

233.9 8.2

4

29343 1041

55

55.

1.9

5

114.9

4.

5

174.9 6.1

5

234.9 641

5

29443 10 J

58

58.

2.

6

115.9

4-

6

175.9 6.1

6

235.9 8.3

6

205.8 10 J

57

57.

9.

7

116.9

4.1

7

176.9 6.2

7

338.9 8.3

7

21643 10.4

58

53.

9.

8

117.9

41

8

177.9 6.9

8

337.9 a3

8

29743 1M

59

59.

9.1

9

118.9

4.9

9

17R9 6.9

9

338.9 8.3

9

29a8 10.4

69

60.

2.1

190

119.9

4.9

189

179.9 6.3

940

339.9 8.4

300

23943 1041

dirt.

dep.

d. lat

dirt.

dtp.

d. lat.

dirt. 1 dep. d. lat.

dirt.

dep. d.lat.

dirt.

dep. d.lat

Dirtaaee, Departure and DUE Latitude.
OerareaSSo

Digitized by VjOOQIC

. TABLE V.

Course 3*.

Distance, DifE Latitude and Departure

113

dist.

d. lat

dep.

diet.

d lat. dep.

dial

d. lat. dep.

dist

d. lat. dep.

df*t.

d. lat. dep.

1

1.

O.l

01

60.9 33

lftt

130.8 6.3

181

180.8 9.5

441

840 7 18.6

3

3.

0.1

12

61.9 3.3

o

131.8 6.4

9

181.8 9.5

2

241. 7 12.7

3

3.

OS

63

62.9 3.3

3

182.8 6.4

3

183 7 9.6

3

242.7 12 7

4

4.

0.3

64

63 9 3.3

4

133.8 6.5

4

1K3 7 9.6

4

2437 12.8

5

5.

0.3

65

64.9 3.4

5

124.8 6 5

5

184.7 9.7

5

244.7 12.8

6

6.

0.3

66

65.9 3.5

6

135.8 6.6

6

185.7 9 7

6

245.7 12.9

7

7.

0.4

67

66.9 3.5

7

126.8 6.6

7

186.7 9.8

7

246.7 12.9

8

8.

0.4

68

67.9 3.6

8

127.8 6 7

8

187 7 98

8

247.7 13.

9.

0.5

09

68.9 3.8

9

128 8 6.8

9

188.7 9.9

9

248.7 13.

10

10.

0.5

70

69.9 37

130

129.8 6.8

190

189.7 9.9

960

249.7 13 1

11

11.

0.6

71

70.9 3 7

1

130.8 6.9

1

190.7 10

1

250.7 13.1

IS

12.

06

73

71.9 3.8

3

131.8 6.9

3

191.7 10.

8

251.7 132

13

13.

0.7

73

78.9 3.8

3

133.8 7.

3

Ib2.7 10.1
193 7 10.3

3

3587 13.8

14

14.

0.7

74

73.9 3.9

4

133.8 7.

4

4

253 7 13.3

15

15.

08

75

74.9. 3.9

5

134.8 7.1

5

104.7 10 3

5

254.7 13 3

16

16.

Oil

76

759 4.

«

135.8 7.1

6

1^5.7 10 3

6

255.6 13.4

17

17.

0.9

77

76.9 4.

7

136.8 7.3

7

1967 10.3

7

256.6 13 5

18

18.

0.9

78

77.9 4.1

8

137.8 7.8

8

1977 10.4

8

257.6 13.5

19

19.

1.

79

78.9 41

9

138.8 7.3

9

1U8.7 10.4

9

258 613.6

90

30.

1.

80

79.9 42

140

139.8 7.3

900

199.7 10.5

960

859.6 13.6

31

81.

1.1

81

80.9 4.3

1

140.8 7.4

1

300.7 10.5

1

31.0.6 13 7

33

33.

13

83

8L9 4 3

8

141.8 7.4

3

801 7 10.6

3

861.6 13.7

33

33.

1.3

83

83.9 4.3

3

143 8 7.5

3

303.7 10.6

3

S(JS.6 13.8

34

34.

1.3

84

83.9 4.4

4

143.8 7.5

4

803.7 10.7

4

3«3.6 13.8

35

35.

1.3

85

84.9 4 4

5

144 8 7.6

5

204.7 107

5

364.6 13.9

3d

36.

1.4

86

85 9 4.5

6

1458 7.6

6

205.7 10.8

6

365.6 13.9

87

37.

1.4

87

86.9 4.6

7

146.8 7.7

7

306.7 10.8

7

866 614.

38

38.

1.5

88

87.9 4.6

8

147 8 7.7

8

807.7 10.9

8

267.6 14.

39

39.

1.5

89

88.9 4.7

9

148.8 7.8

9

208.7 10.9

9

» 868.6 141

30

30.

16

00

89,9 4.7

150

149.8 7.9

910

309.7 11.

970

869.6 14 1

31

31.

1.6

91

90.9 4.8

1

1508 7.9

1

310.7 11.

1

870.6 14 8

33

33.

1.7

93

91.9 4.8

3

151.8 8.

8

3117 11 1

8

371.6 14.2

33

33.

17

93

93.9 4.9

3

152.8 8.

3

3127 11,1

3

272 14.3

34

34.

1.8

94

939 4.9

4

153.8 8.1

4

813 7 113

4

273 6 14.3

35

35.

1.8

95

94.9 5.

5

154.8 8.1

5

314.7 11 3

5

274.6 14.4

38

36.

19

96

95.9 5

6

155.8 8 2

6

315.7 11 3

6

275.6 14.4

37

36.9

1.9

97

96 9 5.1

7

156.8 83

7

8167 '14

7

276.6 14.5

38

379

8.

98

97.9 5.1

8

157.8 8.3

8

317.7 1M

8

277.6 14.5

39,

38.9

3.

99

98.9 5.3

9

158.8 8.3

9

818.7 11.5

9

278.0 14.6

40

39.9

8.1

100

99 9 5.3

160

159.8 8.4

990

319.7 115

980

279.6 14.7

41

409

3.1

1

100.9 5.3

1

160,8 a4

1

830.7 1L6

1

2H> 6 147

43

41.9

3.3

3

101.9 53

3

161.8 8.5

8

321.7 11.6

3

281 6 14.8

43

43.9

2-3

3

1089 5 4

3

162.8 8.5

3

828 7 11.7

3

282 6 14.8

44

439

8.3

4

103.9 5.4

4

163 8 8.0

4

823.7 11.7

4

283.6 14.9

45

44.9

8.4

5

1049 5 5

5

164,8 86

5

324 7 11.8

5

284 14.9

46

459

8.4

6

105 9 5 5

6

165.8 8.7

6

225? 118

6

285.6 15.

47

46.9

8.5

7

1069 5.6

7

166.8 8.7

7

3267 11.9

7

286.6 15.

48

47.9

8.5

8

107.9 5.7

8

167.8 8 8

8

337.7 11.9

8

267.6 15.1

49

48.9

8.8

9

1089 5.7

9

168.8 8.8

9

838.7 13.

9

888.6 15.1

00

49.9

8.6

»»

109.8 5.8

170

169.8 8,9

930

229.7 12.

900

889.6 15.3

51

509

87

110.8 5.8

1

170.8 8.9

1

230.7 13 1

1

310.6 15.3

53

519

S-T

8

Jll.8 5.9

3

171.8 9.

3

331.7 13 1

2

8P1.6 15.3

53

52.9

8.8

3

113 8 5.9

3

173.8 9.1

3

232.7 18 3

3

» 8.6 15.3

54

53.9

8.8

4

113.8 6.

4

173.8 9.1

4

333.7 12 3

4

393.6 15.4

55

549

3.9

5

114.8 6.

5

174,8 9.3

5

334 7 18.3

5

894.6 15.4

56

55.9

3.9

6

115.8 6.1

6

175.8 9.8

6

3357 13.4

6

395-0 15,5

57

569

3.

7

J 16 8 6.1

7

176.8 9.3

7

836.7 13 4

7

396.6 15v5

58

579

3.

8

117.8 6.3

8

177 8 9.3

8

337.718 5

8

397.6 15.6

59

589

3.1

118 8 6.3

9

178 8 9.4

9

838 7 13.5

9

898.6 15 6

60

59.9

3.1

130

119.8 6.3

180

179.8 9.4

840

339.7 136

300

399.6 15.7

dist.

dep.

d.lat

ditt.

dep. d. lat.

ditt.

dep. d. lat

dint.

dep. d. lat.

dist.

dep. d. lat.

Distance, Departure and Dial Latitude.

Co wee 3 TO

9

Digitized by VjOOQIC

114

TABLE V.

Coarse 40.

Dirtauce, DtC Latitude and Departure.

diat.

d. lat.

dep.

dirt.

d. lat.

dep.

dirt.

d. lat.

dep.

dirt.

d. lat.

dep.

dirt.

d. lat. dap.

l

1.

0.1

61

604

44

191

190.7

a4

181

180.6

12.6

341

940.4 164

s

3.

0.1

63

61.8

44

3

191.7

84

3

181.6

12.7

3

241.4 164

3

3.

0.8

63

624

4.4

3

133.7

ae

3

183.8

12.8

3

342.4 17.

4

4.

0.3

64

634

44

4

133.7

a6

4

183.6

12.8

4

2434 17.

5

5.

0.3

65

644

4.5

5

184.7

a7

5

184.5

12.9

5

244.4 17.1

6

6.

04

66

65.8

4.6

6

135.7

84

6

185.5

13.

6

245.4 174

7

7.

0.5

67

66.8

4.7

7

136.7

ao

7

1864

13.

7

2464 174

8

8.

0.6

68

67.8

4.7

8

127.7

a9

8

187.5

13.1

8

247.4 174

9

9.

0.6

69

68.8

4.8

9

I3a7

9.

9

1884

134

9

2484 17.4

10

10.

0.7

70

694

44

130

189.7

9.1

100

1894

134

360

8494 174

11

11.

04

71

704

5.

1

130.7

9.1

1

190.5

134

1

8504 174

18

19.

04

73

71.8

5.

9

181.7

94

9

191.5

13.4

8

8514 174

13

13.

0.9

73

734

5.1

3

138.7

9.3

3

1924

134

3

3524 174

14

14.

1.

74

73.8

54

4

133.7

94

4

193.5

134

4

8534 17,7

IS

15.

1.

75

744

5.3

5

134.7

94

5

1944

13.6

5

8544 174

16

16.

1.1

76

75.8

5.3

6

135.7

9.5

6

1954

13.7

6

8554 174

17

17.

J. 9

77

764

5.4

7

136.7

9.6

7

196.5

13.7

7

8564 174

18

ia

14

78

774

5.4

8

137.7

9.6

I

1974

134

8

8574 ia

19

19.

14

79

784

5.5

9

isa7

9.7

1984

134

9

2584 iai

90

90.

14

80

794

54

140

139.7

94

aoo

1994

14.

060

8594 l&l

91

30.9

1.5

81

804

5.7

1

140.7

9.8

l

2004

14.

1

860.4 184

S3

31.9

14

83

814

5.7

8

141.7

9.9

9

8014

14.1

3

8614 lt>3

93

33.9

16

83

834

5.8

3

143.7

10.

3

8034

144

3

362.4 lt>3

94

33.9

1.7

84

834

5.9

4

143.6

10.

4

8034

14.9

4

963.4 1H.4

95

84.9

1.7

85

84.8

59

5

144.6

10.1

5

8044

144

5

8644 1«4

98

35.9

14

86

85.8

6.

6

145.6

10.8

6

3054

14.4

6

805.4 i84

97

96.9

1.9

87

868

6.1

7

146.6

10.3

7

3064

14.4

7

866.3 ia7

S

37.9

3.

88

874

6.1

8

147.6

10.3

8

8074

144

8

3674 ia7

384

8.

89

884

64

9

14a6

10.4

9

8084

144

9

8684 184

30

394

8.1

00

898

64

160

149.6

104

8310

3095

144

370

8694 184

91

30.9

8.3

91

904

64

1

150.6

104

1

3104

14.7

1

8704 184

93

31.9

84

93

914

6.4

3

151.6

10.6

3

81L5

144

3

8714 19.

93

33.9

8.3

93

93.6

6.5

3

1524

10.7

3

3134

14.9

3

8784 19.

94

33J9

8.4

94

93.8

6.6

4

153.6

10.7

4

3134

14.9

4

8734 19.1

95

344

8.4

95

944

6.6

5

154.6

104

5

3144

15.

5

3743 194

98

35.9

84

96

95.8

6.7

6

155.6

10.9

6

8154

15.1

6

3754 194

97

36.9

8.6

97

964

64

7

1564

11.

7

3164

15.1

7

8764 194

S

37.9

8.7

98

974

64

8

157.8

11.

8

3174

154

8

8774 194

38.9

8.7

99

984

64

9

i5ao

11.1

9

3184

154

9

8784 194

40

99.9

84

100

994

7.

160

1594

114

MO

3194

154

389

8794 194

41

40.9

3.9

1

1004

7.

1

169.6

114

1

3804

15.4

1

3SD4 194

43

41.9

8.9

8

1018

7.1

9

161.6

11.3

9

391.5

154

9

8914 19.7

43

49,9

3.

3

108.7

74

3

163.6

11.4

3

3934

154

3

8884 19.7

44

43.9

3.1

4

103.7

7.3

4

163.6

11.4

4

9834

15.6

4

9434 194

45

44.9

3.1

5

104.7

74

5

164.6

1J4

5

8344

15.7

5

8844 194

48

434

34

6

105.7

7.4

6

165.6

116

6

835.4

154

6

8854 9a

47

4*4

34

7

106.7

74

7

166.6

11.6

7

836.4

154

7

8894 80.

48

47.9

34

8

107.7

74

8

167.6

11.7

8

287.4

154

8

3873 80.1

49

484

. &*

9

108.7

74

9

16a6

114

9

S8&4

16.

9

8884 804

60

494

34

110

109.7

7.7

170

1994

11.9

330

889.4

16.

300

8894 804

51

50.9

34

1

110.7

7.7

1

170.6

114

1

830.4

16.1

1

8994 804

IB

51.9

34

9

111.7

74

9

1714

13.

9

831.4

164

9

891.3 804

53

53.9

3.7

3

113.7

74

3

179.6

12.1

3

839.4

164

3

8984 804

54

53.9

38

4

113.7

8.

4

173.6

12.1

4

833.4

164

4

8934 804

55

54.9

34

5

114.7

8.

5

1744

12.2

5

834.4

16 4

5

8944 804

56

55.9

3.9

6

115.7

8.1

6

1754

124

6

835.4

16.5

6

8354 894

57

56.9

4.

7

116.7

as

7

176.6

12.3

7

836.4

16.5

7

8964 80.7

58

57.9

4.

8

117.7

84

8

177.6

12.4

8

337.4

164

8

3974 804

59

58.9

4.1

9

118.7

8.3

9

17a6

124

9

238.4

16.7

9 3984 804

60

59.9

44

180

119.7

a4

180

179.6

13.6

840

3394

16.7

300 | 8994 904

dirt.

dep.

d. lat

dUt.

dep. d. lat.

dirt.

dep.

i. lat.

dirt.

dep.

d. lat.

dirt. 'dep. d. lat.

Dirtance, Departure and Diff. Latitude
Camrce 860.

Digitized by VjOOQIC

TABLK V.

Course SO,
Dirtanee, DUE Latitude and Departure.

115

dirt.

d.lat

dep.

dirt.

d. ut.

dep.

dirt.

d. lat.

dep.

dirt.

d. lat

dep.

dist.

d. lat.

dep.

1

1.

0.1

61

60.8

5.3

1191

120.5

10.5

181

180.3

15.8

*4l

2401

31.

S

9.

0.2

62

61.8

5.4

2

121.5

10.6

2

181.3

15.9

3

241.1

31.1

3

3.

0.3

63

62.8

5.5

3

122.5

10.7

3

182.3

15.9

3

242.1

314

4

4.

0.3

04

634

5.6

4

123.5

10.8

4

183.3

16.

4

243.1

31.3

5

5.

0.4

65

64.8

5.7

5

1245

10.9

5

184.3

16.1

5

244.1

21.4

e

6.

0.5

66

65.7

5.8

6

125.5

11.

6

185.3

16.2

6

245.1

21.4

7

7.

0.6

67

66.7

5.8

7

126.5

Ill

7

1864

16.3

7

246.1

21.5

8

a

' 0.7

68

67.7

5.9

8

127.5

11.2

8

187.3

16.4

8

2471

21.6

9

9.

0.8

69

68.7

6.

188.5

118

9

18&3

104

9

243.1

21.7

10

10.

04

TO

69.7

6.1

130

199.5

11.3

190

1804

16.6

950

849.

91.8

11

11.

1.

71

70.7

64

1

130.5

11.4

1

190.3

16.6

1

250.

31.9

IS

19.

1.

72

71.7

6.3

9

131.5

11.5

3

191.3

16.7

9

251.

88.

13

13.

1.1

73

72.7

6.4

3

1324

11.6

3

192.3

16.8

3

253.

22.1

14

13.9

19

74

73.7

6.4

4

133.5

11.7

4

193.3

16.9

4

253.

22.1

15

14.9

1.3

75

74.7

6.5

5

134.5

11.8

5

194.3

17

5

254.

22.2

16

15.9

1.4

76

757

6.6

6

135.5

11.9

6

1054

17.1

6

255.

22.3

17

16.9

1.5

77

76.7

6.7

7

136.5

11.9

7

196.3

17.2

7

256.

88.4

18

17.9

1.6

78

77.7

68

8

137.5

12.

8

197.8

17.3

8

257.

82.5

19

184

1.7

79

78.7

6.9

9

1384

12.1

9

1984

17.3

9

258.

88.6

*0

19.9

17

80

79.7

7.

140

139.5

124

8JO0

199.3

17.4

360

859.

£

91

90.9

14

81

80.7

7.1

1

140.5

12.3

1

200.2

175

1

360.

9*

91.9

1.9

88

81.7

7.1

2

141.5

12.4

2

201.2

17.6

9

361.

824

93

99.9

9.

83

83.7

7.2

3

142.5

12.5

3

202.2

17.7

3

868.

884

94

834

2.1

84

83.7

7.3

4

1435

12.6

4

203.2

17.8

4

363.

83.

95

94.9

2.3

85

84.7

7.4

5

144.4

12.6

5

204.2

17.9

5

364.

83.1

96

35.9

3.3

86

85.7

7.5

6

145.4

12.7

6

205.2

18.

6

365.

334

97

96.9

3.4

87

86.7

7.6

7

146.4

12.8

7

206.2

18,

7

366.

83.3

98

97.9

8.4

88

87.7

7.7

8

147.4

12.9

8

2072

181

8

867.

23.4

99

98.9

9.5

89

88.7

7.8

9

148.4

13.

9

8084

M.2

9

868.

83.4

80

99.9

9.6

60

89.7

7.8

150

149.4

13,1

8310

209.3

18.3

970

369.

834

31

30.9

9.7

91

90.7

7.9

1

150.4

13.2

1

210.2

18.4

1

870.

83.6

39

31.9

24

99

91.6

8.

S

151.4

13.2

2

211.2

18.5

3

871.

83.7

33

39.9

9.9

93

92.6

8.1

3

152.4

13.3

3

212.2

18.6

3

873.

834

34

33.9

3.

94

93.6

a2

4

153.4

13.4

4

213.2

18.7

4

873.

83.9

35

34.9

3.1

95

94.6

8.3

5

154.4

13.5

5

214.2

1&7

5

274.

84.

36

35.9

31

96

95.6

8.4

6

155.4

13.6

6

2155

18.8

6

2744

34.1

37

36.9

3.2

97

96.6

8.5

7

156.4

13.7

7

216.2

18.9

7

3754

84.1

38

37.9

3.3

98

97.6

8.5

8

157.4

13.8

8

217.2

19.

8

876.9

344

39

38.0

3.4

90

98.6

8.6

9

15&4

13.9

9

3184

19.1

9

877.9

844

40

394

3.6

100

99.6

8.7

160

159.4

13.9

9830

319.3

19.2

MO

378.9

34.4

41

40.8

3.6

1

100.6

8.8

1

160.4

14.

1

220.2

19.3

1

379.9

244

49

418

3.7

8

101.6

8.9

3

161.4

14.1

2

221.2

19.3

8

8804

24.6

43

49.8

3.7

3

109.6

9.

3

168.4

14.2

3

2224

19.4

3

881.9

24.7

44

43.8

34

4

103.6

9.1

4

163.4

14.3

4

223.1

19.5

4

383.9

24.8

45

44.8

3.9

5

104.6

9.2

5

1644

14.4

5

224.1

19.6

5

383.9

24.9

46

45.8

4.

6

105.6

9.2

6

165.4

14.5

6

225.1

19.7

6

8844

24.9

47

46.8

4.1

7

106.6

9.3

7

166.4

146

7

226.1

194

7

3354

25.

48

474

4.2

8

J07.6

9.4

6

167.4

14.6

8

827.1

19.9

8

386.9

25.1

49

484

4.3

9

108.6

9.5

9

16a4

14.7

9

328.1

80.

9

887.9

854

60

49.8

4.4

110

109.6

9.6

170

169.4

14.8

930

889.1

8a

990

88a9

854

51

504

44

1

110.6

9.7

1

170.3

14.9

1

830.1

80.1

1

389.9

85.4

59

514

4.5

9

111.6

9.8

3

171.3

15.

3

831.1

30.8

3

390.9

85.4

53

594

46

3

112.6

94

3

172.3

15.1

3

332:1

80.3

3

811.9

854

54

534

4.7

4

113.6

9.9

4

173.3

15.2

4

233.1

80.4

4

898.9

35.6

55

544

44

5

1-4.6

10.

5

174.3

15.3

5

234.1

80.5

5

393.9

85.7

56

554

4.9

6

115,6

10.1

6

175.3

15.3

6

835.1

80.6

6

3144

854

57

564

5.

7

116.6

10.2

7

176.3

15.4

7

236.1

30.7

7

395.9

35.9

58

574

5.1

8

117.6

10.3

8

177.3

15.5

8

237.1

80.7

8

316.9

36.

59

584

6.1

9

118 .5

10.4

9

178.3

15.6

9

338.1

804

9

817.9

36.1

60

594

5.9

199

119.5

10.5

180

179.3

15.7

840

839.1

30.9

300

896.9

36.1

di«t

*»!►

d.lat

dtrt.|

dep. d

lat.

dirt.

dep. d .lat

diet.

dap. d. lat

dirt.

dep.

4. lat.

Dtrtaace, Departure and DUE Latituoe.
C*ur*e85°.

Digitized by VjOOQIC

116

TABLET.

Dietanee, Diff Latitude and Departure.

diet.

d,lat.

dep.

diit.

d. let.

dep.

diet.

d. lat

dep.

dirt.

(LlaL

**

diet.

d.lat dep.

1

1.

0.1

61

60.7

6.4

i»i

1303

18.6

181

180.

183

941

839.7 853

s

9.

0.8

68

61.7

63

3

121.3

18.8

3

181.

19.

8

840.7 953

3

3.

0.3

63

69.7

86

3

128.b

13.9

3

188.

19.1

3

841.7 85.4

4

4.

0.4

64

63.6

6.7

4

123.3

13.

4

183.

193

4

848.7 853

5

5.

0.5

65

64.6

6.8

5

124.3

13.1

5

184.

19.3

5

843.7 85.6

6

6.

0.6

66

65.6

6.9

6

125.3

13.8

6

185.

19.4

6

844.7 85.7

7

7.

0.7

67

66.6

7.

7

126.3

13.3

7

186.

19.5

7

845.6 3543

8

8.

0.8

68

67.6

7.1

8

127.3

13.4

8

187.

19.7

8

8463 85.9

9

9.

0.9

69

68.6

7.3

9

1283

1&5

9

188.

193

9

8473 86.

10

9.9

1.

70

69.6

73

180

1893

13.6

190

189.

19.9

950

3483 36.1

11

10.9

1.1

71

70.6

7.4

1

130.3

13.7

1

190.

80.

]

8493 863)

IS

11.9

1.3

78

71.6

73

3

131.3

133

9

190.9

80.1

3

850.6 863

13

19.9

1.4

73

78.6

7.6

3

132.3

13.9

3

1913

80.8

3

8513 96.4

14

13.9

1.5

74

73.6

7.7

4

133.3

14.

4

198.9

803

4

858.6 863

15

14.0

J. 6

75

74.6

7.8

5

1343

14.1

5

193.9

80.4

5

853.6 86.7

16

15.9

1.7

76

75.6

7.9

6

1353

14.8

6

194.9

803

6

854.6 963

17

16.9

1.8

77

76.6

8.

7

136.2

14.3

7

195.9

80.6

7

8553 863

18

17.9

1.9

78

77.6

8.8

8

137.2

14.4

8

196.9

80.7

8

850.6 97.

19

18.9

9.

79

78.6

8.3

9

138J2

143

9

197.9

803

9

8573 87.1

90

19.9

8.1

80

79.6

8.4

140

139.3

14.6

900

196.9

90.9

960

958.6 973!

91

90.9

9.3

81

80.6

83

1

140.8

14.7

1

199.9

31.

1

8593 873

99

81.9

8.3

88

816

8.6

3

141.3

14.8

3

800.9

81.1

3

8603 873

83

93.9

8.4

83

&■*

8.7

3

142.3

149

3

801.9

313

3

861.6 973

94

93*

8.5

84

83.5

8.8

4

143.2

15.1

4

803,9

81.3

4

868.6 873

25

949

3.6

85

84.5

8.9

5

144.2

15.2

5

303.9

81.4

5

8633 87.7

96

95.9

8.7

86

85.5

9.

6

145.2

153

6

804.9

313

6

8643 873

97

363

8.8

87

86.5

9.1

7

146.2

15.4

7

805.9

31.6

7

865.5 873

99

87.9

8.9

88

87.5

93

8

1473

15.5

8

806.9

31.7

8

9863 88.

99

888

a

89

86.5

93

9

148.3

15.6

9

807.9

813

9

8873 8B.1

30

903

3.1

00

893

9.4

150

149.3

15.7

910

8083

83.

970

9683 893

31

30.8

3.9

91

905

93

1

150.3

15.8

1

809.8

82.1

8693 883

39

31.8

3.3

98

91.5

9.6

8

151.2

15.9

3

310.8

83.2

9

8703 98.4

33

39.8

3.4

93

985

9.7

3

152.2

16.

3

311.8

88.3

3

8713 883

34

33.8

3.6

94

93.5

9.8

4

153.2

16.1

4

8183

93.4

4

9783 883

35

343

3.7

95

945

9.9

5

154.2

162

5

313.8

32.5

5

8733 88.7

30

35.8

3.8

96

95.5

10.

6

155.1

16.3

6

9143

93.6

6

8743 883

37

36.8

3.9

97

96.5

10.1

7

156.1

16.4

7

315.8

83.7

7

8753 89.

38

37.8

4.

98

97.5

10.2

8

157.1

16.5

8

816.8

88.8

8

8763 89.1

39

383

4.1

99

983

10.3

9

158.1

16.6

9

917.8

83.9

9

8773 893

40

393

43

too

993

103

160

159.1

16.7

990

318.8

83.

980

8783 893

41

403

4.3

1

1004

10.6

1

160.1

16.8

1

819.6

83.1

1

8793 89.4

49

41.8

4.4

9

101.4

10.7

3

161.1

16.9

3

8803

83.2

9

8803 893

43

£S

4.5

3

102.4

103

3

168.1

17.

3

931.8

83.3

3

881.4 803

44

4.6

4

103.4

10.9

4

163.1

17.1

4

833.8

83.4

4

888.4 89.7

45

44.8

4.7

5

104.4

11.

5

164.1

17.2

5

833.8

833

5

883.4 893

46

45.7

4.8

6

105.4

11.1

6

165.1

174

6

984.8

83.6

6

884.4 893

47

46.7

4.9

7

105.4

11.2

7

166.1

173

7

8853

83.7

7

885.4 30.

48

47.7

5.

8

107.4

11.3

8

167.1

17.6

8

886.8

833

8

886.4 30.1

49

48.7

5.1

9

108.4

11.4

9

168.1

17.7

9

887.7

83.9

9

887.4 393

50

49.7

5.9

110

109,4

113

170

169.1

173

930

888.7

84.

900

888.4 393

51

50.7

5.3

1

J10.4

11.0

1

170.1

17.9

1

889.7

84.1

8894 39.4

59

51.7

5.4

8

111.4

11.7

9

1711

ia

3

830.7

84.3

8

899.4 303

53

53.7

5.5

3

119.4

11.8

3

178.1

18.1

3

831.7

84.4

3

801.4 30.6

54

53.7

5.0

4

113.4

11.0

4

173.

18.2

4

833.7

845

4

893.4 30.7

55

54.7

5.7

5

114.4

18.

5

174.

18.3

5

833.7

84.6

5

893.4 303

58

55.7

5.9

6

1154

191

6

175.

18.4

834.7

84.7

6

804.4 303

57

56.7

0.

7

1164

13.8

7

176.

183

7

835.7

843

7

395.4 3L

58

57.7

6.1

8

U7.4

12.3

8

177.

18.6

8

836.7

843

8

896.4 3L1

59

58.7

8.9

9

118.3

18.4

9

178.

18.7

9

837.7

85.

9

897.4 313

60

53.7

83

190

1193

183

180

179.

183

840

838.7

85.1

300

898.4 313

dep.

d. lat.

dirt.

dep.

d lot

dirt.

dep.

d. lat.

dirt.

dep.

d.let.

dirt.

dep. d. let.

Pittance, Departure and Difi Latitude.
0#uurt*8*o,

Digitized by VjOOQIC

TABLS Y.

Cmuree 7°.

DUU Latitude and Departure.

117

dirt.

d.lat.

dep.

dirt.

d. let.

dep.

dirt.

d. lat.

dep.

dirt.

d. lat.

dep,

dirt.

d. lat.

dep.

1

1.

0.1

61

60.5

7.4

iai

120.1

14.7

181

179.7

22.1

341

2393

29.4

S

9.

0.2

62

615

7.6

2

121.1

14.9

2

1803

22.2

2

240.2

293

3

3.

04

63

623

7.7

3

122.1

15.

3

181.6

22.3

3

241.2

29.6

4

4.

0.5

64

63.5

7.8

4

123.1

15.1

4

182.6

98.4

4

242.2

29.7

5

5.

0.6

65

64.5

7.9

5

124.1

153

5

183.6

295

5

243.2

293

6

6.

0.7

66

653

8.

6

125.1

15.4

6

184.6

22.7

6

244.2

30.

7

6.9

9.9

67

663

83

7

126.1

15.5

7

1856

22.8

7

2453

30.1

8

7.9

1.

68

673

8.3

8

127.

15.6

8

186.6

22.9

8

846.2

303

9

8.9

1.1

69

68.5

8.4

9

128.

15.7

9

1873

93.

9

247.1

30.3

10

9.9

1.2

70

693

83

130

199.

15.8

100

188.6

23.8

350

248.1

303

11

10.9

1.3

71

703

8.7

1

130.

16.

1

189.6

23.3

1

249.1

30.6'

19

11.9

1.5

72

713

63

9

131.

16.1

9

190.6

23.4

2

2501

30.7

13

12.9

1.6

73

72.5

8.9

3

132.

16.2

3

191.6

933

3

251.1

303

14

13.9

1.7

74

73.4

9.

4

133.

16.3

4

192.6

233

4

952.1

31.

15

14.9

1.8

75

74.4

9.1

5

134,

16.5

5

193.5

233

5

253.1

31.1

16

15.9

1.9

76

75.4

93

6

135.

16.6

6

1943

23.9

6

254.1

313

17

16.9

2.1

77

76.4

9.4

7

136.

16.7

7

195.5

'.4.

7

255.1

313

18

17.9

2.2

78

77.4

9.5

8

137.

16.8

8

1963

24.1

8

258.1

31.4

19

18.9

9.3

79

78.4

93

9

138.

16.9

9

1973

243

9

257.1

313

30

19.9

2.4

80

79.4

9.7

140

139.

17.1

300

1983

24.4

360

258.1

31.7

91

90.9

9.6

81

80.4

9.9

1

139.9

17.2

1

1993

94.5

1

259.1

313

98

91.8

9.7

82

81.4

10.

9

140.9

173

9

200.5

243

9

960.

31.9

93

92.8

9.8

83

89.4

10.1

3

141.9

17.4

3

201.5

24.7

3

261.

32.1

94

93.8

9.9

84

83.4

10.2

4

142.9

173

4

202.5

24.9

4

269.

323

99

24.8

a

85

84.4

10.4

5

143.9

17.7

5

203.5

25.

5

263.

323

96

25.8

3.2

86

85.4

105

6

144.9

17.8

6

204.5

25.1

6

264.

32.4

97

96.8

3.3

87

86.4

10.6

7

145.9

17.9

7

205.5

25.2

7

265.

323

98

978

3.4

88

87.3

10.7

8

146.9
^47.9

18.

8

208.4

253

8

266.

32,7

99

98.8

3.5

89

883

10.8

9

183

9

207.4

253

9

267.

323

30

993

3.7

•0

89.3

11.

160

H8.9

18.3

910

208.4

25.6

370

268.

38.9

31

30.8

3.8

91

90.3

11.1

1

149.9

18.4

1

209.4

95.7

1

269.

33.

39

31.8

3,9

92

91.3

J 1.2

2

150.9

183

9

210.4

253

9

27a

33.1

33

32.8

4.

93

92.3

11.3

3

151.9

18.6

3

211.4

26.

3

271.

33.3

34

33.8

4.1

94

93.3

11.5

4

152.9

18.8

4

212.4

96.1

4

272,

33.4

35

34.7

4.3

95

94.3

11.6

5

153.8

18.9

5

213.4

963

5

273.

335

36

35.7

4.4

96

953

11.7

6

1543

19.

6

214.4

96.3

6

273.9

33.6

37

36.7

4.5

97

96.3

113

7

155.8

19.1

7

215.4

96.4

7

274.9

33.8

38

37.7

4.6

98

97.3

11.9

8

156.8

19.3

8

216.4

96.6

8

275.9

33.9

39

38.7

43

99

983

12.1

9

157.8

19.4

9

217.4

26.7

9

2763

34.

40

39.7

4.9

100

993

124

160

1583

193

330

2184

963

880

277.9

34.1

41

40.7

5.

1

100.2

12.3

I

159.8

19.6

1

219.4

26.9

I

278.9

34.9

49

41.7

5.1

2

101.2

12.4

2

160.8

19.7

9

220.3

27.1

2

879.9

34.3

43

42.7

5.2

3

102.2

12.6

3

161.8

19.9

3

2213

97.8

3

8803

34.5

44

43.7

5.4

4

103.2

12.7

4

162.8

90.

4

2223

273

4

281.9

34.6

45

44.7

53

5

1049

12.8

5

163.8

90.1

5

2233

97.4

5

282.9

34.7

46

45.7

5.6

6

1059

12.9

6

1643

903

6

2243

27.5

6

283.9

34.9

47

46.7

5.7

7

108.8

13.

7

1653

90.4

7.

225.3

97.7

7

284.9

35.

48

47.6

5.8

8

107.9

13.2

8

166.7

903

8

2963

87.8

8

385.9

35.1

49

48.6

6.

1083

133

9

167.7

20.6

9

9973

27.9

9

2863

353

50

49.6

6.1

116

1095

13.4

1T0

168.7

90.7

330

2283

28.

300

2873

35.3

51

50.6

6.9

1

110.9

133

1

169.7

90.8

1

2993

28.9

1

298.8

35.5

59

51.6

63

2

111.2

133

2

170.7

21.

2

9303

28.3

9

289.8

35.6

53

52.6

6.5

3

112.2

133

3

171.7

91.1

3

2313

28.4

3

290.8

35.7

54

53.6

6.6

4

113.2

13.9

4

172.7

913

4

2323

285

4

291.8

353

55

54.6

6.7

5

114.1

14.

5

173-7

91.3

5

233.2

28.6

5

2923

36.

56

55.6

6.8

6

115.1

14.1

6

174.7

91.4

6

234.2

28.8

6

293.8

36.1

57

56.6

0.9

7

116.1

14.3

7

175.7

21.6

7

235.2

28.9

7

294.8

363

59

57.6.

7.1

8

117.1

14.4

8

176.7

21.7

8

2363

29.

8

295.8

36.3

59

58.6

7.9

9

118.1

143

9

177.7

21.8

9

237.2

29.1

9

2983

36.4

60

59.6

73

190

119.1

143

180

178.7

21.9

240

2383

293

300

8973

36.6

dirt.

dep.

d. let.

dirt.

dep.

d. lat.

dirt.

dep.

d. lat.

dirt. | dep.

d. lat.

dirt.

dep.

d. lat.

Distance, Departure and DUE Latitude.
Course 83°.

Digitized by VjOOQIC

118

TA1L1V.

Coufffte 80.

Dietaflee, DUE Latitude and Departure.

dlit.

d.lat.

de|>.

dirt.

d. lat.

dep.

dirt.

d. lat.

dep,

dirt-

d. lat.

dep,

dirt.

d.lat. dep.

1

1.

0.1

61

60.4

85

181

1194

164

181

179.9

254

841

238.7 334

2

2.

0.3

62

61.4

86

2

120.8

17.

9

1804

254

3

239.6 33.7

3

a

0.4

63

62.4

8.8

3

1214

17.1

8

181.2

25.5

3

9406 334

4

4.

0.6

64

63.4

89

4

1228

17.3

4

1824

25.6

4

241.6 34.

5

5.

0.7

65

64.4

9.

5

123.8

17.4

5

183.2

257

5

243.6 34.1

6

5.9

04

66

65.4

92

6

124.8

174

6

184.2

259

6

943-6 344

7

6.9

1,

67

66.3

9.3

7

1254

17.7

7

185.2

26.

7

944.6 34.4

8

7.9

1.1

68

67.3

9.5

8

1268

17.8

8

186.2

364

8

9454 344

9

a9

1.3

69

68.3

9.6

9

127.7

18.

9

1872

364

9

946.6 34.7

10

9.9

1.4

70

69.3

9.7

130

1287

18.1

100

188.2

99.4

9)50

9474 344

11

10.9

1.5

71

70.3

99

1

129.7

18.2

1

189.1

96.6

1

848.6 348

12

11.9

1.7

72

71.3

10.

9

130.7

18.4

9

190.1

98.7

9

3494 33J

13

12.9

1.8

73

723

10.2

3

131.7

184

3

191.1

96.9

3

9504 35.9

14

13.9

19

74

73.3

10.3

4

132.7

18.6

4

193.1

27.

4

9514 35 3

15

14.9

9.1

75

74.3

10.4

5

133.7

188

5

193.1

37.1

5

252.5 3S4

Id

15.8

9.2

76

763

10.6

6

134.7

18.9

6

1941

37.3

6

253.5 358

17

164

2.4

77

763

10.7

7

135.7

19.1

7

195.1

974

7

2544 3S4

18

17.8

9.5

78

77.2

10.9

8

136.7

19.9

8

196.1

876

8

3554 354

19

184

2.6

79

784

11.

9

137.7

194

9

197.1

27.7

9

3564 36.

30

19.8

2.8

80

79.9

11.1

140

139.6

195

aoo

198.1

974

800

3574 364

31

90.8

2.9

81

802

11.3

1

139.6

19.6

i

199.

28.

1

358.5 384

22

21.8

3.1

82

81.2

11.4

2

140.6

19.8

9

300.

281

8

2594 364

23

224

3.2

63

82.2

11.6

3

14H6

19.9

3

301.

284

3

280.4 36,6

24

23.8

3.3

84

839

117

4

142.6

90.

4

309.

88.4

4

261.4 367

25

24.7

3.5

85

844

114

5

1496

90.2

5

903.

28.5

5

962.4 364

20

25.7

3.6

86

854

li.

6

144.6

20.3

6

304.

88.7

6

963.4 37.

27

26.7

3.8

87

864

121

7

145.6

20.5

7

305.

284

7

9644 374

28

27.7

&9

88

87.1

184

8

146.6

20.6

8

son.

284

8

965.4 374

29

28,7

4.

89

88.1

12.4

9

1475

90.7

9

307.

29.1

9

9664 374

30

29.7

4.9

00

891

12.5

150

148.5

20.9

810

908.

994

870

967.4 374

31

30.7

4.3

91

90.1

12.7

1

149.5

21.

1

308.9

99.4

1

968.4 37.7

32

31.7

44

92

911

124

2

150.5

21.9

2

209.9

994

9

369.4 374

33

32.7

4.6

93

99.1

12.9

3

151.5

213

3

910.9

99.6

3

9704 38.

34

33.7

4.7

94

931

13.1

4

152.5

214

4

911.9

298

4

9714 38.1

35

34.6

49

95

94.1

134

5

153.5

21.6

5

912.9

894

5

273.3 384

36

356

5.

98

95.1

13.4

6

154.5

21.7

6

913.9

30.1

6

973.3 38.4

37

366

5.1

97

06.1

13.5

7

155.5

21.9

7

914.9

304

7

9744 384

38

37.6

5.3

98

97.

13.6

8

156.5

22.

8

9159

304

8

9754 38.7

39

386

5.4

99

98.

13.8

9

157.5

23.1

9

916.8

304

9

9764 384

40

39.6

5.6

100

99.

13.9

160

158.4

29.3

880

9179

304

880

3774 39.

41

406

5.7

1

100.

14.1

1

159.4

99.4

1

9184

304

1

278.3 39.1

49

416

5.8

2

101.

144

2

160.4

224

9

3198

30.9

3

9794 394

43

496

6.

3

103.

14.3

3

161.4

22.7

3

3304

31.

3

9809 39.4

44

43.6

6.1

4

103.

14.5

4

162.4

218

4

3914

31.9

4

981.9 394

45

44.6

6.3

5

104.

14.6

5

163.4

23.

5

393.8

31.3

5

9822 394

46

455

6.4

6

105

144

6

164.4

23.1

6

9934

314

6

9834 394

47

404

6.5

7

106.

14.9

7

165.4

234

7

994.8

31.6

7

3848 394

48

475

6.7

8

1069

15.

8

166.4

23.4

8

995.8

317

8

3858 40.1

49

485

6.8

9

107.9

154

9

167.4

234

9

9964

314

9

3864 404

50

49.5

7.

110

1089

15.3

170

168.3

23.7

880

9974

32

800

9874 40.4

51

50.5

7.1

1

109.9

15.4

1

1694

938

1

3988

32.1

1

9883 404

52

515

7.2

2

110.9

156

9

170.3

23.9

3

399.7

39.3

9

389.9 40.6

53

52.5

7.4

3

J 11.9

15.7

3

17J4

94.1

3

9397

39.4

3

990.1 404

54

53.5

7.5

4

1139

15.9

4

172.3

24.2

4

331.7

33.6

4

891.1 404

55

545

7.7

5

1139

16.

5

173.3

94.4

6

339.7

387

5

999.1 41.1

58

55.5

7.8

6

114.9

16.1

6

174.3

245

6

9337

324

6

993.1 414

57

544

7.9

7

115.9

16.3

7

175 3

24.6

7

334.7

33.

7

994.1 414

58

574

ai

8

1169

164

8

176.3

94.8

8

9357

331

8

9951 418

59

584

8.2

9

117.8

16.6

9

177 3

94.9

9

936.7

334

9

9961 41.0

60

59.4

84

190

1184

16.7

180

1789

95.1

940

937.7

334

300

897.1 414

dirt.

dep.

d. lat.

dirt.

dep. d. lat .

dirt.

dep. d. let

dirt.

dep.

d. lat.

dirt.

dep. d. let.

Dietaaee, Departan and Diff Latitude.
Ce*a**e 88*.

Digitized by VjOOQIC

TABUS T.

Cemrea 90.

Dietaace, Diff Latitude mad Departure.

119

diet.
1 "

d. lat.

dep.

dirt.

d.lat.

dep.

dirt.

d. lat.

dep.

dirt.

d. lat. dep.

dirt.

d. lat. dep.

1.

0.2

61

00.9

94

191

1194

18.9

181

178.8 28.3

941

238. 37.7

9

9.

0.3

69

61.9

9.7

9

1204

19.1

9

179.8 284

3

239. 37.9

3

3.

0.5

63

69.9

9.9

3

1214

19.2

3

180.7 28.6

3

240. 38.

4

4.

0.6

64

63Ji

10.

4

1225

19.4

4

181.7 284

4

241. 38.2

5

4.9

0.8

65

64.2

10.3

5

1334

19.6

5

182.7 38.9

5

242. 3S4

6

5.9

0.9

66

65.9

104

6

134.4

19.7

6

183.7 39.1

6

243. 38.5

7

6.9

1.1

67

66.9

104

7

125.4

19.9

7

184.7 29.3

7

344. 38.6

8

7.9

14

68

67.3

10.6

8

126.4

30.

8

185.7 29.4

8

244.9 38.8

8.9

1.4

69

684

10.8

9

127.4

20.2

9

186.7 29.6

9

245.9 39.

10

9.9

1.6

70

69.1

11.

130

138.4

20.3

190

187.7 99.7

950

246.9 39.1

11

10.9

1.7

71

70.1

11.1

1

129.4

204

1

188.8 294

1

247.9 39.3

12

11.9

19

72

71.1

11.3

9

130.4

20.6

9

189.6 30.

3

24.19 39.4

13

12.8

9.

73

72.1

11.4

3

131.4

20.8

3

100.6 304

3

249.9 39.6

14

i3.8

22

74

73.1

11.6

4

132.4

21.

4

191.6 304

4

350.9 39.7

15

14.8

9.3

75

74.1

11.7

5

1334

21.1

5

192.6 304

5

251.9 39.9

16

15.8

95

76

75.1

11.9

6

1344

213

6

193.6 30.7

6

252.8 40.

17

104

9.7

77

76.1

12.

7

1354

21.4

7

194.6 304

7

2534 40.3

19

178

9.8

78

77.

12.2

8

1364

21.6

8

105.6 31.

8

354.8 40.4

19

184

3.

79

'7a

12.4

9

1374

21.7

9

1964 31.1

9

355.8 40.5

SO

19.7

3.1

80

79.

194

140

1384

21.9

900

1974 314

960

3564 40.7

91

90.7

3.3

81

80.

19.7

1

139.3

92.1

1

1984 31.4

1

3574 40.8

S3

21.7

3.4

82

81.

12.8

9

140.3

92,2

2

199.5 31.0

2

358.8 41.

S3

22.7

3.6

83

82.

13.

3

141.9

29.4

3

200.5 314

3

359.8 41.1

S4

23.7

3.8

84

83.

13.1

4

142.9

994

4

2014 31.9

4

360.7 41.3

35

24.7

3.9

85

84.

13.3

5

143.2

99.7

5

2024 32.1

5

961.7 414

96

25.7

4.1

86

84.9

134

6

144.2

93.8

6

2034 334

6

263.7 41.6

87

26.7

4.9

87

85.9

13.6

7

145.2

23.

7

2044 32.4

7

363.7 41.8

38

27.6

4.4

88

86.9

13.8

8

146.2

93.9

8

205.4 324

8

3647 41.9

98

28.6

44

89

87.9

13L9

9

147.2

934

9

206.4 32.7

9

365.7 43.1

30

99.6

4.7

00

88.9

14.1

150

148.9

934

910

907.4 39.9

970

966.7 494

31

39.6

4.8

91

80.9

14.2

1

149.1

23.6

]

908.4 33.

1

367.7 43.4

38

31.6

5.

92

90.9

14.4

9

150.1

934

9

9094 334

3

36a7 43.6

33

39.6

5.2

93

91.9

144

3

151.1

93.9

a

910.4 334

3

369.6 43.7

34

33.6

5.3

94

92.8

147

4

152.1

94.1

4

211.4 334

4

2704 42.8

35

34.6

54

ft

93.8

14.9

5

153.1

94.2

5

212.4 33.6

5

271.6 43.

39

35.5

5.6

96

94.8

15.

6

154.1

94.4

6

213.3 33.8

6

272.6 434

37

364

5.8

97

95.8

15.2

7

155.1

94.6

7

214.3 33.9

7

273.6 43.3

38

374

5.9

98

98.8

15.3

8

156.1

94.7

8

215.3 34.1

8

2746 434

38

384

6.1

99

97.8

154

9

157.

94.9

9

2184 344

9

275.6 43.6

40

394

6.3

100

984

154

160

158.

35. .

MO

217.3 344

980

276.6 434

41

404

6.4

1

99.8

15.8

I

159.

35^

1

318.3 34.6

1

2774 44.

49

415

6.6

9

100.7

16.

9

160.

35.3

3

319.3 34.7

2

278.5 441

43

494

6.7

3

101.7

10.1

3

161.

954

3

390.3 34.9

3

279.5 444

44

43.5

6.9

4

102.7

16.3

4

169.

95.7

4

221.2 35.

4

2E0.5 44.4

45

44.4

7.

5

103.7

16.4

5

163.

95.8

5

832.2 35.2

5

281.5 44.6

46

45.4

7.9

6

104.7

16.6

6

164.

96.

6

223.2 35.4

6

382.5 44.7

47

46.4

7.4

7

105.7

16.7

7

164.9

96.1

7

924.2 35.5

7

283.5 44.9

48

47.4

74

8

106.7

16.9

8

165.9

96.3

8

9354 35.7

8

284.5 45.1

49

4a4

7.7

9

107.7

17.1

9

166.9

96.4

9

9964 35.8

9

285.4 454

50

49.4

7.8

110

108.6

17.9

170

1674

96.6

9330

9374 36.

900

986.4 45.4

51

50.4

8.

1

169.6

17.4

1

168.9

96.8

1

9384 36.1

1

287.4 454

59

51.4

ai

9

na6

174

9

169.9

96.9

9

939.1 363

9

268.4 45.7

53

52.3

84

3

111.6

17.7

3

170.9

97.1

3

930.1 36.4

3

389.4 454

54

53.3

8.4

4

112.6

17.8

4

171.9

27.2

4

931,1 36.6

4

200.4 46.

55

54.3

8.6

5

113.6

18.

5

172.8

97.4

5

933.1 36.8

5

391.4 46.1

56

55.3

8.8

6

1*4.6

18.1

6

1734

974

6

933.1 36.9

6

293.4 46.3

57

56.3

8.9

7

115.6

183

7

174.8

97.7

7

934.1 37.1

7

393.3 46.5

58

57.3

9.1

8

116.5

184

8

175.8

97.8

8

335.1 373

8

394.3 46.6

59

58.3

9.2

9

1174

18.6

9

176.8

98.

9

936.1 37.4

9

395.3 46.8

60

59 4

9.4

ISO

1184

18.8

180

177.8

38.9

940

937. 374

300

9964 464

*•:

|dep.

d.lat.

dift

dep. d. lat.

dirt.

dep.

d .lat.

dirt.

dep. d. lat.

dirt. |dep. d. let.

Dartanee Departure end Diff Latitude.
>0*o.

Digitized by VjOOQIC

120

TABLE V.

Cou*se 10°.

Distance, Diff Latitude and Departure.

dist.

d. lat.

dep.

dist.

d. lat.

dep.

dist.

d. lat.

dep.

dist.

d. lat.

dep.

dist.

d. lat.

depi

1

1.

02

61

60.1

10.6

191

119.2

21.

181

178.3

31.4

941

237.3

415 I

2

2.

0.3

62

61.1

10.8

2

120.1

21.2

2

1795

31.6

2

238.3

43.

3

3.

0.5

63

62.

10.9

3

121.1

214

3

180.9

31.8

3

239.3

425

4

3.9

0.7

64

63.

11.1

4

122.1

21.5

4

161.2

32.

4

2403

42.4

5

4.9

0.9

65

64.

11.3

5

123.1

21.7

5

1825

32.1

5

2415

425

6

5.9

1.

66

65.

11.5

6

124.1

21.9

6

163.2

32.3

6

2425

42.7

7

6.9

1.2

67

66.

11.6

7

1251

22.1

7

184.2

32.5

7

2435

42.9

8

7.9

14

06

67.

11.8

8

126.1

22.2

8

185.1

32.6

8

2445

431

9

8.9

1.6

69

68.

12.

9

127.

22.4

9

186.1

32.8

9

2455

435

10

9.8

1.7

70

68.9

12.2

130

128.

22.6

100

187.1

33.

950

2465

434

11

10.8

1.9

71

69.9

12.3

1

129.

22.7

1

188.1

33.2

1

2475

43.6

IS

11.8

2.1

72

70.9

12.5

2

130.

22.9

2

189.1

33.3

2

2485

435

13

12.8

2.3

73

71.9

12.7

3

131.

23.1

3

190.1

33.5

3

2495

43.9

14

13.8

2.4

74

72.9

12.8

4

132.

23.3

4

191.1

33.7

4

250.1

44.1

15

14.8

2.6

75

73.9

13.

5

132.9

23.4

5

192.

33.9

5

25L1

445

16

15.7

2.8

76

74.8

135

6

133.9

23.6

6

193.

34.

252.1

445

17

16.7

3.

77

75.8

13.4

7

134.9

23.8

7

194.

345

7

253.1

445

18

17.7

3.1

78

76.8

13.5

8

135.9

24.

8

195.

34.4

8

254.1

445

19

18.7

3.3

79

77.8

13.7

9

136.9

24.1

9

198. *

34.6

9

255.1

45.

ao

19.7

3.5

80

78.8

13.9

140

137.9

24.3

900

197.

34.7

960

256.1

45.1

21

20.7

&6

81

79.8

14.1

1

136.9

24.5

i

1979

349

1

257.

455

22

21.7

3.8

83

80.8

14.2

2

130.8

24.7

2

198.9

35.1

2

258.

455

23

22.6

4.

83

81.7

14.4

3

140.8

24.8

3

199:9

35.3

3

259.

45.7

24

23.6

4.2

84

82.7

14.6

4

141.8

25.

4

200.9

35.4

4

260.

455

25

24.6

4.3

85

83.7

14.8

5

142.8

25.2

5

201.9

35.6

5

261.

46.

26

25.6

4.5

86

84.7

14.9

6

143.8

25.4

6

202.9

35.8

6

262.

465

27

26.6

4.7

87

85.7

15.1

7

144.8

25.5

7

2035

35.9

7

262.9

46.4

28

27.6

4.9

88

867

15.3

8

145.8

257

8

204.8

36.1

8

2635

465

29

28.5

5.

89

87.6

15.5

9

146.7

25.9

9

205.8

36.3

9

2645

46.7

30

29.5

5.2

00

88.6

15.6

150

147.7

26.

810

206.8

36.5

970

2655

465

31

30.5

5.4

91

89.6

15.8

1

148.7

262

1

207.8

36.6

1

266.9

47.1

32

31.5

5.6

92

90.6

16.

2

149.7

26.4

2

208.8

36.8

S

2675

475

33

32.5

5.7

93

91.6

16.1

3

150.7

26.6

3

209.8

37.

3

268.9

47.4

34

33.5

5.9

94

92.6

16.3

4

151.7

26.7

4

210.7

375

4

369.8

47.6

35

34.5

6.1

95

93.6

16.5

5

152.6

26.9

5

211.7

37.3

5

2705

475

36

35.4

63

96

94.5

16.7

6

153.6

27.1

6

212.7

37.5

6

271.8

475

37

36.4

6.4

97

95.5

16.8

7

154.6

273

7

213.7

37.7

7

272.8

48.1

38

37.4

6.6

98

96.5

17.

8

155.6

27.4

8

214.7

37.9

8

2735

48.3

39

38.4

6.8

99

97.5

17.2

9

156.6

27.6

9

215.7

38.

9

374.8

48.4

40

39.4

6.9

100

98.5

17.4

160

157.6

27.8

990

216.7

38.2

980

375.7

48.6

41

40.4

7.1

1

99.5

17.5

1

158.6

2a

i

217.6

38.4

1

378.7

485

42

41.4

7.3

2

100.5

17.7

2

159.5

28.1

8

218.6

36.5

3

377.7

49.

43

42.3

7.5

3

1014

17.9

3

160.5

28.3

3

219.6

38.7

3

278.7

49.1

44

43.3

7.6

4

102.4

18.1

4

J61.5

28.5

4

220.6

38.9

4

379.7

495

45

44.3

7-8

5

103.4

18.2

5

162.5

28.7

5

221.6

39.1

5

380.7

495

46

45.3

8.

6

104.4

18.4

6

163.5

28.8

6

222.6

39.2

6

381.7

49.7

47

46.3

8.2

7

105.4

18.6

7

164.5

29.

7

223.6

39.4

7

382.6

495

48

47.3

8.3

8

106.4

18.8

8

165.4

29.2

8

224.5

39.6

8

983.6

50.

49

48.3

8.5

9

107.3

16.9

9

166.4

29.3

9

225.5

39.8

9

384.6

503

30

49.2

8.7

110

108.3

19.1

170

167.4

29.5

930

226.5

39.9

990

385.6

50.4

51

50.2

8.9

1

109.3

19.3

1

168.4

29.7

1

227.5

40.1

1

986.6

505

52

51.2

9.

2

110.3

19.4

2

169.4

29.9

2

228,5

40.3

2

387.6

50.7

53

52.2

9.2

3

111.3

19.6

3

170.4

30.

3

229.5

40.5

3

3885

50,9

54

53.2

9.4

4

112 3

19.8

4

171.4

30.2

4

230.4

40.6

4

889.5

51.1

55

54.2

9.6

5

113.3

20.

5

172.3

30.4

5

231.4

40.8

5

8905

515

56

55.1

9.7

6

114.2

20.1

6

173.3

30.6

232.4

41.

6

291.5

51.4

57

56.1

9.9

7

115.2

20.3

7

174.3

30.7

7

233.4

415

7

8925

51.6

58

57.1

10.1

8

1165

20.5

8

175.3

30.9

8

234.4

41.3

8

8935

51.7

59

58.1

10.2

9

117.2

20.7

9

176.3

31.1

9

235.4

41.5

9

894.5

51.9

60

59.1

10.4

120

118.2

20.6

180

177.3

31.3

240

236.4

41.7

300

395.4

52.1

dist.

dep.

d. lat

dist.

dep.

1. lat.

dirt.

dep.

1 lat.

dist.

dep.

d. lat.

dist i dep.

d. tat.

Distance, Departure and Diff Latitude.
Cowrie 80°.

Digitized by VjOOQIC

121

Dirtanca, IMC Latitude and Daaartoia.

dirt.

<Llat.

atp.

dirt.

d.Ut.

dap.

dirt.

d.Ut. dap.

dirt.

d.Ut.

dap.

dirt.

d.lat. dap.

1

L

04

01

599

114

181

1184 83.1

181

177.7

344

Ml

836.6 46.

9

8.

0.4

68

604

114

8

1104 83.3

8

178.7

34.7

8

837.6 464

3

84

0.6

63

614

18.

3

180.7 934

179.6

344

3

8384 464

4

3.9

04

64

684

184

4

191.7 83.7

180.6

35.1

4

8394 464

5

49

1.

65

634

18.4

5

199.7 83.9

181.6

354

5

8404 46.7

6

54

LI

66

644

13.6

6

183.7 84.

183.6

354

6

8414 464

7

64

14

67

65.8

184

7

194.7 844

183.6

35.7

7

8484 47.1

e

7.9

14

68

664

13.

8

1854 34.4

1844

354

8

343.4 474

»

84

1.7

09

67.7

134

9

1864 94.6

185.5

36.1

844.4 474

10

9J8

1.9

TO

68.7

134

130

187.6 844

100

1864

364

840

945.4 47.7

n

104

SI

71

69.7

134

1

1884 85.

1874

36.4

1

8464 474

IS

114

84

79

70.7

13.7

9

1994 854

1884

364

3

847.4 48.1

IS

184

84

73

71.7

13.9

3

1304 S5.4

1804

364

3

8484 484

14

13.7

8.7

74

78.6

14.1

4

1314 85.6

190.4

37.

4

8494 464

13

14.7

8.9

75

73.6

144

5

1334 854

101.4

374

5

8504 48.7

16

15.7

3.1

76

74.6

144

6

1334 86.

198.4

37.4

8514 484

17

16.7

3.3

77

754

14.7

7

1344 96.1
1354 86.3

193.4

37.6

7

8594 49.

18

17.7

3.4

78

764

144

8

194.4

374

8

9534 494

19

ia7

ao

79

774

15.1

9

138.4 864

1954

38.

8544 494

ao

194

34

SO

764

154

140

1374 36.7

aoo

1964

38.8

900

8554 494

81

804

4.

81

794

154

1

138.4 86.9

1974

36.4

1

8564 494

99

814

44

88

804

15.6

3

139.4 87.1

1984

384

S

8574 50.

S3

88.6

4.4

83

814

154

3

140.4 874

1994

36.7

3

8584 504

34

83.6

4.6

84

834

16.

4

141.4 874

8004

38.9

4

859.1 50.4

39

844

4.8

85

83.4

164

5

1484 87.7

801.9

39.1

5

860.1 504

96

854

5.

66

844

16.4

6

1434 87.9

303.3

394

6

861.1 50.8

37

864

54

87

854

164

7

1444 88.

3034

394

7

863.1 50.9

38

874

54

88

86.4

164

8

1454 88.9

804.9

39.7

8

863.1 51.1

89

8B4

54

89

874

17.

9

1464 98.4

8054

39.9

9

864.1 514

30

89.4

5.7

00

88.3

174

ISO

1474 984

910

866.1

40.1

a to

895. 514

31

30.4

54

91

894

17.4

1

1484 984

807.1

404

1

966. 51.7

as

31.4

6.1

93

904

174

3

1494 99.

308.1

40.5

9

867. 614

33

38.4

64

93

914

17.7

3

150.9 99.9

809.1

404

3

868. 53.1

34.

33.4

64

94

934

17.9

4

1514 90.4

910.1

40.8

4

869. 59.3

35

34.4

6.7

95

93.3

iai

4

1594 994

911.

41.

5

809.9 534

36

354

6.9

96

94.3

18.3

6

153.1 994

918.

414

6

9704 53.7

37

364

7.1

97

954

184

7

154.1 30.

813.

41.4

7

8714 584

38

374

74

98

664

18.7

8

155.1 30.1

314.

41.6

8

3784 53.

39

384

7.4

99

974

18.9

9

156.1 304

815.

414

9

8734 534

40

39.3

74

100

984

19.1

100

157.1 304

aao

816.

48.

5180

8744 534

41

40.3

74

1

99.1

193

1

158. 30.7

816.9

484

1

8754 534

49

41.3

&

9

100.1

194

9

159. 30.9

317.9

48.4

3

8764 534

43

484

84

3

101.1

19.7

3

160. 31.1

918.9

484

3

8774 54.

44

434

8.4

4

108.1

194

4

161. 31.3

9194

48.7

4

8784 544

45

444

8.6

5

103.1

90.

5

169. 314

830.9

48.0

5

8794 54.4

46

454

84

6

104.1

894

6

163. 31.7

881.8

43.1

6

989.7 54.6

47

48.1

9.

7

105.

90.4

7

163.0 31.9

8894

434

7

981.7 544

48

47.1

94

8

106.

80.6

8

1644 39.1

9934

43.5

8

883.7 55.

49

48.1

94

9

107.

80.8

9

1654 334

8344

43.7

9

883.7 55.1

50

49.1

94

116

109.

91.

170

1664 39.4

J630

9954

43.0

aoo

984.7 554

51

50.1

9.7

1

109.

314

1

167.9 33.6

996.8

44.1

i

885.7 554

58

51.

9.9

9

109.0

91.4

3

1684 33.8

897.7

44.3

3

886.6 55.7

53

58.

J0.1

3

1104

914

3

1694 33.

838.7

444

3

8874 554

54

53.

10.3

4

1114

814

4

1704 333

889.7

44.6

4

8884 56.1

55

54.

104

5

1134

81.0

5

1714 33.4

930.7

444

5

S894 564

56

AS-

10.7

6

113.9

83.1

6

1734 33.6

931.7

45.

6

8904 56.5

57

56.

10.9

7

114.9

894

7

173.7 334

833.6

454

7

9914 56.7

58

56.9

11.1

8

115.8

894

8

174.7 34.

8

833.6

45.4

8

9994 56.9

59

579

114

9

1168

89.7

175.7 34.3

8344

45.6

9

9934 57.1

00

J 58.9

114

ISO

1174

83.9

180

176.7 34.3

840

835.6

454

300

8944 57.8

dirt.

Idep.

d. lat.

dirt.

dap.

d.iat

Idirt. dap. d.1at.|dirt.

dap.

d.lat.

dirt.

Idep. d.Ut.

Dirtanca, Dapartaia and Diff. Latitoda.

Digitized by VjOOQIC

12*

TABLE ▼.

C©1

18B.

Dituaet, DUE Latitude and Departure.

ditt.

d.tat.

dap.

diet

d.tat.

dep.

diet.

d.tat

dep.

diet

d. let

dep.

diet

d. tat dep. (

1

L

0.8

61

59.7

18.7

1S1

118.4

9S4

181

177.

374

841

835.7 50.1

S

S.

0.4

68

60.6

18.9

9

1194

85.4

3

178.

374

S

236.7 504

3

8.6

0.6

63

61.6

13.1

3

1904

95.6

3

179.

38.

3

237.7 504

4

3.9

04

64

634

13.3

4

1214

854

4

180.

384

4

238.7 50.7

5

4.3

1.

65

63.6

134

5

133.3

96.

5

181.

384

5

2394 504

e

5.9

14

66

644

13.7

6

193.3

964

6

181.9

38.7

6

8104 51.1

7

64

1.5

67

654

13.9

7

1844

90.4

7

183.0

384

7

3414 51.4

8

7.8

17

68

664

14.1

8

135.3

96.6

8

163.9

39.1

8

2434 514

9

84

14

69

674

144

9

1964

964

8

1844

394

9

2434 5UB

io

9.8

8.1

TO

684

14.6

180

197.3

87.

100

1858

394

880

8444 58.

11

104

8.3

71

09.4

144

1

138.1

974

1

1864

39.7

1

2454 584

IS

11.7

84

78

70.4

15.

9

199.1

87.4

8

1674

394

3

9464 52.4

13

12.7

9.7

73

71.4

154

3

130.1

87.7

3

1884

40.1

3

9474 534

14

13.7

8.9

74

78.4

15.4

4

131.1

87.9

4

1894

404

4

948.4 524

15 '

14.7

3.1

75

73.4

15.6

6

133.

98.1

5

190.7

404

5

249.4 53.

16

15.6

34

76

744

154

6

133.

88.3

6

191.7

404

6

2304 53.2

17

16.6

34

77

75.3

16.

7

134.

984

7

193.7

41.

7

2514 534

18

17.6

3.7

78

76.3

16.8

8

135.

88.7

8

193.7

414

8

953.4 534

13

18.6

4.

79

774

16.4

9

136.

98.9

9

194.7

41.4

9

9534 534

SO

194

44

80

784

16.6

140

1364

99.1

SOO

195.6

414

860

2544 544

91

904

4.4

81

794

16.8

1

137.9

99.3

1

1964

414

1

9554 544

33

914

44

88

804

17.

3

138.9

994

9

197.6

49.

8

9564 544

33

334

44

83

81.3

17.3

3

139.9

29.7

3

198.6

494

3

9574 54.7

34

334

5.

84

834

17.5

4

140.9

99.9

4

1994

49.4

4

9584 544

35

84.4

5.3

85

83.1

17.7

5

141.8

30.1

5

8004

484

5

2594 55.1

38

85.4

5.4

86

84.1

17.9

6

148.8

30.4

6

9014

494

6

960.9 554

87

96.4

5.6

87

85.1

18.1

7

143.8

30.6

7

9034

43.

7

9614 554

SB

87.4

5.8

88

86.1

18.3

8

144.8

30.8

8

9034

434

e

362.1 55.7

33

984

6.

83

87.1

184

9

145.7

31.

9

904.4

434

9

963.1 554

SO

994

64

•0

88.

18.7

150

146.7

31.3

S10

9054

43.7

810

964.1 56.1

31

30.3

6.4

91

89.

iao

1

147.7

31.4

1

906.4

4?.9

1

965.1 564

as

31.3

6.7

98

90.

19.1

8

148.7

31.6

3

907.4

44.1

9

966.1 564

33

33.3

6.9

93

91.

19.3

3

149.7

31.8

3

9084

444

3

967. 564

34

33.3

7.1

94

914

19.5

4

1504

33.

4

9094

444

4

968. 57.

35

34.8

74

95

98.9

19.8

5

151.6

334

5

9104

44.7

5

869. 574

33

354

74

96

93.9

80.

6

1584

33.4

6

9114

44.9

6

WO. 57.4

87

364

7.7

97

94.9

80.9

7

153.6

334

7

818.3

45.1

7

9704 574

38

37.8

7.9

08

954

20.4

8

1544

38.9

8

813.9

454

8

971.9 574

33

38.1

8.1

99

964

80.6

9

1554

33.1

9

3144

454

9

8784 58.

40

89.1

84

too

974

808

160

1564

334

880

9154

45.7

880

973.9 584

41

40.1

8.5

1

984

81.

1

1575

33.5

1

9164

45.9

t

874.9 58.4

43

41.1

8.7

3

994

31.3

9

1584

33.7

8

917.1

464

9

2754 584

43

48.1

8.9

3

100.7

81.4

3

159.4

33.9

3

818.1

46.4

a

9764 584

44

43.

9.1

4

101.7

81.6

4

160.4

34.1

4

319.1

46.6

4

9774 59.

45

44.

9.4

5

103.7

914

5

161.4

34.3

5

990.1

46.8

5

9784 594

46

45.

9.6

6

103.7

88.

6

163.4

344

6

931.1

47.

6

9794 594

47

40.

9.8

7

104.7

88.3

7

163.4

34.7

7

239.

47.9

7

989.7 59.7

48

47.

10.

6

105.7

994

8

1644

34.9

8

233.

47.4

8

981.7 594

43

474

104

8

106.6

98.7

9

165.3

33.1

9

934.

474

9

983.7 60.1

80

48.9

10.4

110

167.6

88.9

190

1664

354

830

995.

474

880

983.7 604

51

49.9

10.6

1

108.6

93.1

1

1674

354

1

936.

48.

1

984.6 604

53

50.9

10.8

3

1094

93.3

3

1684

35.8

3

936.9

484

8

985.6 60.7

53

51.8

11.

3

110.5

93.5

3

1694

36.

3

937.9

48.4

3

986.6 604

54

584

114

4

1114

93.7

4

1704

36.9

4

938.9

48.7

4

2874 61.1

55

534

11.4

5

1184

93.9

5

171.3

36.4

5

989.9

48.9

5

V84 614

56

54.8

11.6

6

1134

94.1

6

178.3

36.6

6

9304

49.1

6

2994 614

57

554

11.9

7

114.4

94.3

7

173.1

36.8

7

931.8

49.3

7

9904 61.7

58

56.7

18.1

8

115.4

944

8

174.1

37.

8

933.8

49.5

8

2914 62.

53

57.7

18.3

9

116.4

94.7

9

175.1

37.3

9

933.8

49.7

9

9934 694

63

58.7

184

190

117.4

94.9

180

176.1

37.4

940

9344

494

300

993.4 634

dirt.

dep.

d.lat

diflt.

dep.

d. tat.

dirt.

dep.

1 tat

ditt.

dep.

I. tat.

diet

dep. d.tat

Distance, Departare end DiC Latitude.
Course 98°.

Digitized by VjOOQIC

TAILS V.

Centre* 13°.

Diatajice, DUE Latitude and Departure.

123

dirt.

d.lat.

dep.

dirt.

dial.

dep.

diflt.

d.lat.

dep.

dirt.

<Llat.

dep.

dirt.

cLlat

oep.

1

L

0.3

61

SI

13.7

iai

117-0

375

181

176.4

40.7

Ml

234.8

543

8

1.0

0.4

68

13.0

3

118.0

27.4

9

177.3

40.0

8

2355

54.4

3

3.0

0.7

63

61.4

14.3

3

1193

27.7

3

178.3

413

3

236*

54.7

4

3.0

0.9

64

63.4

14.4

4

130.8

27.9

4

179.3

41.4

4

237.7

545

5

4.0

1.1

65

633

14.6

5

121.8

28.1

5

1803

413

5

238.7

55.1

6

5.8

1.3

66

64.3

148

6

1338

283

6

1813

41*

6

239.7

553

7

6.8

1.6

67

65.3

15.1

7

133:7

286

7

183.2

48.1

7

240.7

55.6

8

7.8

1.8

68

66.3

15.3

8

134.7

28*

8

1833

433

8

241.6

55.8

8.8

a.

69

675

15.5

135.7

89.

184.3

4S.5

248.6

56.

10

0.7

35

70

68.3

15.7

130

130.7

895

100

185.1

48.7

MO

3436

563

11

10.7

8.5

71

60.8

16.

1

137.6

805

1

186.1

43.

I

844.6

563

13

11.7

8.7

72

70.3

16.8

3

138.6

29.7

8

187.1

433

8

845.5

56.7

13

13.7

8.9

73

71.1

16.4

3

130.6

89.9

3

188.1

43.4

3

8463

56.9

14

13.6

3.1

74

73.1

16.6

4

130.6

30.1

4

180.

43.6

4

8473

57.1

15

14.6

3.4

75

73.1

16.9

5

131.5

50.4

5

190.

43.9

5

8483

57.4

16

15.6

3.8

76

74.1

17.1

6

1333

30.6

6

191.

44.1

6

840.4

575

17

16.6

3.8

77

75.

173

7

1333

30.8

7

193.

44.3

7

850.4

575

18

175

4.

78

76.

17.5

8

1343

31.

8

198.9

44.5

8

251.4

58.

10

185

43

70

77.

173

1354

31.3

9

1039

44*

353.4

583

ao.

195

43

SO

773

18.

14V0

13&4

313

900

104.9

45.

000

853.3

583

21

305

4.7

81

78.0

18.3

1

137.4

31.7

1

105.8

453

1

8543

587

S3

31.4

4.9

82

705

18.4

8

138.4

31.0

3

106*

45.4

8

8553

58.9

23

82.4

5.2

83

80.0

18.7

3

139.3

393

3

107*

45.7

3

3563

493

24

23.4

5.4

84

81*

18.9

4

140.3

32.4

4

196*

45.9

4

8573

50.4

39

34.4

5.6

85

83.8

19.1

5

1413

32.6

5

190.7

46.1

5

8585

505

36

253

5.8

86

83*

19.3

6

1433

33*

6

800.7

463

6

3595

505

27

263

6.1

87

84.8

19.6

7

1435

33.1

7

801.7

46.6

7

8605

60.1

28

273

63

88

85.7

193

8

1445

33.3

8

203.7

46.8

8

861.1

603

29

883

85

80

86.7

30.

1455

333

803.6

47.

368.1

603

30

803

6.7

00

87.7

305

150

1465

33.7

SjlO

804.6

473

»90

363.1

60.7

31

305

7.

01

88.7

303

1

1471

34.

1

205.6

475

1

a 1

61.

33

315

7.8

03

80.6

90.7

8

148.1

34.3

8

206.6

47.7

8

613

33

33.1

7.4

03

004

80.0

3

140.1

34.4

3

2075

47.9

3

366.

6X4

34

33.1

7.6

94

01j6

31.1

4

150J

34.6

4

208.5

481

4

367.

61.6

35

34.1

7.9

05

98.6

31.4

5

151.

340

5

209.5

48.4

5

86a

&!

36

35.1

ai

06

93.5

31.6

6

153.

35.1

6

210.5

48.6

6

868.9
3685

37

36.1

8.3

07

94.5

813

7

153.

35.3

7

211.4

48*

7

683

38

37.

8.5

08

953

S3.

8

154.

35.5

8

2124

40.

8

870.9

683

30

38.

8.8

00

963

833

1545

35*

213.4

403

871*

68*

4V0

30.

0.

too

073

883

100

1553

36.

MO

814.4

40.5

SjOO

3793

63.

41

30.0

9.3

1

08.4

83.7

1

1565

365

1

3153

40.7

1

873*

635

43

40.0

9.4

8

00.4

33.0

8

157*

36.4

8

3163

40.0

3

8743

63.4

43

41.0

0.7

3

100.4

33.3

3

158*

36.7

3

3173

503

3

875.7

68.7

44

480

0.0

4

1013

83.4

4

159*

36.0

4

3183

50.4

4

876.7

633

45

43.8

10.1

5

103.3

83.6

5

160*

37.1

5

8193

50.6

5

377.7

64.1

46

44.8

10.3

6

103.3

833

6

161.7

37.3

6

8305

50*

6

878J

643

47

45.8

106

7

104 3

84.1

7

163.7

37.6

7

8815

51.1

7

8705

64.6

48

46*

10.8

8

1053

843

8

163.7

37*

8

8335

513

8

880.6

645

40

47.7

11.

1065

84.5

164.7

38.

9

833a

313

381.6

65.

50

487

11.3

110

1075

34.7

170

1653

385

3j30

334.1

5X7

OOO

3883

653

51

40.7

11.5

1

1085

85.

1

166*

383

1

835.1

53.

1

888.5

655

53

50.7

11.7

8

100.1

853

8

1675

38.7

3

935.1

583

8

384.5

65.7

53

51.6

11.0

3

110.1

85.4

3

1683

36.9

3

837.

58.4

3

3853

65.0

54

58.6

13.1

4

111.1

85.6

4

1603

39.1

4

338.

58.6

4

386.4

66.1

55

534

12.4

5

113.1

85.0

5

1703

39.4

5

839.

52 9

5

387.4

66.4

56

54.6

13.6

6

113.

86.1

6

17J3

30.6

6

830.

531

6

988.4

663

57

555

18.8

7

114.

863

7

1783

30*

7

830.9

53.3

7

369.4

663

58

56.5

13.

8

115.

863

8

173.4

40.

8

8315

53.5

8

390.4

67.

50

575

13.3

116.

86.8

174.4

403

833.9

53*

9

8013

673

60

58.5

1*5

130

116.0

87.

180

1754

403

340

833*

54.

300

3983

675

dirt.

|dep.

dJat.

dirt.

dep.

A. lat .

dirt.

dep.

d. lat.

dirt.

dep.

d. let.

diet.

Idep.

d. Ut,

Dietaace, Departure aad DUE Latitude.
1 7T°-

Digitized by VjOOQIC

1*4

TABLE ▼.

Orane 149.
Dietanee, DUE Latitude and Departure.

diit.

d. tat.

dep.

dift.

d.lat.

dep.

diet.

d. lat.

dep.

diit.

d. tat.

dep.

diet.

d. lat. dep.

1

1.

0.3

01

594

144

181

117.4

29.3

181

175.6

434

941

9334 584

9

1.9

0.5

63

804

15.

9

118.4

394

9

176.6

44.

3

9344 584

3

9.9

0.7

63

61.1

15.3

3

119.3

994

3

177.6

444

3

9354 584

4

3.9

1.

64

68.1

15.5

4

1304

30.

4

1784

444

4

3364 59.

5

4.9

1.9

65

63.1

15.7

5

131.3

30.9

5

1794

444

5

937.7 594

6

5.8

1.5

66

64.

16.

6

1934

304

6

180.5

45.

6

938.7 594

7

6.8

1.7

67

65.

164

7

183.3

30.7

7

181.4

45.9

7

939.7 594

8

7.8

1.9

68

66.

16.5

8

134.3

31.

8

183.4

454

8

9404 00.

8.7

94

69

67.

16.7

9

1954

314

9

183.4

45.7

9

9414 084

10

9.7

9.4

TO

67.9

164

180

136.1

31.4

180

184.4

46.

950

9494 884

11

10.7

9.7

71

664

174

1

197.1

31.7

1

185.3

464

1

9434 80.7

19

11.6

9.9

79

69.9

17.4

8

198.1

31.9

9

1864

46.4

9

9444 6L

13

19.6

3.1

73

70.8

17.7

3

199.

33.3

3

187.3

46.7

3

9454 614

14

13.6

3.4

74

714

17.9

4

130.

33.4

4

188.3

46.9

4

9484 614

13

14.6

3.6

75

784

18.1

5

131.

33.7

5

189.3

474

5

9474 6L7

16

154

3.9

76

73.7

18.4

6

131.

33.9

6

190.9

47.4

6

948.4 614

17

164

4.1

77

74.7

184

7

1319

33.1

7

191.1

47.7

7

9494 634

18

17.5

4.4

78

75.7

18.9

8

133.9

33.4

8

198.1

474

8

9504 69.4

19

18.4

4.6

79

76.7

19.1

9

134.9

33.6

9

193.1

48.1

9

2514 69.7

90

19.4

44

80

77.6

19.4

140

1354

33.9

900

194.1

48.4

980

9594 894

91

90.4

5.1

81

78.6

19.6

1

1364

34.1

1

195.

48.6

1

9534 63.1

99

31.3

54

83

79.6

19.8

9

1374

34.4

9

196.

48.9

9

9544 634

83

9*4

5.6

83

804

90.1

3

1384

344

3

197.

49.1

3

9554 634

94

93.3

54

84

81.5

30.3

4

139.7

34.8

4

197.9

49.4

4

9564 634

95

844

6.

85

824

90.6

5

140.7

35.1

5

1984

49.6

5

957.1 64.1

98

95.9

6.3

88

83.4

904

6

141.7

354

6

199.9

494

6

958.1 64.4

37

964

6.5

87

84.4

91.

7

1436

35.6

7

300.9

50.1

7

959.1 644

98

374

64

88

85.4

91.3

8

143.6

354

8

901.8

504

8

900. 644

99

98.1

7.

80

86.4

914

9

1444

36.

9

909.8

50.6

9

961. 65,1

80

99.1

7.3

00

874

814

150

1454

36.3

910

903.8

504

990

969. 654

31

30.1

7.5

91

884

93.

1

1464

364

1

904.7

51.

1

963. 654

39

31.

7.7

99

89.3

93.3

9

147.5

364

9

205.7

514

8

963.9 654

33

39.

a

93

90.9

934

3

1484

37.

3

908.7

514

3

9644 66.

34

33.

84

94

91.3

93.7

4

149.4

37.3

4

907.6

514

4

965.9 664

33

34.

84

95

93.3

93.

5

150.4

374

5

906.6

53.

5

9064 884

38

34.9

8.7

96

93.1

33.3

6

151.4

37.7

6

9094

534

6

9674 664

37

35.9

9.

97

94.1

935

7

153.3

38.

7

910.6

534

7

9684 67.

38

38.9

9.3

98

95.1

93.7

8

1534

384

8

9114

53.7

8

969.7 674

39

374

9.4

99

96.1

94.

9

1544

385

9

9194

53.

9

870.7 674

40

384

9.7

100

97.

94.9

180

1554

38.7

990

9134

53.9

980

971.7 67.7

41

394

9.9

1

98.

94.4

1

156.3

38.9

1

914.4

534

1

273.7 68.

49

404

104

9

99.

94.7

9

157.3

39.3

3

915.4

53.7

9

9734 684

43

41.7

10.4

3

99.9

94.9

3

1584

39.4

3

916.4

53.9

3

9744 684

44

43.7

10.6

4

100.9

954

4

159.1

39.7

4

9174

544

4

9754 68.7

45

43.7

10.9

5

1014

95.4

5

160.1

39.9

5

9184

54.4

5

3764 684

48

44.6

11.1

6

103.9

35.6

6

161.1

40.3

6

919.3

54.7

6

9774 894

47

45.6

11.4

7

1034

95.9

7

103.

40.4

7

930.3

54.9

7

9784 894

48

49.6

114

8

1044

99.1

8

163.

40.6

8

9914

55.9

8

979.4 69.7

49

47.5

11.9

9

1054

96.4

9

164.

40.9

9

9924

55.4

9

980.4 694

50

484

131

110

106.7

98.6

lit)

165.

41.1

990

933.9

554

980

981.4 704

51

49.5

133

1

107.7

96.9

1

165.9

41.4

1

9*4.1

55.9

1

989.4 70.4

59

504

134

9

108.7

97.1

9

166.9

414

9

295.1

56.1

9

9834 704

53

51.4

138

3

1694

374

3

1674

41.9

3

996.1

56.4

3

9844 704

54

53.4

13.1

4

1104

974

4

1684

49.1

4

937.

584

4

9854 71.1

55

53.4

13.3

5

1114

974

5

1694

494

5

998.

56.9

5

9984 714

56

54.3

134

6

1136

98.1

6

1704

494

6

999.

57.1

6

9874 714

57

55.3

134

7

1134

98.3

7

171.7

494

7

930.

57.3

7

9984 71.9

58

564

14.

8

1144

984

8

173.7

43.1

8

930.9

574

8

989.1 79.1

59

57.3

14.3

9

1154

98.8

9

173.7

434

9

9314

574

9

990.1 794

60

58.9

144

130

116.4

99.

189

174.7

434

940

938.9

58.1

308

991.1 734 |

diet.

dep.

d. lat.

diit.

dep.

1 tat.

diet.

dep. d. lat.

diet.

dep.

Hat

diet.

dep. d. lac|

Dtitanee, Departure and DiC Latitude.
C*wrm* fO».

Digitized by VjOOQIC

TABLET.

125

.Dirtanoe, Diff Latitude and Departure.

dirt.

d.lal

dap,

dirt.

d.Iat

dap.

dirt.

d.lat.

aap.

dirt.

d.lat

dap,

dirt.

d.lat.

«•*

1

1.

03

61

58.9

153

1931

116.9

313

181

1743

463

041

9393

694

S

1.9

0.5

69

593

16.

9

1173

31.6

9

1753

47.1

9

9333

693

3

9.9

03

63

00.9

163

3

1183

313

3

1763

47.4

3

934.7

693

4

3.9

1.

64

613

16.6

4

1193

38-1

4

177.7

47.6

4

935.7

633

3

4.8

1.3

65

69.8

163

5

190.7

394

5

178.7

47.9

5

936.7

634

e

5*

1.6

66

633

17.1

6

191.7

39.6

6

179.7

48.1

6

937.6

63.7

7

6.8

13

67

64.7

17.3

7

199.7

39.9

7

180.6

484

7

938.6

633

6

7.7

9.1

68

65.7

17.6

8

1933

33.1

8

181.6

48.7

8

9393

643

9

&7

93

69

663

17.9

9

194.6

334

9

1893

483

9403

644

10

9.7

93

70

673

iai

180

1953

333

109

1833

493

SJOO

9413

64.7

11

10.6

93

71

683

184

1

1963

333

1

1843

49.4

1

9494

65.

19

11.6

3.1

79

693

18.6

9

1973

343

9

1853

49.7

9

943.4

653

13

19.6

3.4

73

703

163

3

1983

344

3

186.4

50.

3

9444

653

14

133

3.6

74

713

193

4

1994

34.7

4

1874

503

4

9453

65.7

15

143

3.9

75

79.4

194

5

130.4

343

5

1884

503

5

9463

66.

ie

153

4.1

76

734

19.7

6

1314

353

6

189.3

50.7

6

9473

663

17

16.4

4.4

77

74.4

19.9

7

1393

353

7

190.3

51.

7

9483

683

18

17.4

4.7

78

753

903

8

1333

35.7

8

191.3

513

8

9493

663

19

18.4

43

79

763

894

9

1343

36.

9

1993

513

9

9503

67.

ao

19.3

5.9

80

773

90.7

140

1353

363

800

1933

513

MO

251.1

673

si

90.3

5.4

81

783

91.

1

1363

383

1

1943

59.

1

959.1

67.6

99

813

5.7

89

793

913

9

1373

38.8

9

105.1

593

S

953.1

673

93

993

6.

83

80.9

913

3

138.1

37.

3

196.1

593

3

954.

68.1

94

933

69

84

81.1

91.7

4

139.1

373

4

197.

593

4

955.

68.3

95

941

63

85

89.1

99.

5

140.1

373

5

198.

531

5

956.

683

96

95.1

6.7

86

83.1

99.3

6

141.

373

6

199.

58.3

6

9563

683

97

96.1

7.

87

84.

993

7

149.

38.

7

199.9

53.6

7

9573

69.1

98

97.

73

88

85.

993

8

143.

383

8

9003

53.8

8

958.9

694

99

98.

73

89

80.

93.

9

1433

38.6

9

9013

54.1

9

9593

693

ao

99.

73

•0

86.9

933

160

144.9

383

9310

9093

544

« 70

9603

69.9

31

99.9

8.

91

87.9

93.6

1

1453

39.1

]

9033

54.6

1

9613

70.1

39

30.9

83

93

883

933

9

1463

393

9

9043

543

s

969.7

70.4

33

31.9

83

93

893

94.1

3

1473

393

3

905.7

55.1

3

963.7

70.7

34

393

83

94

903

94.3

4

1483

39.9

4

908.7

554

4

964.7

70.9

35

333

9.1

95

913

94.6

5

149.7

40.1

5

907.7

55.6

5

9653

713

38

343

93

96

99.7

943

6

150.7

404

6

9083

55.9

6

9663

714

37

35.7

9.6

97

93.7

95.1

7

151.7

40.6

7

9093

563

7

9673

71.7

38

36.7

9.8

98

94.7

954

8

159.6

40.9

8

910.6

564

8

9683

79.

39

37.7

10.1

99

953

95.6

9

153.6

413

9

9113

56.7

9

9693

793

OO

383

104

100

963

953

100

1543

41.4

8SJ0

9193

56.9

»80

9703

793

41

393

10.6

1

97.6

96.1

1

1553

41.7

1

9133

573

1

9714

79.7

49

403

10.9

9

983

964

9

1563

41.9

9

914.4

573

8

979.4

73.

43

41.5

11.1

3

993

96.7

3

157.4

493

3

9154

57.7

3

973.4

733

44

493

11.4

4

J003

96.9

4

1584

494

4

916.4

58.

4

9743

733

45

433

11.6

5

10L4

973

5

1594

49.7

5

9t73

583

5

9753

733

46

44.4

11.9

6

109.4

974

6

1603

43.

6

918.3

583

6

9763

74.

47

454

19.9

7

1034

97.7

7

1613

433

7

9193

563

7

9773

743

48

464

19.4

8

1043

98.

8

1693

433

8

9903

59.

8

9783

743

49

473

19.7

9

1053

983

9

1633

43.7

9

9913

593

9

9793

743

50

483

193

110

1063

983

170

1643

44.

980

9993

593

900

980.1

75.1

51

493

133

1

1073

987

1

165-9

443

1

993.1

593

1

981.1

753

59

593

133

9

1083

99.

9

166.1

443

9

994.1

69.

9

989.1

75.6

53

519

13.7

3

109.1

99.9

3

167.1

443

3

995.1

603

3

983.

75.8

54

59.9

14

14

110.1

993

4

168.1

45.

4

996.

60.6

4

984.

76.1

55

53.1

143

15

111.1

993

5

169.

45.3

5

997.

603

5

9843

764

56

54.1

143

16

J 19.

30.

6

170.

453

6

998.

61.1

6

985.9

76.6

57

55.1

143

17

113.

30.3

7

171.

453

7

998.9

613

7

'988.9

76.9

58

56.

15.

18

114.

305

8

1713

46.1

8

999.9

613

8

9873

77.1

59

57.

153

19

1143

303

9

179.9

463

9

939.9

61.9

9

9883

774

69

58.

153

199

115.9

31.1

180

173.9

46.6

940

9313

69.1

300

9893

773

dirt.

dap.

d. lat.

diit.

dap. d.lat.

dirt.

dep*

d.lat.

dirt.

dap.

d.lat.

diit.

dep.

d.lat.

Dirtance, Departure

and DUE
750.

Digitized by VjOOQIC

126

TABLE W.

Omtm tea

Distance, Diff LatKade and Deptrtare.

diet.

d.lat.

dep.

diflt

d.lat.

dep.

diet.

d.Ut.

dep.

diflt.

d. Ut.

*P.

diflt.

<Llat.

dep

1

1.

0.3

61

58.6

163

191

1163

33.4

181

174.

49.9

941

331.7

664

3

1.9

0.G

63

59.6

17.1

8

1173

33.6

8

1743

503

3

333.6

66.7

3

8.9

0.8

63

60-6

17.4-

3

1183

33.9

3

175.9

504

3

333.6

67.

4

33

1.1

64

61.5

17.6

4

119.3

34.3

4

1763

50.7

4

8343

673

5

4.8

1.4

65

633

17.9.

5

1803

34.5

5

1773

51.

5

3353

673

6

53

1.7

66

63.4

183

6

131.1

34.7

6

1783

513

6

8363

673

7

6.7

1.9

67

644

183

7

133.1

35.

7

1793

513

7

337.4

6ai

8

7.7

3.3

68

65.4

18.7

8

133.

35.3

8

189.7

513

8

3384

68.4

9

&7

3.5

69

663

19.

9

134.

35.6

9

181.7

53.1

9

3394

683

10

9.6

83

TO

673

19*3

180

135.

353

190

1893

53.4

950

3403

63.9

11

10.6

3.

71

683

19.6

1

1353

36.1

1

1833

533

1

8413

693

IS

113

3.3

79

693

19.8

3

1363

36.4

3

1843

533

9

8433

693

13

13.5

3.6

73

703

30.1

3

1373

36.7

3

185.5

533

3

3433

69.7

M

133

3.9

74

71.1

80.4

4

138.8

36.9

4

1863

533

4

3443

7a

15

144

4.1

75

73.1

80.7

5

1393

37.8

5

187.4

53.7

5

845.1

703

W

15.4

4.4

76

73.1

80.9

6

130.7

37.5

6

18M

54.

6

346.1

703

17

163

4.7

77

74.

313

7

131.7

37.8

7

1694

543

7

347.

703

18

17.3

5.

78

75.

31.5

8

133.7

38.

8

199.3

543

8

34a

71.1

19

18.3

53

79

753

31.8

9

1333

38.3

9

1913

54.9

9

349.

714

90

19.9

5.5

80

763

33.1

140

1343

383

900

1993

55.1

960

3493

7L7

31

30.3

5.8

81

77.9

33.3

1

1353

38.9

1

193.9

554

1

350.9

713

38

31.1

'6.1

68

783

33.6

3

13S3

39.1

8

1943

55.7

3

S5L9

793

83

83.1

6.3

63

79.8

33.9

3

1373

39.4

3

195.1

56.

3

8533

733

34

33.1

6.6

84

80.7

33.2

4

138.4

39.7

4

196.1

56.9

4

353.8

733

35

34.

6.9

85

81.7

33.4

5

139.4

40.

5

197.1

56.5

5

854.7

7a

36

35.

7.3

86

83.7

33.7

6

140.3

403

6

19a

56.8

6

855.7

733

37

86.

7.4

87

83.6

34.

7

141.3

40.5

7

199.

57.1

7

856.7

73.6

38

36.9

7.7

88

843

343

8

143.3

40.8

8

1993

57.3

8

857.6

733

39

37.9

8.

89

853

84.5

9

143.3

41.1

9

9993

573

9

85a6

74.1

30

38.8

83

00

863

343

150

1443

413

910

301.9

573

979

3593

744

31

39.8

8.5

91

673

35.1

1

145.3

41.6

1

3033

58.8

1

S693

74.7

33

30.8

as

98

88.4

35.4

3

146.1

41.9

3

303.8

584

3

8613

75.

33

31.7

9.1

93

89.4

35.6

3

147.1

43.3

3

394.7

58.7

3

8634

753

34

32.7

9.4

94

90.4

35.9

4

148.

43.4

4

305.7

59.

4

9634

753

35

333

9.G

95

913

36.3

5

149.

43.7

5

306.7

59.3

5

8643

753

36

34.6

9.9

96

933

36.5

6

159.

43.

6

3073

593

6

8653

76.1

37

3S.6

19.3

97

933

36.7

7

150.9

43.3

7

808.6

593

7

8663

764

38

36.5

10.5

98

943

37.

8

151.9

43.6

8

9093

60.1

8

3673

76.6

39

37.5

10.7

99

953

87.3

9

1533

43.8

9

9193

69.4

9

9683

783

40

38.5

11.

100

98.1

373

160

1533

44.1

990

3113

69.6

960

8693

773

41

39.4

11.3

1

97.1

37.8

1

154.8

44.4

1

8184

60.9

1

376.1

773

48

40.4

11.6

8

98.

38.1

3

155.7

44.7

3

8134

613

3

871.1

77.7

43

41.3

11.0

3

99.

38.4

3

158.7

44.9

3

814.4

61.5

3

879,

7a

44

43.3

13.1

4

100.

33.7

4

157.6

45.3

4

315.3

61.7

4

873.

783

45

43.3

13.4

5

100.9

38.9

5

158.6

453

5

316.3

63.

s

871

783

46

44.3

18.7

6

101.9

39.3

6

159.6

45.8

6

3173

63.3

6

8749

783

47

45.3

13.

7

103.9

89.5

7

1603

46.

7

3183

63.6

7

875.9

79.1

48

46.1

13.3

8

103.8

39.8

8

181.5

46.3

8

319.8

63.8

8

376.8

794

49

47.1

13.5

9

1043

30.

9

1683

46.6

9

399.1

63.1

9

3773

79.7

50

48.1

J3.8

110

105.7

30.3

190

163.4

463

980

991.1

634

990

siae

79.9

51

49.

14.1

1

106.7

30.6

1

164.4

47.1

1

833.1

63.7

1

879.7

803

53

50.

14.3

9

107.7

30.9

3

165.3

47.4

3

833.

63.9

3

980.7

803

53

50.9

14.6

3

1063

31.1

3

1663

47.7

3

834.

643

3

361.6

80.8

54

51.9

14.9

4

109.6

31.4

4

1673

48.

4

834.9

643

4

883.6

61.

55

53.9

15.9

5

110.5

31.7

5

168.3

48.3

5

835.9

643

5

883.6

613

56

53.8

15.4

6

1113

33.

6

169.8

48.5

6

836.9

65.1

8843

813

57

54.8

15.7

7

1135

333

7

170.1

48.8

7

837.8

653

7

385.5

813

58

55.8

16.

8

1114

33.5

8

171.1

49.1

8

898.8

65.6

8

88.1

59

56.7

16.3

9

114.4

33.8

9

173.1

49.3

9

839.7

65.9

9

8874

834

60

57.7

165

190

1154

33.1

180

173.

493

840

330.7

663

300

388.4

83.7

diflt.

dep.

tLlat

diflt.

dep.

d. tat.

diflt.

dep.

d. tat.

diet.

dep.

d. lat.

dlet.1 dep.

1. Iflt ,

DtiUMe, Departere and DUE Letitode.
Coarie W.

Digitized by VjOOQIC

TABLET.

OOUTM lf°.

Dirtance, DUE Latitude mad Departure.

127

ditt.

d. Ut.

dep.

dirt.

d. let.

dep.

diet.

d. let.

dep.

ditt.

d. let.

dep.

dirt.

d.lat.

dep.

1

1.

0.3

61

58.3

17.8

191

115.7

35.4

181

173.1

53.9

941

230.5

70.5

3

1.9

0.6

03

59.3

iai

8

110.7

35.7

3

174.

53.3

3

831.4

70.8

3

3.9

0.9

63

60.8

18.4

3

117.0

30.

3

175.

53.5

3

238.4

71.

4

S3

1.3

64

61.3

18.7

4

118.0

30.3

4

176.

53.8

4

333.3

71.3

5

4.8

1.5

65

63.3

19.

5

119.5

30.5

5

176.9

54.1

5

834.3

71.0

6

5.7

1.8

66

63.1

19.3

130.5

36.8

6

177.9

54.4

235.3

71.9

7

6.7

8.

67

64.1

19.0

7

121.5

37.1

7

178.8

54.7

7

230.2

73.3

8

7.7

8.3

68

65.

19.9

8

138.4

37.4

8

179.8

55.

8

237.3

73.5

9

8.6

8.0

69

66.

30.3

9

133.4

37.7

9

180.7

55.3

9

838.1

73.8

10

9.6

8.9

70

66.9

80.5

190

1845

38.

100

181-7

55.0

990

339.1

73.1

11

10.5

3.3

71

67.9

80.8

1

1853

38.3

1

188.7

55.8

1

340.

73.4

13

11.5

3.5

73

68.9

31.1

3

186.3

36.0

3

183.6

50.1

3

841.

73.7

13

13.4

3.8

73

69.8

31.3

3

137.3

38.9

3

184.6

56.4

3

841.9

74.

14

13.4

4.1

74

70.8

31.0

4

138.1

39.3

4

185.5

56.7

4

8439

74.3

15

14.3

4.4

75

71.7

31.9

5

139.1

39.5

5

186.5

57.

5

843.9

74.8

10

15.3

4.7

76

78.7

83.3

130.1

39.8

6

187.4

57.3

844.8

74.8

17

10.3

5.

77

73.0

83.5

7

131.

40.1

7

188.4

57.6

7

845.8

75.1

18

17.8

5.3

78

74.0

83.8

8

133.

40.3

8

189.3

57.9

8

840.7

75.4

19

18.3

5.0

79

75.5

83.1

9

138.9

40.0

9

190.3

58.3

9

847.7

75.7

90

19.1

5.8

80

76.5

83.4

140

133.9

40.9

aoo

191.3

58.5

960

848.6

76.

31

30.1

6.1

81

77.5

83.7

1

134.8

41.3

l

193.3

58.8

1

849.6

763

33

81.

6.4

83

78.4

84.

3

135.8

41.5

8

193.3

59.1

■3

850.6

7641

33

S3.

6.7

63

79.4

84.3

3

130.8

41.8

3

194.1

59.4

3

351.5

76.9

34

33.

7.

84

80.3

34.0

4

137.7

43.1

4

195.1

59.6

4

853.5

773

35

33.9

7.3

85

813

34.9

5

138.7

48.4

5

190.

59.9

5

353.4

77.5

36

34.9

7.0

86

83.3

35.1

6

J39.0

43.7

6

197.

60.3

354.4

773

37

35.8

7.9

87

835

35.4

7

140.0

43.

7

198.

60.5

7

255.3

78.1

88

30.8

8.3

88

84.3

35.7

8

141.5

43.3

8

198.9

60.8

8

256.3

78.4

39

37.7

6.5

89

85.1

80.

9

148.5

43.0

9

199.9

61.1

9

3573

783

30

88.7

as

00

86.1

80.3

150

143.4

43.9

910

300.8

61.4

970

8583

78.9

31

39.6

9.1

91

87.

80.0

1

144.4

44.1

1

301.8

61.7

1

359.2

793

33

30.6

9.4

98

88.

86.9

8

145.4

44,4

8

803.7

63.

8

260.1

79.5

33

31.6

9.0

93

88.9

87.8

3

140.3

44.7

3

803.7

63.3

3

301.1

79.8

34

33.5

9.9

94

89.9

87.5

4

147.3

45.

4

804.0

68.6

4

808.

80.1

35

33.5

105

95

90.8

87.8

5

148.3

45.3

5

305.0

68.9

5

263.

80.4

36

34.4

10.5

96

91.8

sai

6

149.3

45.0

6

806.6

63.8

6

363.9

80.7

37

35.4

108

97

93.8

88.4

7

150.1

45.9

7

207.5

63.4

7

264.9

81.

38

36.3

JM

98

93.7

88.7

8

151.1

46.3

6

208.5

63.7

8

805.9

81.3

39

37.3

11.4

99

94.7

88.9

9

158.1

40.5

9

809.4

64.

9

366.8

81.0

40

3P.3

11.7

100

95.0

89.9

160

153.

48.8

990

310.4

64.3

980

367.8

81.9

41

39.3

13.

1

96.0

39.5

1

154.

47.1

1

311.3

64.6

1

268.7

883

43

40.3

13.3

9

97.5

89.8

8

154.9

47 4

8

313.3

64.9

8

869.7

83.4

43

41.1

13.0

3

98.5

30.1

3

155.9

47.7

3

313.3

65.2

3

370.0

83.7

44

43.1

13.9

4

99.5

30.4

4

150.8

47.9

4

8145

65.5

4

371.0

83.

45

43.

13.3

5

100.4

80.7

5

157.8

48.3

5

215.3

658

5

372.5

83.3

46

44.

13.4

6

101.4

31.

158.7

48.5

6

316.1

66.1

273.5

83.0

47

44.9

13.7

7

103.3

31.3

7

159.7

48.8

7

217.1

66.4

7

274.5

83.9

48

45.9

14.

e

103.3

31.0

8

160.7

49.1

8

318.

66.7

8

275.4

843

49

46.9

14.3

9

1045

31.9

9

101.6

49.4

9

319.

67.

9

376.4

84.5

SO

47.8

14.0

110

105.3

38.9

170

163.6

49.7

990

330.

67.8

990

3773

643

51

48.8

14.9

1

100.1

33.5

1

163.5

50.

1

830.9

67.5

1

878.3

85.1

53

49.7

15.8

8

107.1

32.7

8

164.5

503

3

331.9

67.8

3

8793

85.4

53

50.7

15.5

3

108.1

33.

3

105.4

50.6

3

333.8

68.1

3

380.3

85.7

54

51.0

15.8

4

109.

33.3

4

160.4

50.9

4

223.8

68.4

4

881.3

80.

55

53.6

16.1

5

110.

33.0

5

167.4

51.3

5

234.7

68.7

5

282.1

80.3

56

53.6

10.4

6

110.9

33.9

6

168.3

51 .5

6

225.7

69.

383.1

88.5

57

54.5

10.7

7

111.9

34.9

7

169.3

51.7

7

226.0

69.3

7

284.

80.8

58

55.5

17.

8

113.8

34.5

8

170.9

53.

8

337.0

69.6

8

285.

871

59

56.4

17.3

9

113.8

34.8

9

171.8

53.3

9

328.0

69.9

9

285.9

87.4

60

57.4

17.5

130

114.8

35.1

180

173.1

53.6

340

239.5

70.9

300

880.9

87.7

dirt.

dep.

d. Ut.

dirt.

dep.

d. let

dirt.

dep.

d. let.

dirt.

dep.

<Llat.

dirt.

dep.

llat.

Dtetanee, Departure and Diff. Latitude.
Cvnivft 730.

Digitized by VjOOQIC

128

TABLE V.

COUTM ISO.

Distance, DUE Latitude mad Departure.

dirt.

d. lat.

dep.

dirt.

d. lat. dep.

dirt.

d. lat. dep.

dirt-

d. lat.

dep.

diet.

<LUt.

dep.

1

1.

0.3

61

58. 18.9

1*1

115.1 37.4

181

173.1

55.9

941

8993

744

2

1.9

0.6

63

59. 19.3

3

116. 37.7

3

173.1

563

8

3303

744

3

3.9

0.9

63

v 59.9 19 4

3

117. 38.

3

174.

56.6

3

331.1

75.1

4

34

1.3

64

60.9 19.8

4

1173 384

4

175.

56.9

4

8331

75.4

5

4.8

1.5

65

814 80.1

5

118.9 384

5

175.9

573

5

333.

75.7

6

5.7

,1.9

66

68.8 80.4

6

1194 383

6

1763

57.5

6

334.

76.

7

6.7

8.8

67

63.7 80.7

7

1304 393

7

177.8

57.8

7

3343

76.3

8

7.6

8.5

68

64.7 31.

8

131.7 39.6

8

178.8

581

8

3353

764

9

8.6

8.8

69

65.6 81.3

9

128.7 39.9

9

179.7

58.4

9

3364

763

10

9.5

3.1

70

664 31.6

180

1334 403

100

180.7

58.7

950

3374

774

11

10.5

3.4

71

67.5 81.9

1

134.6 40.5

1

181.7

59.

1

838.7

77.6

19

11.4

3.7

78

68.5 833

3

1354 404

3

188.6

594

8

839.7

773

13

13.4

4.

73

69.4 816

3

1364 41.1

3

183.6

594

3

840.6

783

14

134

4.3

74

70.4 83.9

4

187.4 41.4

4

184.5

593

4

8414

784

15

14.3

4.6

75

71.3 33.3

5

188.4 41.7

5

1854

603

5

8484

784

16

15.2

4.9

76

73.3 835

6

139.3 43.

6

186.4

60.6

6

8434

79J

17

16.3

5.3

77

73.8 33.8

7

1304 433

7

187.4

603

7

344.4

79.4

18

17.1

5.6

78

743 34.1

8

1313 426

8

188.3

61.8

8

345.4

70.7

19

18.1

5.9

79

75.1 84.4

9

1333 43.

9

1894

614

9

8464

80.

ao

19.

63

80

76,1 84.7

140

133.1 43.3

900

1903

614

960

8474

804

81

30.

6.5

81

77. 85.

1'

134.1 43.6

1

1913

63.1

1

9483

80.7

83

80.9

6.8

83

78. 85.3

3

135.1 43.9

3

193.1

63.4

8

8493

8L

83

81.9

7.1

83

78.9 85.6

3

136. 443

3

193.1

63.7

3

850.1

813

84

33.8

7.4

84

79.9 36.

4

137. 44.5

4

194.

63.

4

851.1

814

85

33.8

7.7

85

80.8 86.3

5

1373 44.8

5

195.

63.3

5

358.

813

86

34.7

8.

86

81.8 36.6

6

138.9 45.1

6

195.9

63.7

6

353.

823

37

35.7

8.3

87

83.7 86.9

7

1394 45.4

7

196.9

64.

7

9533

894

88

36.6

8.7

68

83.7 873

8

140.8 45.7

8

1974

644'

8

9543

894

89

27.6

9.

89

84.6 37.5

9

14L7 46.

9

1984

64.6

9

9554

83.1

30

984

9.3

00

85.6 87.8

150

148.7 46.4

910

199.7

649

970

9564

83.4

31

39.5

9.6

91

86.5 88.1

1

143.6 46.7

1

800.7

653

1

957.7

83.7

33

30.4

9.9

93

87.5 38.4

3

144.6 47.

3

8016

654

3

958.7

84.1

33

31.4

10.3

93

88.4 98.7

3

1454 474

3

803.6

65.8

3

9594

84.4

34

33.3

10.5

94

89.4 89.

4

146.5 47.6

4

8034

66.1

4

960.6

84.7

35

33.3

10.8

95

90.4 89.4

5

147.4 47.9

5

8044

66.4

5

9614

85.

36

34.3

11.1

98

91.3 89.7

6

148.4 483

6

805.4

66.7

6

9694

854

37

353

11.4

97

93,3 30.

7

149.3 484

7

806.4

67.1

7

963.4

854

38

36.1

11.7

98

93.3 30.3

8

150.3 48.9

8

807.3

67.4

8

964.4

853

39

37.1

13.1

99

943 30.6

9

1513 49.1

9

808.3

67.7

9

9654

863

40

38.

13.4

100

95.1 30.9

160

1533 49.4

9*0

8093

68.

980

9064

884

41

39.

13.7

1

98.1 31.3

1

153.1 49.8

1

8103

684

1

9073

864

43

39.9

13.

3

97. 314

3

154.1 50.1

3

311.1

684

3

9683

87.1

43

40.9

13.3

3

98. 314

3

155. 50.4

3

313.1

68.9

3

969.1

874

44

418

13.6

4

989 33.1

4

156. 50.7

4

813.

693

4

970.1

874

45

43.8

13.9

5

99.9 38.4

5

156.9 51.

5

314.

694

5

971.1

88.1

46

43.7

14.3

6

1004 33.8

6

157.9 514

6

814.9

69.8

6

979.

88.4

47

44.7

14.5

7

1014 33.1

7

1584 51.6

7

315.9

70.1

7

973.

88.7

48

45.7

14.8

8

103.7 33.4

8

1594 51.9

8

3168

70.5

8

973.9

89.

49

46.6

15.1

9

103.7 33.7

9

160.7 533

9

3174

704

9

8743

894

50

47.6

15.5

110

104.6 34.

110

161.7 534

930

818.7

71.1

900

9754

894

51

484

15.8

1

1054 34.3

1

1634 534

1

819.7

71.4

1

9704

893

53

49.5

16.1

3

1064 34.6

3

163.6 533

3

8804

71.7

3

977.7

963

53

50.4

16.4

3

107.5 34.9

3

1644 534

3

8314

73.

3

978.7

964

54

51.4

16.7

4

108.4 35.3

4

1654 534

4

9834

73.3

4

979.6

963

55

53.3

17.

5

109.4 35.5

5

166.4 54.1

5

8334

73.6

5

980.6

913

56

53.3

17.3

6

110.3 35.8

107.4 54.4

6

934.4

78.9

6

9814

914

57

54.3

17.6

7

U14 36.3

7

1684 54.7

7

835.4

733

7

9894

914

58

553

17.9

8

1183 36.5

8

1694 55.

8

836.4

734

8

989.4

03.1

59

56.1

183

9

1133 36.8

9

1703 554

9

8374

73.9

9

984.4

99.4

60

57.1

ias

130

114.1 37.1

180

1713 554

840

328.3

743

300

9854

•3.7

dirt.

dap.

d.lat.

dirt.

dep. d. lat .

dirt.

dep. d. lat.

dirt. | dep.

d. lat.

dirt, dep

d.lat

Distance, Departure and DUE Latitude.
Ceam Y&o.

Digitized by VjOOQIC

TABLS T.
Dilt latitude sod Denton.

129

diet.

lilt.

*P-

dist.

d. lat.

dap

dtat.

(Llat.

*P.

dilt.

d.lat.

dap.

dJst.

d. lat.

dap.

1

0.9

04

61

57.7

19.9

141

1144

39.4

181

171.1

58.9

941

2974

784

9

1.9

0.7

69

58.6

90.2

2

115.4

39.7

9

179.1

594

9

9984

784

3

94

1.

63

594

904

3

1164

40.

3

173.

59.6

3

299.8

79.1

4

3.8

1.3

64

604

90.8

4

117.9

404

4

174.

59.9

4

930.7

79.4

5

4.7

1.6

65

614

214

5

118.9

40.7

5

1744

604

5

931.7

79.8

6

5.7

9.

66

69.4

214

6

119.1

41.

6

175.9

60.6

6

9396

80.1

7

6.6

9.3

67

634

91.8

7

190.1

41.3

7

176.8

60.9

7

9334

80.4

8

74

9.6

68

644

99.1

8

191.

41.7

8

177.8

61.9

8

934.5

80.7

9

84

9.9

69

654

294

9

199.

A'L

9

178.7

614

9

935.4

81.1

10

9.5

3.3

70

66.9

994

130

199.9

494

100

179.6

61.9

950

936.4

81.4

11

10.4

3.6

71

67.1

93.1

1

193.9

424

1

1806

69.2

1

937.3

81.7

1*

11.3

8.9

79

68.1

93.4

2

1444

43.

2

1814

62.5

9

2384

89.

13

J '.'4

4.9

73

69.

93.8

3

1954

43.3

3

1894

024

3

239.2

89.4

14

134

4.6

74

70.

94.1

4

146.7

43.6

4

163.4

63.9

4

V404

89.7

15

14.9

4.9

75

70.9

24.4

5

1*7.6

44.

5

184.4

634

5

241.1

83.

16

15.1

54

76

71.9

94.7

6

1986

444

6

185.3

634

6

242.1

834

17

16.1

54

77

79.8

95.1

7

1994

44.6

7

186.3

64.1

7

943.

83.7

18

17.

5.9

78

73.8

25.4

8

1304

44.9

8

1874

644

6

943.9

84.

19

18.

64

79

74.7

25.7

9

1314

453

9

1884

64.8

9

244.0

844

90

ia»

64

80

75.6

26.

140

132.4

45.6

200

189.1

65.1

9«0

2454

844

*1

19.9

6.8

81

76.6

26.4

1

1334

45.9

1

190.

65.4

1

2464

85.

99

904

7.2

89

775

98.7

2

1344

469

2

191.

654

2

247.7

85.3

93

9 J .7

7.5

83

78.5

97.

3

1354

46.6

3

191.9

66.1

3

948.7

85.6

94

99.7

7.8

84

79.4

974

4

136.9

46.9

4

192.9

66.4

4

94D.6

8tt.

99

93.6

8.1

85

80.4

97.7

5

137.1

47.9

5

193.8

66.7

5

950.6

86.3

v8

•-•4.6

84

86

M.3

98.

6

138.

474

6

194.8

67.1

6

9514

66.6

97

954

8.8

87

8*4

98.3

7

139.

47.9

7

195.7

67.4

7

2594

864

98

96.5

9.1

88

83.9

98.7

6

139.9

48.9

8

196.7

67.7

8

953.4

67.3

99

97.4

9.4

89

844

29.

9

1404

464

9

197.6

68.

9

954.3

87.6

80

98.4

94

00

85.1

99.3

150

1414

484

9J10

198.6

68.4

•370

9553

87.9

31

90.3

10.1

91

86.

99.6

1

142.8

49 9

1

199.5

68.7

1

•-56.9

88.9

39

30.3

10.4

99

87.

30.

9

143.7

49.5

9

900.4

69.

2

9574

88.6

33

91.9

10.7

93

87.9

30.3

3

144.7

494

3

201.4

69.3

3

958.1

88.9

34

89.1

11.1

94

88.9

30.6

4

145.6

50.1

4

909.3

69.7

4

259.1

89.*

35

83.1

11.4

05

89.8

30.9

5

1464

504

5

903.3

70.

5

9C0.

894

36

34.

11.7

96

90.8

31.3

6

147.5

50.8

6

904.9

70.3

6

961.

89.9

37

35.

19.

97

91.7

31.6

7

148.4

51.1

7

9115.9

70.6

7

9614

90.9

38

35.9

19.4

98

92.7

31.9

8

149.4

51.4

8

906.1

71.

8

9194

904

39

36.9

19.7

99

93.6

39.9

9

1504

51.8

9

907.1

71.3

9

9634

904

40

37.8

13.

100

94.6

9?.6

160

1514

59.1

aao

908.

71.6

980

964.7

914

41

S*.8

13 3

1

95.5

39.9

1

159.9

59.4

i

909.

79.

1

965.7

91.5

49

39.7

13.7

9

96.4

339

9

1534

59.7

2

909.9

794

9

966.6

91.8

43

40.7

14.

3

97.4

33.5

3

1541

53.1

3

210.9

794

3

967.6

99.1

44

41.6

144

4

98.3

33.9

4

155.1

53.4

4

911.8

79.9

4

9684

994

45

49.5

14.7

5

994

34.9

5

156.

53.7

5

912.7

733

5

9694

99.8

46

43.5

15.

6

100.9

34.5

6

157.

54.

6

213.7

73.6

6

370.4

93.1

47

444

154

7

1019

34.8

7

157.9

54.4

7

214.6

73.9

7

971.4

93.4

48

45.4

15 6

8

109.1

354

8

158.8

54.7

8

915.6

74.9

8

9794

934

49

46.3

16.

9

103.1

354

9

159.8

55.

9

2164

74.6

9

273.3

94.1

50

473

164

110

104.

35.8

170

160.7

55.3

930

2174

74.9

900

974.9

94.4

51

48.9

16.6

1

105.

36.1

1

161.7

55.7

1

918.4

754

1

975.1

94.7

59

49.9

16.9

9

105.9

364

9

169.6

56.

9

219.4

75.5

9

976.1

951

53

50.1

17.3

3

1064

36.8

3

163.6

56.3

3

2904

75.9

3

977.

95.4

54

51.1

17.6

4

1074

37.1

4

1644

56.6

4

9914

764

4

978.

95.7

55

59.

17.9

5

108.7

37.4

5

165.5

57.

5

999.9

76.5

5

978.9

98.

56

59.9

18.9

6

109.7

37.8

6

166.4

57.3

6

923.1

76.8

6

979.9

96.4

57

53.9

18.6

7

110.6

38.1

7

167.4

57.6

7

2*4.1

77.9

7

9804

98.7

58

51.8

18.9

8

1114

38.4

8

168.3

58.

8

995.

77.5

8

981.8

97.

59

55.8

19.9

9

1194

38.7

9

169.9

58.3

9

998.

77.8

6

989.7

974

60

5J.7

19.5

190

113.5

39.1

180

170.9

58.6

940

996.9

78.1

300

983.7

97.7

dist.

iep.

d. lat

ditt.

dep.

d. lat.

diet.

dep.

d. lat.

dist.

dap.

d. lat.

diit.

dap-

d. lat.

Diatanee, Departure and Ditt Latitude.
Course 71°.

Digitized by VjOOQIC

130

TABUS T.

Canree 90©.
Diitaaee, DUE Latitude and Departure.

diet

d. lat.

*•*.

diet.

d. let-

dep.

diet.

d. lit. dep.

diet.

d. lat.

dep.

diet.

d.lat.

dep.

1

0.9

03

61

573

90.9

191

113.7 41.4

181

170.1

01.9

941

9963

83.4

9

1.9

0.7

09

583

91.3

9

114.6 41.7

3

171.

69.9

3

997.4

833

3

9.8

1.

03

59.3

913

3

115.6 49.1

3

179.

63.6

3

9883

83.1

4

33

1.4

64

00.1

91.9

4

1163 43.4

4

179.9

63.9

4

3493

833

5

4.7

1.7

65

61.1

93.3

5

117.5 48.8

5

1733

63.3

5

9393

833

6

5.0

9.1

66

09.

93.6

6

118.4 43.1

6

174.8

63.6

6

9313

84.1

7

0.6

9.4

67

63.

93.9

7

119.3 43.4

7

175.7

64.

7

339.1

843

8

7.5

9.7

68

63.9

33.3

8

130.3 43.8

8

176.7

64.3

8

333.

843

9

as

3.1

69

643

33.6

9

131.3 44.1

9

177.6

643

9

334.

853

10

9.4

3.4

70

65.8

93.9

130

1993 443

190

1W3

65.

950

9343

853

11

10.3

33

71

60.7

94.3

1

133.1 -44.8

1

1793

653

1

9353

853

19

113

4.1

79

67.7

94.6

3

194. 45.1

8

180.4

65.7

9

9303

863

13

134

4.4

73

68.6

35.

3

195. 453

3

181.4

66.

3

337.7

863

14

13.3

4.8

74

693

953

4

135.9 453

4

183-3

66.4

4

938.7

883

15

14.1

5.1

75

703

95.7

5

190.9 48.9

5

1833

66.7

5

9393

873

ie

15.

5.5

76

71.4

96.

6

137.8 463

1843

67.

6

3404

87.6

17

16.

5.8

77

73.4

36.3

7

138.7 46.9

7

185.1

67.4

7

3413

87.9

18

10.9

6.2

78

733

98.7

8

139.7 47.9

8

186.1

67.7

8

843.4

883

19

17.9

0.5

79

743

97.

9

1303 473

9

187.

68.1

9

343.4

883

ao

18.8

63

80

75.3

97.4

140

1313 47.9

900

1873

68.4

960

9443

883

91

19.7

7.9

81

76.1

97.7

1

1333 48.3

1

188.9

68.7

1

9453

893

99

90.7

73

89

77.1

98.

9

133.4 48.6

9

189.8

69.1

9

9463

893

93

91.0

7.9

83

78.

28.4

3

134.4 48.9

3

1903

69.4

3

947.1

90.

94

99.0

8.3

84

78.9

98.7

4

1353 493

4

191.7

09.8

4

948.1

903

95

333

6.0

85

79.9

99.1

5

1363 49.6

5

199.6

70.1

5

948,

903

90

94.4

8.9

86

80.8

99.4

6

1373 49.9

6

193.6

703

6

950.

91.

97

95.4

9.9

87

813

39.8

7

138.1 503

7

1943

703

7

9503

913

98

96.3

9.0

88

83.7

30.1

8

139.1 50.6

8

1953

711

8

J513

91.7

99

973

9.9

89

83.6

30.4

9

140. 51.

9

19M

713

9

9593

99,

30

383

103

00

84.6

303

150

141. 513

910

1973

713

91*

353.7

993

31

99.1

103

91

853

31.1

1

141.9 513

]

1983

733

1

354.7

93.7

33

30.1

10.9

99

86.5

31.5

9

1493 59.

3

1993

783

8

3553

98.

33

31.

11.3

93

87.4

31.8

3

143.8 53.3

3

9003

73.9

3

3563

93.4

34

31.9

113

94

883

39.1

4

144.7 53.7

4

901.1

733

4

9573

93.7

35

39.9

19.

95

89.3

39.5

5

145.7 53.

5

909.

733

5

958.4

94.1

36

33.8

19.3

96

90.3

33.8

6

146.6 53.4

6

903.

73.9

6

959.4

94.4

37

34.8

19.7

97

91.3

33.3

7

1473 53.7

7

*w>

743

7

9903

94.7

38

35.7

13.

98

93.1

333

8

1483 54.

8

at*

74.6

8

9613

95.1

39

363

13.3

99

93.

33.9

9

149.4 54.4

9

SB

T43

9

9893

95.4

40

37.6

13.7

100

94.

343

160

150.4 54.7

990

906,7

753

960

983.1

953

41

383

14.

1

94.9

343

1

1513 55.1

i

907.7

75.6

1

364.1

96J

49

393

14.4

9

95.8

34.9

9

153.3 55.4

8

306.6

753

S

365.

96.4

43

40.4

14.7

3

903

35.3

3

1533 55.7

3

3093

763

3

9653

983

44

413

15.

4

97.7

35.6

4

154.1 58.1

4

310.5

76.6

4

9083

97.1

45

493

15.4

5

98.7

35.9

5

155. 56.4

5

311.4

77.

5

9873

973

46

43.9

15.7

6

993

363

156. 563

6

813.4

773

6

9683

973

47

443

10.1

7

1003

36.6

7

156.9 57.1

7

813.3

77.6

7

969.7

983

S

45.1

10.4

8

1013

363

8

157.9 573

8

3143

78.

8

8703

983

46.

16.8

9

103.4

37.3

9

1583 573

9

3153

783

9

8713

983

80

47.

17.1

110

103.4

37.6

170

159.7 58.1

930

316.1

78.7

900

3793

993

51

47.9

17.4

1

1043

38.

1

160.7 58.5

1

917.1

79.

1

9733

993

59

48.9

173

9

1053

38.3

9

1613 583

9

318.

79.3

9

374.4

993

53

49.8

l&l

3

1063

38.6

3

109.6 59.9

3

3183

79.7

3

9753

1093

54

50.7

183

4

197.1

39.

4

1633 593

4

9193

80.

4

976.3

1093

55

51.7

163

5

108.1

39.3

5

164.4 593

5

990.8

80.4

5

9773

1093

56

59.6

103

6

109.

39.7

6

1654 60.9

6

991.8

80.7

6

378.1

1013

57

53.6

193

7,

1093

40.

7

1063 60.5

7

993.7

81.1

7

9794

1013

58

543

198

8'

1103

40.4

8

1673 60.9

8

9333

81.4

8

980.

101.9

59

514

90.9

9

1113

40.7

9

1683 613

9

9343

81.7

9

981.

1033

00
diit.

56.4

903

190

119.8

41.

180

109.1 616

940

3353

89.1

300

9813

1093

dej».

d. lat.

diet.

dep. d. lak

diat.

dep d .lat.

diet.

dep

d.lal.

diat.

dep.

diet.

Pittance, Departure and Diff Latitude.
Courae70°.

Digitized by VjOOQIC

TABLET.

C a* u r — 01°.
Diff Latitude and Departure.

131

dirt.

d.tat.

dep.

dirt.

d.tat.

dep.

dirt. 'd. tat.

dep,

dirt.

d-let.

dep,

dirt.

d.tat.

dep,l

1

04

0.4

01

56.9

914

1*1

na

434

181

169.

644

041

995.

88.4

9

1.9

0.7

69

574

934

3

113.9

43.7

9

169.9

654

9

9954

86.7

3

94

1.1

63

584

934

3

114.8

44.1

3

1704

65.6

3

9984

87.1

4

3.7

1.4

64

59.7

99.9

4

115.8

444

4

171.8

65.9

4

997.8

87.4

5

4.7 1

14

65

60.7

934

5

116.7

444

5

179.7

66.3

5

998.7

874

6

5.6

94

66

61.6

93.7

6

1174

454

6

173.6

66.7

6

999.7

884

7

64

94

67

694

94.

7

1184

454

7

174.6

67.

7

9304

884

6

74

9.9

68

634

944

8

1194

454

8

1754

674

8

9314

884

9

84

34

69

64.4

94.7

9

1904

464

9

176.4

67.7

9

9334

894

10

9.3

3.6

TO

65.4

95.1

130

1914

46.6

100

1774

68.1

000

333.4

894

11

10.3

34

71

664

95.4

1

1994

46.9

1

1784

684

1

9344

90.

18

114

4.3

79

674

95.8

3

1934

474

9

1794

684

8

9354

904

13

19.1

4.7

73

684

964

3

1944

47.7

3

1804

094

3

9964

90.7

14

13.1

5.

74

69.1

98.5

4

195.1

48.

4

181.1

694

4

937.1

91.

15

14.

5.4

75

70.

96.9

5

198.

484

5

189.

09.9

5

238.1

914

16

144

5.7

76

71.

974

6

1*7.

48.7

6

183.

704

6

939.

91.7

17

159

6.1

77

71.9

974

7

197.9

49.1

7

183.9

704

7

9394

99.1

18

164

64

78

794

98.

8

198.8

494

8

1844

71.

8

940.9

994

19

17.7

64

79

734

984

9

1994

494

9

1854

714

9

9414

99.8

00

1&7

74

80

74.7

98.7

140

130.7

504

000

186.7

71.7

OOO

849.7

934

91

194

74

81

75.6

99.

1

131.6

504

1

187.6

79.

1

943.7

934

93

904

74

89

76.6

994

3

139.6

50.9

9

188.6

794

9

944.6

934

93

914

84

83

774

39.7

3

1334

514

3

189.5

79.7

3

3454

944

94

99.4

84

84

78.4

30.1

4

1344

514

4

1904

73.1

4

9464

944

95

934

9.

85

794

304

5

135.4

59.

5

1914

734

5

8474

95.

96

944

94

86

80.3

304

6

1364

59.3

6

1994

73.8

6

9484

954

97

954

9.7

87

814

314

7

1374

59.7

7

1934

744

7

9494

95.7

98

96.1

10.

88

894

314

8

1384

53.

8

1944

744

8

9509

96.

99

97a

10.4

89

83.1

314

9

139.1

534

9

195.1

744

9

951.1

964

SO

9*

104

00

84.

394

160

140.

534

010

196.1

75.3

090

958.1

964

31

98.9

11.1

91

85.

39.6

1

141.

54.1

1

197.

75.6

1

95a

97.1

39

994

114

93

85.9

33.

9

1414

544

9

197.9

76.

9

9534

974

33

304

114

93

86.8

33.3

3

1494

54.8

3

198.9

764

3

954.9

974

34

31.7

18.9

94

874

33.7

4

1434

554

4

1994

76.7

4

855.8

984

35

39.7

194

95

88.7

34.

5

144.7

554

5

900.7

77.

5

956.7

98.6

38

334

19.9

98

894

344

6

145.6

554

6

901.7

77.4

6

857.7

984

37

344

134

97

904

344

7

1464

564

7

903.6

774

7

9584

994

SB

354

136

96

&

35.1

8

1474

56.6

8

9034

£i

8

9594

994

39

364

14.

99

364

9

1484

57.

9

9044

9

9604

100.

40

374

144

100

93.4

354

100

149.4

574

OOO

9054

784

080

9614

1004

41

394

14.7

1

94.3

384

1

1504

57.7

1

9064

79.3

I

9694

100.7

49

394

15.1

9

954

36.6

9

1514

58.1

9

9074

79.6

9

963.3

101.1

43

40.1

154

3

964

384

3

1594

584

3

908.9

79.9

3

9644

101.4

44

41.1

154

4

97.1

374

4

153.1

58.8

4

909.1

80.3

4

965.1

1014

45

49.

16.1

5

Oft

374

5

154.

59.1

5

910.1

806

5

966.1

109.1

46

49.9

164

6

99.

38.

6

155.

59.5

6

911.

81.

6

987.

1094

47

434

16.8

7

99.9

384

7

1554

594

7

911.9

814

7

9074

1094

48

444

174

8

1004

38.7

8

156.8

604

8

9194

81.7

8

9684

1034

49

45.7

17.6

9

1014

39.1

9

1574

60.6

9

813.8

83.1

9

9094

103.6

50

46.7

17.9

110

109.7

394

170

158.7

694

000

314.7

834

000

970.7

1034

51

47.6

183

1

103.6

39.8

1

159.6

614

1

915.7

894

1

971.7

104.3

59

484

18.6

9

104.6

40.1

9

160.6

61.6

3

916.6

83.1

9

9794

1044

53

494

19.

3

1054

404

3

1614

69.

3

9174

834

3

9734

105.

54

50.4

19.4

4

106.4

40.9

4

169.4

694

4

9184

63.9

4

9744

1054

55

514

19.7

5

107.4

414

5

1634

69.7

5

9194

844

5

9754

105.7

56

594

90.1

6

1084

414

6

1644

63.1

6

9904

846

6

9764

106.1

57

534

904

7

1094

41.9

7

1654

634

7

991.3

844

7

9774

1064

58

54.1

904

8

1104

49.3

8

1664

634

8

9334

854

8

9784

1064

59

56.1

91.1

9

111.1

494

9

167.1

64.1

9

993.1

85.6

9

879.1

1074

69

56.

914

190

119.

43.

180

168.

644

940

994.1

86.

300

980.1

1074

JdUt.

dep.

d. kit.

(dirt.

Idep.

d. tat.

dirt. 1 dep.

d.tat.

dirt.

dep,

d-tat.

dirt.

dep.

d. tat.

Mrtaaoe, Departure end Dift Latitude.

Digitized by VjOOQIC

132

TAB1E T.

DMilM, DUE Latitat* and Departure.

dirt.

d. lat.

dep.

dirt.

d. tat

dep.

dirt

d.lat.

dap.

dirt.

d. lat. dep.

dirt.

(Llat.

de*.

1

0.9

04

61

56.6

889

15*1

1135

454

181

1675 675

941

9834

905

S

1.9

0.7

69

574

83.8

3

113.1

45.7

9

168.7 685

8

884.4

89.7

3

88

1.1

63

58.4

83.6

9

114.

46.1

3

169.7 686

3

3855

91.

4

3.7

1.5

64

59.3

84.

4

115.

465

4

1704 085

4

8895

915

5

4.6

1.9

65

00.3

845

5

115.9

48.8

5

1715 69.3

5

8875

914

6

5.6

8.3

69

61.3

84.7

6

1165

47.8

6

1794 69.7

6

398.1

985

7

6.5

86

67

681

85.1

7

1175

47.6

7

173,4 70.1

7

389.

885

8

7.4

3.

68

63.

854

8

118.7

47.9

8

174.3 70.4

6

8894

985

ft

85

3.4

69

04.

955

9

1194

48.3

9

175.8 105

9

3395

835

10

9.3

9.7

70

64.9

965

130

1305

48.7

190

1765 715

990

9915

9X7

11

195

4.1

71

65.8

86.6

1

1314

49.1

1

177.1 715

1

9987

94.

IS

11.1

45

73

664

97.

3

1384

49.4

3

13fc 715

8

933.7

94,4

13

181

4.9

73

67.7

87.3

3

133.3

495

3

178.9 78.3

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DiC Latitat aadDepartaie.

138

dirt.

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dep

dUt

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9324) 94.6

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Digitized by VjOOQIC

134

TABUS T.

Canrm 9J4P.
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d. lat.

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1

0.9

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55.7 243

191

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2483

1103

33

30.1

134

93

85. 37.8

3

139.8

693

3

1943

86.6

3

9494

111.

34

31.1

133

94

85.9 383

4

140.7

62.6

4

1953

87.

4

9593

1114

39

32.

14.2

05

86.8 38.6

5

141.6

63.

5

196.4

874

5

2513

1113

36

33.9

143

98

87.7 39.

6

1423

633

6

1973

873

6

252.1

1123

37

33.8

15.

97

88.6 393

7

143.4

63.9

7

1983

883

7

253.1

113.7

39

34.7

153

98

893 39.9

8

144.3

643

8

1993

88.7

8

254.

113.1

39

354

153

99

90.4 40.3

9

1453

64.7

9

9003

89.1

9

3543

1133

40

36.5

163

100

914 40.7

160

1463

65.1

aao

201.

893

980

2553

1133

•41

373

16.7

1

92.3 41.1

1

147.1

653

i

201.9

89.9

1

256.7

1143

49

38.4

17.1

2

933 413

2

148.

65.9

2

202.8

903

3

257.6

114.7

43

39.3

173

3

94^ 41.9

3

148.9

663

3

203.7

90.7

3

3583

1154

44

40.9

17.9

4

•«r«

4

1493

66.7

4

2043

91.1

4

3594

1153

45

41.1

183

5

5

150.7

67.1

5

2053

913

5

9894

1153

46

49,

18.7

6.

963 43.1

6

151.6

673

6

9063

913

6

2913

1163

47

49.9

19.1

7

97.7 433

7

152.6

67.9

7

2074

923

7

2623

116.7

48

43.9

193

e

98.7 433

8

1533

683

8

2083

92.7

8

363.1

117.1

49

443

.193

9

993 443

9

1544

08.7

9

2093

93.1

9

264.

1173

90

45.7

203

110

1003 44.7

170

1553

69.1

93)0

910.1

933

900

264.9

na

51

46.6

20.7

1

101.4 45.1

1

156.9

69.6

1

211.

94.

1

9653

na4

52

47.5

21.2

2

1023 45.6

2

157.1

70.

2

211.9

944

2

2663

1183

53

46.4

21.6

3

1033 46.

3

158,

70.4

3

912.9

94.8

3

367.7

119.9

54

49.3

22.

4

104.1 46.4

4

159.

703

4

2133

953

4

3683

1193

55

50.9

224

5

105.1 463

5

159.9

713

5

214.7

95.6

5

3893

120.

56

312

22.8

6

106. 473

6

160.8

713

6

215.6

96.

6

8794

129.4

57

52.1

23.2

7

106.9 473

7

161.7

72.

7

2163

964

7

2713

1903

58

53.

233

8

1073 48.

8

162.6

724

8

2174

983

8

3733

1913

59

53.9

24.

9

108.7 4*4

9

163.5

723

9

2183

97.2

9

273.2

1213

60

543

24.4

120

1093 483

180

164.4

73.2

240

2193

973

300

974,1

122.

dirt.

dep.

d. lat.

dirt

dep. d. lat

dirt.

dep.

4.1at.

dirt.

dep. d. lat.

dirt.

dep. d. lat.

Dirtanee, Departure and DiH Latitude.

Digitized by VjOOQIC

135

DUE Latitude and Departure.

diet.

d.lat.

ddp.

dirt.

d.lat.

dep.

diet

d.lat.

dep.

dirt

d. lat.

dep.

dirt-

d. lat.

dep.

1

9.9

0:4

61

55.3

85.8

191

109.7

51.1

181

164.

765

941

818.4

101.9

9

1.8

0.8

63

503

96.3

8

110.6

51.6

8

1644

76.9

3

3194

103 3

3

8.7

14

63

57.1

96.6

3

111.5

58.

3

165.9

774

3

930.3

108.7

4

3.6

1.7

64

58.

97.

4

113.4

53.4

4

166.8

77.8

4

331.1

103.1

5

44

8.1

65

58.9

975

5

1134

53.8

5

167.7

78.3

5

333.

103.5

6

5.4

85

66

59.8

97.9

6

1143

53.3

6

1684

78.6

6

333.

104.

7

6.3

3.

67

60.7

98.3

7

115.1

53.7

7

1695

79.

7

333.9

104.4

6

7.3

3.4

68

61.6

98.7

8

116.

54.1

8

170.4

795

8

3344

104.8

9

83

3.8

69

63.5

993

9

116.9

545

9

171.3

79.9

9

335.7

105,8

10

9.1

4.3

70

63.4

99.6

130

1174

544

100

1733

804

900

936.6

105.7

11

10.

4.6

71

644

30.

1

11&7

55.4

1

173.1

80.7

1

2*7.5

106.1

19

10.9

5.1

79

65.3

30.4

3

119.6

55.8

9

174.

81.1

3

338.4

1065

13

11.8

55

73

66.3

30.9

3

1805

56.3

3

174.9

81.6

3

3394

108.9

14

18.7

5.9

74

67.1

314

4

131.4

56.6

4

1754

88-

4

3393

107.3

15

13.6

6.3

75

68.

31.7

5

133.4

57.1

5

176.7

83.4

5

331.1

1074

16

144

6.8

76

68.9

33.1

6

133.3

57.5

6

177.6

834

6

333.

108.3

17

15.4

73

77

69.8

33.5

7

184.3

57.9

7

1785

83.3

7

333.9

1084

18

16.3

7.6

78

70.7

33.

8

135.1

58.3

8

179.4

83.7

8

3334

109.

19

17.3

a

79

71.6

33.4

9

136.

58.7

9

180.4

84.1

9

834.7

1094

90

18.1

&5

80

734

33.8

140

1964

593

aoo

1814

845

900

8354

1094

SI

19.

8.9

81

73.4

34.3

1

1374

59.6

i

183.9

84.9

1

9364

1104

S3

19.9

9.3

88

744

34.7

8

138.7

69.

3

183.1

85.4

3

9374

110.7

S3

80.8

9.7

83

75.3

35.1

3

1394

69.4

3

184.

85.8

3

338.4

1U.I

84

31.8

10.1

84

76.1

354

4

130.5

60.9

4

184.9

86.8

4

8394

1114

89

82.7

10.6

85

77.

35.9

5

131.4

614

5

185.8

86.6

5

8403

118.

88

33.6

11.

86

77:9

364

6

133.3

61.7

6

186.7

87.1

6

841.1

118.4

87

84.5

11.4

87

78.8

36.8

7

133.8

63.1

7

187.6

874

7

843.

1184

84

85.4

11.8

88

79.8

373

8

134.1

634

8

188.5

87.9

8

3434

113.3

89

86.3

134

89

80.7

37.6

9

135.

63.

9

189.4

884

9

3434

113.7

30

873

13.7

00

81.6

38.

150

1354

63.4

910

1904

88.7

970

944.7

114.1

31

38.1

13.1

91

885

384

1

139.9

63.8

1

1913

893

1

94S4

1144

38

89.

13.5

93

834

38.9

9

137.8

64.3

3

193.1

894

a

9464

115.

33

394

13.9

93

84.3

394

3

138.7

64.7

3

193.

90.

3

947.4

115.4

34

304

14.4

94

85.3

39.7

4

139.6

65.1

4

193.9

90.4

4

948.3

1158

35

31.7

14.8

95

86.1

40.1

5

140.5

65.5

5

194.9

90.9

5

9493

1163

36

33.6

15.3

98

87.

40.6

6

141.4

65.9

6

195.8

914

6

950.1

1164

37

334

15.0

97

87.9

41.

7

1434

66.4

7

196.7

91.7

7

951.

117.1

38

34.4

16.1

98

88.8

41.4

8

1433

66.8

8

197.6

93.1

8

953.

1174

39

354

165

99

89.7

414

9

144.1

673

9

1984

934

9

953.9

1174

40

364

16.9

100

90.6

494

100

145.

67.6

990

199.4

93.

980

353.8

118.3

41

37.3

17.3

1

91.5

48.7

1

145.9

68.

i

3004

93.4

1

854.7

i&

48

38.1

17.7

3

93.4

43.1

3

146.8

685

9

3013

934

3

955.6

43

39.

18.3

3

934

43.5

3

147.7

68.9

3

803.1

94.9

3

956.5

1194

44

39.9

18.6

4

944

44.

4

148.8

69.3

4

303.

94.7

4

957.4

180.

45

404

19.

5

95.3

44.4

5

1494

69.7

5

303.9

95.1

5

9584

1804

46

41.7

19.4

6

96.1

44.8

6

150.4

70.3

6

994.8

955

6

9593

1804

47

43.6

19.9

7

97.

45.3

7

151.4

70.6

7

905.7

95.9

7

980.1

1814

48

43.5

30,3

8

97.9
98.8

45.6

8

153.3

71.

8

306.6

96.4

8

961.

181.7

49

44.4

80.7

9

46.1

9

1533

71.4

9

307.5

964

9

9614

133.1

50

45.3

81.1

110

99.7

465

170

154.1

714

930

3084

97.3

900

9694

1394

51

463

81.6

1

100.6

46.9

1

155.

73.3

1

309.4

974

1

963.7

183.

53

47.1

88.

3

101.5

47.3

3

155.9

78.7

3

310.3

98.

8

964.6

133.4

53

48.

33.4

3

103.4

474

3

156.8

73.1

3

3113

984

3

9654

183.8

54

484

83.8

4

103.3

48.8

4

157.7

73.5

4

313.1

98.9

4

9664

194.3

55

49.8

83.3

5

104.3

48.6

5

158.6

74.

5

313.

994

5

967.4

184.7

56

50.8

93.7

6

105.1

49.

6

1595

74.4

6

3134

99.7

6

9684

185.1

57

51.7

34.1

7

106.

49.4

7

160.4

74.8

7

3144

100.3

7

9693

1954

58

53.6

84.5

8

106.9

49.9

8

161.3

75.3

8

315 7

1004

8

970.1

195.9

59

53.5

849

9

107.9

50.3

9

163.3

75.6

9

3166

101.

9

971.

1964

60

54.4

«4

180

108.8

50.7

180

163.1

76.1

840

817.5

101.4

300

971.9

1864

din.

dep.

d.lat. J

diit.

dep. d.lat.

dirt.

dep. d. lat

dirt.

dep. d. lat.

diet.

dep. d. lat.

Distance, Departure and Dili Latitude.
Cowraa 05°.

Digitized by VjGOQIC

136

TAHIS V.
<fe*rn»900.

INaUnea, DUE Latitude and Departure,

dirt.

d.lat.

dap.

dirt.

d.lat.

dap.

dirt.

d. lat.

dap.

disc

d. lat.

dap.

dirt.

d.lat.

1
dap.

1

0.9

0.4

61

54.8

86.7

iai

108.8

53.

181

168.7

795

3*1

816.6

195.6

3

1.8

0.9

62

55.7

87.2

3

109.7

53.5

3

1635

79.8

*

8175

10b. i

3

8.7

1.3

63

56.6

27.6

8

110.6

53.9

3

1645

805

3

21H.4

1065

4

8.6

1.8

64

57.5

88.1

4

111.5

54.4

4

165.4

80.7

4

219.3

»07.

5

4.5

8*3

65

58.4

28.5

5

112.3

548

5

106.3

81.1

5

2905

107.4

6

5.4

8.6

66

59.3

38.9

6

113.9

55.3

6

1675

815

6

291.1

107^

7

6.3

3.1

67

605

39.4

7

114.1

55.7

7

168.1

83.

7

222

1085

8

7.9

3.5

68

61.1

SJ.8

8

U5.

56.1

8

169.

83.4

8

292.9

1087

9

ai

3.9

69

68.

305

9

115.9

565

9

169.9

83.9

9

3235

104.2

10

9.

4.4

90

68.9

30.7

130

1165

57.

100

1705

835

950

324.7

109.6

11

9.9

4.8

71

63.9

3J.1

1

117.7

574

1

171.7

83.7

1

395.6

119.

IS

10.8

5.3

78

64.7

31.6

9

1185

57.9

9

1785

84.9

9

8365

1105

13

11.7

5.7

73

65.6

38.

3

1195

58.3

3

1735

84.6

3

897.4

110»

14

13.6

6.1

74

66.5

39.4

4

120.4

58.7

4

174.4

85.

4

8985

1115

15

13.5

6.6

75

674

38-9

5

1315

59.2

5

1755

855

5

395

1115

16

14.4

7.

76

685

33.3

6

1335

53.6

6

176.3

65.9

6

830.1

1125

17

15.3

7.5

77

695

33.8

7

133.1

60.1

7

177.1

86.4

7

831.

1127

18

19.3

7.9

78

70.1

343

8

134.

60.5

8

17&

86.8

8

8315

113.1

19

17.1

&3

79

71.

34.6

9

1345

60.9

9

1785

875

9

8385

1135

ao

18.

8.8

80

715

35.1

140

1355

614

900

1795

87.7

360

833.7

114.

si

18.9

9.9

81

735

35.5

1

196,7

61.8

1

180.7

88.1

1

834.6

114.4

S3

19.8

9.6

89

73.7

35.9

3

S3

685

3

1815

88.6

8

8355

114.9

33

30.7

10.1

83

745

36.4

3

68.7

3

1885

89.

3

336.4

1155

84

3141

19.5

84

755

36.8

4

139.4

63.1

4

183.4

89.4

4

8375

115.7

S

88.5

11.

85

76.4

37.3

5

1305

63.6

5

1845

80.9

5

3385

1165

83,4

11.4

86

77.3

37.7

6

1315

64.

6

185.3

905

6

839.1

116.6

87

843

11.8

87

785

38.1

7

133.1

64.4

7

186.1

90.7

7

340.

117.

88

854

19.3

88

79.1

38.6

6

133.

64.9

8

1865

915

8

8409

1175

89

86.1

18.7

89

80.

39.

9

133.9

65.3

9

1875

91.6

9

8415

1175

80

87.

135

00

80.9

395

100

134.8

655

919

188.7

99,1

3T0

343.7

11M

31

87.9

iae

91

815

39.9

1

135.7

66.9

1

189.6

935

1

843.6

iias

38

33.6

14.

93

83.7

49.3

3

136,6

66.6

3

190.5

93.9

3

9445

1195

33

9J.7

14.5

93

83.6

40.8

3

1375

67.1

3

191.4

93.4

3

845.4

119.7

34

30.6

14.9

94

84.5

41.2

4

138.4

67.5

4

193.3

93.8

4

8465

180.1

35

31.5

153

95

85.4

416

5

139.3

67.9

5

193.9

945

5

8475

1895

36

33.4

15.8

96

86.3

491

6

140.8

68.4

6

194.1

94.7

6

848.1

191.

37

33.3

16.3

97

875

48.5

7

141.1

68.8

7

195.

95.1

7

849.

191.4

38

34.9

16.7

98

88.1

43.

8

143.

69.3

8

195.9

95.6

8

849.9

181.9

39

35.1

17.1

99

89.

43.4

9

1435

69.7

9

1985

96.

9

8595

1895

40

36.

17.5

100

89.9

435

100

1435

70.1

330

197.7

96.4

380

351.7

183.7

41

36.9

18.

1

905

44.3

1

144.7

70.6

1

198.6

96.9

1

858.6

1335

43

37.7

18.4

3

91.7

44.7

3

145.6

71-

9

199.5

975

9

8535

1335

43

3B.6

18.8

3

98.6

455

3

1485

71.3

3

300.4

975

3

854.4

194.1

44

39.5

19.3

4

93.5

45.6

4

147.4

71.9

4

301.3

98.9

4

8555

1845

45

40.4

19.7

5

94.4

46.

5

148.3

72.3

5

902.2

98.6

5

2565

1845

46

41.3

80.3

6

955

465

6

149.3

72.8

6

303.1

99.1

6

857.1

185,4

47

49.3

90.6

7

96.3

46.9

7

150.1

73.9

7

304.

995

7

958.

1855

48

43.1

81.

8

97.1

47.3

8

151.

736

8

304.9

99.9

8

858.9

1865

49

44.

Slid

9

98.

47.8

9

1515

T4.1

9

305.8

100.4

9

8595

196.7

50

44.9

81.9

110

98.9

485

19B

1595

745

330

806.7

1005

300

969.7

187.1

51

45.8

93.4

1

995

48.7

1

139.7

75.

1

3075

1015

1

9615

1875

58

46.7

83.8

9

100.7

49.1

9

1545

754

3

8085

101.7

9

968-4

18a

53

47.6

83.3

3

101.6

495

3

1555

75.8

3

809.4

109.1

3

9635

198.4

54

48.5

83.7

4

103.5

50.

4

159.4

76.3

4

910.3

103.6

4

8643

1885

55

49.4

84.1

5

103.4

50.4

5

157.3

76.7

5

911.9

103.

5

865.1

1995

59

50.3

945

6

104.3

50.9

6

158.9

77.9

6

918.1

1035

6

38$.

1*5

57

51.9

85.

7

105.9

51.3

7

159.1

77.8

7

313.

103.9

7

886.9

1395

58

59.1

85.4

8

106.1

51.7

8

160.

78.

8

3119

1045

8

8675

1395

59

53.

85.9

9

107.

53.3

9

160.9

785

9

914.8

1045

9

888.7

131.1

60

53.9

96.3

190

107.9

58.6

180

161.8

78.9

840

dirt.

915.7

105.3

300

8695

1315

1 dUt. dap.

d. lat. I dirt.

dap.

d. lat.

dirt.

dap.

d.lat.

dep.

d. lat. 1 dirt.

dap.

d.lat.

Dtrtanoa, Dtpartora and DUE Latitoda.

OOP.

Digitized by VjOOQIC

TABLE V.

Course »T°.
DUE Latitude tod Departure.

18?

dirt.

d. tat.

dap.

dirt.

d. lat

dep

dirt.

d. lat. dep.

dirt.

d.lat.

dep.

dist.

d. lat

dep.

1

0.9

0.5

61

54.4

97.7

131

1073 545

181

161.3

82.2

341

314.7

109.4

9

1.8

0.9

69

55.3

28.1

2

106.7 55.4

2

162-2

82.6

2

315.6

1093

3

9.7

1.4

63

56.1

98.6

3

109.6 55.8

3

163.1

83.1

3

2163

1103

4

3.6

L8

64

57.

29.1

4

1103 56.3

4

163.9

83.5

4

217.4

1105

5

4.5

9.3

65

57.9

29.5

5

111.4 56.7

5

1645

84.

5

2183

1115

6

5.3

9.7

66

58.8

30.

6

1193 575

6

165.7

84.4

6

2193

111.7

7

6.9

3.2

67

59.7

30.4

7

1135 57.7

7

1666

84.9

7

220.1

112.1

8

7.1

3.6

68

60.6

30.9

8

114. 58.1

8

1673

85.4

8

221.

112.6

9

&

4.1

69

6L5

313

9

114.9 '58.6

9

168.4

853

9

2815

113.

io

ao

44

70

694

313

130

1153 59.

1O0

1693

863

950

9295

1133

11

9.8

5.

71

633

39.2

1

116.7 593

1

170.9

86.7

1

223.6

114.

19

10.7

5.4

79

64.9

33.7

3

117.6 59.9

2

171.1

873

2

2945

114.4

13

11.6

5.9

73

65.

33.1

3

1183 60.4

3

179.

87.6

3

225.4

1145

14

19.5

6.4

74

65.9

33.6

4

119.4 603

4

1725

88.1

4

2263

1153

15

13.4

65

75

66.8

34.

5

1903 613

5

173.7

885

5

9273

1155

16

14.3

7.3

76

67.7

344

6

1915 61.7

6

174.6

89.

6

228.1

1165

17

151

7.7

77

68.6

35.

7

192.1 62.2

7

1753

89.4

7

299.

116.7

18

16.

85

78

69.5

35.4

8

123. 62.7

8

176.4

89.9

8

999.9

117.1

19

16.9

84

79

70.4

35.9

9

193.8 63.1

9

1773

903

9

2305

1174

ao

173

9.1

80

713

36.3

140

194.7 63.6

800

1785

903

960

231.7

118.

91

tt.7

9.5

81

79.9

365

1

195.6 64.

1

179.1

91.3

1

9395

1185

99

19.6

10.

83

73.1

375

3

1963 643

3

180.

91.7

3

933.4

1185

93

90.5

10.4

83

74.

37.7

3

197.4 64.9

3

180.9

995

3

9343

1194

94

21.4

10.9

84

74.8

38.1

4

128.3 654

4

1815

925

4

235.2

119.9

95

99.3

115

85

75.7

38.6

5

199.9 65.8

5

182.7

93J

5

936.1

1903

96

235

11.8

86

76.6

39.

6

130.1 66.3

6

1833

935

6

937.

1905

97

94.1

123

87

77 .5

394

7

131. 66.7

7

184.4

94.

7

9373

191.9

98

94.9

12.7

68

78.4

40.

8

131.9 675

8

1853

94.4

8

238.8

121.7

99

95.8

13.2

89

793

40.4

9

1395 67.6

9

186.3

94.9

9

239.7

199.1

30

96.7

13.6

00

80S

40.9

150

133.7 68.1

910

187.1

953

970

940.6

1224

31

976

14.1

91

81.1

41.3

1

1343 68.6

1

188.

955

1

941.5

193.

39

98.5

145

99

82.

41.8

3

135.4 69.

3

188.9

965

2

949.4

1933

33

99.4

15.

93

82.9

49.9

3

1363 693

3

1895

96.7

3

2435

1935

34

30.3

15.4

94

833

42.7

4

137.3 69.9

4

1P0.7

975

4

244.1

194.4

35

31.2

15.9

95

84.6

43.1

5

138.1 70.4

5

191.6

97.6

5

245.

1245

36

33.1

16.3

96

854

43.6

6

139. 70.8

6

1P93

98.1

6

2455

1953

37

33.

16.8

97

86.4

44.

7

139.9 713

7

193.3

985

7

2465

195.8

38

33.9

17.3

98

87.3

44.5

8

140.8 71.7

8

1945

99.

8

247.7

196.2

39

34:7

17.7

99

88.9

44.9

9

141.7 725

9

195.1

99.4

9

248.6

196.7

40

35.6

18.9

100

80.1

45.4

160

142.6 72.6

930

196.

99.9

980

2495

197.1

41

36.5

18.6

1

90.

45.9

1

1433 731

1

196.9

1003

I

2504

m.6

43

37.4

19.1

3

90.9

463

3

144.3 733

3

1975

100.8

8

25>3

198.

43

38.3

19.5

3

91.8

463

3

1455 74.

3

19&7

1013

3

9595

1985

44

39.2

90.

4

92.7

475

4

146.1 743

4

199.6

101.7

4

953.

1984

45

40.1

90.4

5

93.6

47.7

5

147. 74.9

5

2003

103.1

5

953.9

199.4

46

41.

90.9

6

94.4

48.1

6

147.9 75.4

6

201.4

102.6

6

2545

1993
1303

47

41.9

91.3

7

95.3

48.6

7

1483 753

7

2023

103.1

7

255.7

48

49.8

91.8

8

96.9

49.

8

149.7 76.3

8

203.1

1033

8

258.6

130.7

49

43.7

225

9

97.1

49.5

9

150.6 76.7

9

204.

104.

9

257.5

1313

00

444

99.7

110

98.

495

170

1513 775

930

2043

104.4

900

258.4

131.7

51

45.4

93.2

1

98.9

50.4

1

152.4 773

1

205.8

1045

1

2593

132.1

59

465

93.6

3

99.8

50.8

3

153.3 78.1

3

206.7

1053

2

2603

1324

53

47.9

94.1

3

100.7

51.3

3

154.1 783

3

207.6

105.8

3

261.1

133.

54

48.1

945

4

101.6

513

4

155. 79.

4

2085

1063

4

262.

133.5

55

49.

95.

5

1025

59.2

5

155.9 79.4

5

209.4

106.7

5

2623

133.9

56

49.9

95.4

6

103.4

52.7

6

1565 79.9

6

210.3

107.1

6

263.7

134.4

57

50.8

95.9

7

104.9

53.1

7

157.7 80.4

7

2115

1076

7

964.6

1345

58

51.7

983

8

105.1

53.6

8

158.6 80.8

8

212.1

10a

8

9655

1353

59

59.6

96.8

9

106.

54.

9

159.5 81.3

9

213.

1083

9

966.4

135.7

60

535

97.2

120

106.9

543

180

160.4 81.7

240

2135

109.

300

9873

1363

dirt.

dep.

d. lat.

dirt.

dep.

d. lat.

dirt. I dep. d. lat.

dist.

dep.

d. lat.

dirt.

dep.

d. lat.

Distance, Departure and Diff Latitude.
Course 63°.

Digitized by VjOOQIC

138

TABLE T.
Wetanee, Biff Latitade and Depnrtara.

diet.

d. tat.

dep

ditt.

d. lat. dap.' ditt.

d.lat

dep.

disc.

d.lat

dep.

diet.

d.lat

**

1

9.9

04

01

53.9 88.6

191

1084

56.8

181

1594

85.

3941

9134

113.1

9

1.8

0.9

69

54.7 S9.1

2

107.7

574

3

M0.7

85.4

3

313.7

1134

3

9.6

1.4

63

55.6 89.6

3

108.6

57.7

3

161.6

85.9

9

3144

114.1

4

3.5

1.9

64

56.5 30.

4

1094

584

4

1624

86.4

4

215.4

1144

5

4.4

24

65

57.4 30.5

5

110.4

58.7

5

1634

864

5

2164

115.

e

6.3

9.8

66

58.3 31.

6

1114

594

6

164.3

87.3

6

2174

1154

?

64

34

67

599 31.5

7

1191

59.6

7

165.1

87.8

7

218.1

116.

8

7.1

34

68

69. 31.9

8

.113.

60.1

8

166.

88.3

8

219.

1164

9

7.9

44

69

60.9 33.4

9

1134

60.6

9

1664

88.7

9

2194

1194

10

84

4.7

TO

614 39.9

130

1144

61.

100

1674

894

059

299.7

117.4

11

9.7

54

71

69.7 334

1

115.7

614

1

1684

89.7

1

8314

1174

n

10.6

5.6

79

634 334

2

116.5

68.

9

1094

90.1

3

2894

1104

13

1M

6.1

73

644 344

3

117.4

69.4

3

170.4

994

3

2S&4

1184

14

19.4

64

74

654 34.7

4

1184

68.9

4

1714

91.1

4

3944

1194

15

134

7.

75

664 354

5

1194

63.4

5

178.3

914

5

3394

119.7

16

14.1

74

76

67.1 35.7

6

120.1

634

6

173.1

93.

6

899.

1294

17

15.

8.

77

68. 36.1

7

121.

643

7

1734

984

7

8964

139v7

18

15.9

84

78

094 36.6

8

1214

644

8

1744

93.

8

8974

191.1

19

16.8

84

79

604 37.1

9

188.7

654

9

175.7

93.4

9

828.7

1314

00

17.7

9.4

80

794 37.6

140

1934

65.7

9300

1764

994

009

8394

133.1

91

18.5

9.9

81

714 38.

1

134.5

664

1

1774

94.4

1

830.4

1394

»

19.4

10.3

89

73.4 384

2

195.4

66.7

9

178.4

944

9

3314

133.

S3

80.3

10.8

83

734 39.

3

1964

67.1

3

1794

954

3

8394

1334

94

21.9

11.3

84

744 39.4

4

197.1

67.6

4

180.1

95.8

4

833.1

1934

95

99.1

11.7

85

75.1 39.9

5

188.

68.1

5

181.

96.-2

5

834.

1344

99

93.

124

86

75.9 40.4

6

138.9

684

6

181.9

96.7

6

3344

1944

97

93.8

12.7

87

76.8 40.8

7

129.8

69.

7

1824

97.9

7

8317

1354

98

94.7

13.1

88

77.7 414

. 8

130.7

694

8

183.7

97.7

8

8364

1354

99

95.6

13.6

89

784 414

9

131.6

70.

9

1844

981

9

3374

1904

30

96.5

14.1

00

794 494

150

138.4

70.4

SJlO

185.4

98.6

9JT0

33M

1304

31

97.4

14.6

91

804 42.7

1

133.3

70.9

1

1864

99.1

1

8394

1374

39

283

15.

99

814 434

9

1344

71.4

9

1874

994

9

3494

127.7

33

99.1

154

93

89.1 43.7

3

135.1

714

3

188.1

109.

3

341.

1394

91

39.

16.

94

83. 44.1

4

136.

784

4

189.

100.5

4

3414

1394

39

30.9

16.4

95

83.9 44.6

5

1364

794

5

1894

100.9

5

3494

1394

39

31.8

16.9

98

844 45.1

6

137.7

734

6

190.7

101.4

6

843,7

1394

97

39.7

17.4

97

85.6 454

7

138.6

73.7

7

191.6

101.9

7

8444

139.

38

33.6

17.8

99

864 46.

8

1394

74.2

8

1934

1034

8

3454

1394

39

34.4

184

99

87.4 464

9

140.4

74.6

9

193.4

1038

9

8464

131.

40

35.3

184

100

884 464

100

1414

73.1

asjo

1944

1034

009

3474

1314

41

36.9

194

1

894 47.4

1

148.3

75.6

i

195.1

1034

1

348.1

1314

49

37.1

19.7

9

90.1 474

9

143.

76.1

8

196.

1044

9

349.

139.4

43

3a

90.9

3

90.9 48.4

3

143.9

764

3

196.9

104.7

3

3494

1334

44

38.8

90.7

4

91.8 484

4

1444

77.

4

1974

1054

4

8594

1334

45

39.7

91.1

5

92.7 4»4

5

145.7

77.5

5

198.7

105.6

5

8514

1334

49

40.6

91.6

6

93.6 494

6

146.6

774

6

199.5

106.1

6

8584

1344

47

41.5

99.1

7

944 509

7

1474

78.4

7

900.4

106.6

7

853.4

134.7

48

49.4

224

8

95.4 50.7

8

1484

78.9

8

201.3

107.

8

8544

1354

49

434

S3.

9

964 514

9

149.9

794

9

209.9

1074

9

2554

135.7

50

44.1

334

110

97.1 51.6

190

159.1

794

9330

903.1

168.

000

859.1

139.1

51

45.

93.9

1

98. 52.1

1

151.

804

1

904.

108.4

1

9564

1364

59

45.9

944

9

984 52.6

2

151.9

80.7

3

9044

108.9

9

8574

137.1

53

46.8

24.9

3

99.9 53.1

3

159.7

81.9

3

995.7

169.4

3

85B.7

1374

54

47.7

254

4

100.7 534

4

153.6

81.7

4

806.6

10fr.9

4

3594

139.

55

48.6

354

5

1014 54.

5

1544

884

5

8074

110.3

5

9604

1394

56

49.4

964

6

163.4 544

6

155.4

82.6

6

808.4

1104

6

361.4

139.

57

50.3

864

7

1034 54.9

7

156.3

83.1

7

909.3

1114

7

369.9

130.4

58

519

974

8

1644 55.4

8

1574

83.6

8

210.1

1117

8

869.1

1394

99

59.1

97.7

9

105.1 554

9

158.

84.

9

911.

1194

9

864.

149.4

60

53.

98.2

190

106. 564

180

158.9

844

340

911.9

112.7

399

994.9

1494

dirt.

dap.

<Llat.

din.

dep. d lat.

ditt.

dep.

d. lat.

diet.

dep.

d. lat.

diet.

dep.

lied

Distance, Departure and Diff. Latitude.
Couree 09)°.

Digitized by VjOOQIC

TABLE T.

OoureeOO?.

Dirtance, Diff Latttede and Departare.

139

din.

d.lat

dep.

dirt.

d.lat dep. 1 dlir.

d. lat dap.

diet.

d. lat. dep.

diet.

d. lat. dep.

1

0.9

93

61

93.4 99.6

101

M5.8 98.7

181

158.3 873

041

9103 1103

9

1.7

].

69

543 30.1

9

106.7 59.1

9

1593 88.9

9

911.7 1173

3

9.6

13

63

55.1 303

3

107.6 59.6

3

169.1 88.7

3

9193 1173

4

33

1.9

64

56. 31.

4

1083 60.1

4

169.9 693

4

913.4 1183

5

4.4

9.4

65

56.9 313

5

169.3 60.6

5

1613 85K7

5

9143 1183

6

53

9.9

66

57.7 39.

6

1103 61.1

6

169.7 903

6

9153 1193

7

6.1

3.4

67

59.6 393

7

111.1 61.6

7

163.6 90.7

7

916. 1197

8

7.

3.9

68

595 33.

8

119. 69.1

8

164.4 91.1

8

916.9 190.9

9

7.9

4.4

69

60.3 333

9

1193 993

9

1693 91.6

9

9173 190.7

16

8.7

4.8

TO

613 33.9

ISO

113.7 63.

1O0

1663 99.1

060

918.7 1913

11

9.6

5.3

71

69.1 34.4

1

114.6 633

1

167.1 99,6

1

9193 191.7

19

10.9

5.8

79

63. 34.9

9

115.4 64.

9

1673 93.1

9

990.4 1993

13

114

6.3

73

63.8 35.4

3

116.3 645

3

168.8 93.6

3

9313 199.7

14

19.9

6.8

74

647 35.9

4

117.9 65.
118.1 65.4

4

169.7 94.1

4

999.3 193.1

IS

111

7.3

75

65.6 36.4

5

5

170.6 94.5

5

993. 1933

14.

7.8

76

663 36.8

6

118.9 65.9

6

171.4 95.

6

993.9 1941

17

14.9

83

77

673 373

7

1193 66.4

7

1793 953

7

994.8 194.6

18

15.7

R7

IS

683 373

8

190.7 66.9

8

1733 96.

8

995.7 195.1

id

16.6

9.9

79

69.1 38.3

9

191.6 67.4

9

174. 963

9

0183 195.6

00

17.5

9.7

60

79. 38.8

140

199.4 67.9

900

1743 97.

060.

937.4 1961

91

18.4

10.9

81

70.8 39.3

1

193.3 68.4

1

175.8 97.4

1

9983 1963

99

19.9

10.7

89

71.7 39.8

9

194.9 68.8

9

176.7 97.9

9

9993 197.

93

90.1

113

83

79.6 403

3

195.1 09.3

9

1773 98.4

3

930. J973

94

91.

113

84

733 40.7

4

195.9 69.8

4

178.4 983

4

930.9 198.

99

91.9

19.1

85

74.3 413

5

196.8 70.3

5

1793 99.4

5

9313 1983

98

99.7

19.6

86

733 417

6

197.7 70.8

6

1803 99.9

6

939.6 199.

97

93.6

13.1

87

76.1 49.9

7

193.6 713

7

181. 100.4

7

9333 199.4

98

943

13.6

88

77. 49.7

8

199.4 713

8

1813 1003

8

934.4 199.9

99

99.4

14.1

80

77.8 43.1

9

1303 793

9

1893 1013

9

935.3 130.4

30

963

14.5

00

78.7 43.6

160

1313 79.7

010

183.7 1013

090

9311 130.9

31

97.1

15.

91

79.6 44.1

1

133.1 73.9

1

184.5 1093

1

937. 131.4

39

93.

153

99

803 44.6

9

139.9 73.7

9

185.4 1093

9

9373 1313

33

98.9

16.

93

813 45.1

3

133.8 743

3

189.3 103.3

3

938.8 139.4

34

99.7

163

94

893 45.6

4

134.7 74.7

4

1873 103.7

4

930.6 1333

35

30.6

17.

99

83.1 46.1

5

135.6 75.1

5

188. 1043

5

9403 1333

36

313

173

96

84. 463

6

136.4 75.6

1883 104.7

6

941.4 1333

37

32.4

173

97

843 47.

7

1373 76.1

7

1893 1053

7

9493 134.3

38

333

18.4

98

85.7 473

8

138.9 76.6

8

190.7 105.7

8

943.1 1343

33

34.1

18.9

99

86.6 48.

9

139.1 77.1

9

1913 1063

9

944. 1353

40

35.

19.4

100

873 483

160

1393 77.6

9*0

199.4 106.7

060

344.9 135.7

41

39.9

19.9

1

883 49.

1

1403 781

1

193.3 107.1

1

9453 1363

49

36.7

30.4

9

893 49.5

9

141.7 783

9

1943 107.6

9

946.6 136.7

43

37.6

90.8

3

90.1 49.9

3

149.6 79.

3

195. 108.1

3

9473 1373

44

384

913

4

91. 50.4

4

143.4 793

4

195.9 103.6

4

948.4 137.7

49

39.4

91.8

5

913 30.9

5

1443 60.

5

1963 109.1

5

9493 1383

46

403

933

6

99.7 51.4

6

145.9 803

6

197.7 109.6

6

950.1 138.7

47

41.1

993

7

93.6 51.9

7

146.1 81.

7

198.5 1J0.1

7

951. 139.1

48

42.

93.3

8

943 59.4

8

146.9 81.4

8

199.4 1103

8

9513 1303

49

49.9

933

9

953 59.8

9

1473 813

9

900.3 111.

9

9593 140.1

40

43.7

943

110

963 53.3

lit)

148.7 89.4

030

9013 1113

ooo

953.6 140.6

31

443

94.7

1

97.1 93.8

1

1493 89.9

1

909. 119.

1

9943 141.1

99

493

95.9

9

96. 943

9

150.4 83.4

9

909.9 ,1193

9

955.4 1413

S3

4&4

95.7

3

983 54.8

3

1513 83.9

3

903.8 113.

3

9563 149.

54

473

98.9

4

99.7 55.3

4

1593 84.4

4

9047 113.4

4

957.1 1493

99

48.1

98.7

5

1093 55.8

5

153.1 843

5

905.5 113.9

5

953. 143.

96

49.

97.1

6

1013 563

6

153.9 85.3

6

906.4 114.4

9

958.9 1433

97

493

97.6

7

1093 96.7

7

1543 85.8

7

907.3 1143

7

9593 144.

98

50.7

98.1

8

1033 57.9

8

155.7 86.3

8

908.9 115.4

8

9616 1443

99

51.6

983

9

104.1 97.7

9

198.6 86.8

3

909. 115.9

9

9613 145.

60

99.9

99.1

190

109. 583

180

157.4 873

940

9093 116.4

300

969.4 1453

di«.

«•*

d.lat

dirt.

dep. d. lat.

dM.

dep. d. lat.

dirt.

dep. d.lat.

dirt.

dep. d. lat

IMrtanee, Departure and Diff Latitude.
i Olo.

Digitized by VjOOQIC

140

TABLE V.

Distance, DUE Latitude and Departure.

diet.

d. let.

dep.

diet.

d.lat.

dep.

dirt.

d.lat.

dep.

dtat.

d.lat.

dep.

dirt.

d.lat.

dtp.

1

0.9

04

61

58.8

304

191

1043

604

181

1563

904

Ml

908.7

1304

S

1.7

1.

69

53.7

31.

9

105.7

61.

3

1576

91.

9

909.6

131.

3

3.6

14

63

54.6

314

3

1064

61.5

3

1584

914

3

810.4

1314

4

34

8.

64

55.4

33.

4

107.4

8*.

4

159.3

98.

4

3113

133.

5

4.3

84

65

56.3

334

5

108.3

634

5

100.3

684

5

313.3

1834

6

5.9

3.

66

57.8

33.

6

109.1

63.

6

161.1

93.

6

313.

193.

7

6.1

3.5

67

58.

33.5

7

110.

63.5

7

161.9

934

7

8133

1834

8

6.0

4.

68

58.9

34.

8

110.9

64.

8

168.8

94.

8

3143

134.

9

7.8

4.5

69

593

344

9

111.7

64.5

9

163.7

944

9

3153

1345

10

8.7

5.

90

60.6

35.

130

118.6

65.

190

1644

95.

950

9164

185.

11

9.5

54

71

614

354

1

na4

65.5

1

165.4

954

1

917.4

1854

IS

10.4

6.

78

62.4

36.

9

114.3

66.

8

1663

98.

8

9183

196.

13

11.3

64

73

63.3

364

3

1153

664

3

167.1

964

3

919.1

J964

14

13.1

7.

74

64.1

37.

4

116.

67.

4

168.

97.

4

980.

197.

15

13.

74

75

65.

374

5

1163

67.5

5

168.9

974

5

8803

1974

16

13.9

a

76

653

3ft.

6

1173

68.

6

169.7

98.

6

831.7

198.

17

14.7

84

77

66.7

384

7

118.6

684

7

170.6

984

7

983.6

1984

18

15.6

0.

78

674

39.

8

1194

69.

8

1714

99.

8

883.4

199.

19

16.5

94

79

68.4

394

9

190.4

694

9

178.3

994

9

8943

1994

90

173

10.

80

693

40.

140

131.3

70.

900

1733

100.

960

8953

139.

SI

183

10.5

81

70.1

404

1

188.1

704

1

174.1

1004

1

336.

1394

S3

10.1

11.-

83

71.

41.

3

183.

71.

9

174.9

101.

9

8863

131.

83

19.9

11.5

83

71.9

414

3

1833

714

3

175.8

1014

3

8873

1314

34

20.8

13

84

78.7

48.

4

184.7

72.

4

176.7

108.

4

8283

133.

85

31.7

13.5

65

73.6

43.5

5

125.6

784

5

1774

1684

5

2294

1394

96

33.5

13.

86

744

43.

6

186.4

73.

6

178.4

103.

6

930.4

133.

87

33.4

134

87

75.3

434

7

1273

73.5

7

1793

1034

7

8313

1334

SB

84.8

14.

88

70.8

44.

8

1883

74.

8

180.1

104.

8

838.1

134.

89

35.1

144

89

77.1

444

9

189.

744

9

181.

1044

9

833.

1344

30

36.

15.

00

77.9

78.8

45.

150

189.9

75.

910

181.9

105.

970

3333

135.

31

363

154

91

45.5

1

130.8

755

1

188.7

1054

1

834.7

1354

33

87.7

16.

93

79.7

46.

3

131.6

76.

9

183.6

108.

9

8353

136.

33

88.6

164

93

804

464

3

1384

764

3

1844

1064

3

236.4

1364

34

30.4

17.

94

81.4

47.

4

133.4

77.

4

185.3

107.

4

2373

137.

35

30.3

174

95

833

474

5

134.3

774

5

186.8

1074

3

2383

1374

36

31.3

18.

96

83.1

48.

6

135.1

78.

6

187.1

108.

6

839.

138.

37

38.

184

97

84.

484

7

136

78.5

7

187.9

1084

7

839.9

1384

38

33.9

19.

98

84.9

49.

8

1363

79.

6

188.8

109.

8

8403

139.

38

3X8

194

99

85.7

494

9

137.7

794

9

189.7

1094

9

841.6

1394

40

34.6

90.

109

86.6

50.

100

138.6

89.

990

1904

110.

980

9484

14a

41

354

804

1

874

504

1

139.4

804

i

191.4

1104

1

843.4

1404

48

36.4

81.

9

883

51.

9

140.3

81.

3

198.3

111.

9

8443

141.

43 1

37.9

814

3

89.9

514

3

1413

814

3

193.1

1114

3

845.1

1414

44

38.1

93.

4

90.1

58.

4

148.

89.

4

194.

118.

4

846,

149.

45

39.

884

5

90.9

594

5

148.9

894

5

194.9

1124

5

9463

1484

46

39.8

93.

6

913

53.

6

143.8

83.

6

105.7

113.

6

947.7

143.

47

40.7

834

7

98.7

53.5

7

144.6

«B4

7

196.6

1134

7

8484

1434

48

416

84.

8

934

54.

8

1454

84.

8

1974

114.

8

849.4

144.

49

48.4

84.5

9

94.4

544

9

146.4

844

8

198.3

1144

9

850.3

1444

50

433

85.

110

95.3

55.

199

1473

85.

930

1993

115.

900

951.1

145.

51

44.3

854

1

96.1

555

1

148.1

854

1

900.1

1154

1

859.

1454

58

45.

86.

9

97.

56.

9

149.

86.

9

900.9

116.

9

359.9

149.

53

45.9

964

3

97.9

56.5

3

1493

864

3

8013

1164

3

853.7

1464

54

463

87.

4

98.7

57.

4

150.7

87.

4

808.6

117.

4

8543

147.

55

47.6

974

5

90.6

574

5

151.6

874

5

8094

1174

5

8554

1474

56

484

88.

6

1004

58.

6

158.4

88.

6

204.4

118.

6

856.3

148.

57

49.4

884

7

1013

584

7

1533

884

7

8053

1184

7

8573

14*4

58

50.3

89.

8

109,8

59.

8

1543

89.

8

906.1

119.

8

958.1

149.

59

51.1

894

9

103.1

594

9

155.

89.5

6

907.

1194

9

858.9

1494

60

53.

30.

180

103.9

69.

180

155.9

90.

840

8073

190.

300

9593

159.

diet

dep.

d.lat.

ditt.

dep. d. lat.

dirt.

dep.

d .lat.

•diet.

dep.

d.lat

dirt.

dep.

d.lat.

Urtanee, Departure and Diff Latitude.
C*wee60°.

Digitized by VjOOQIC

T/BLS V.

141

Coarse 310.

Distance, INC Latitude and Departure.

dirt.

d. Ut.

dep.

dirt

d. lat. dep.

dirt.

i. lit. dep.

dirt.

1 lat. dep.

dirt

d. lat. dep.

1

0.9

0.5

61

58.3 31.4

131

103.7 63.3

181

155.1 934

5141

806.6 134.1

3

1.7

L

63

53.1 31.9

3

104.6 684

3

156. 93.7

3

307.4 184.6

• 3

8.6

1.5

63

54. 33.4

3

105.4 63 3

3

156.9 944

3

3084 135.2

4

3.4

3.1

64

54.9 33.

4

106.3 63.9

4

157.7 94.8

4

809.1 135.7

5

4.3

3.6

65

55.7 334

5

107.1 04.4

5

158.6 95.3

5

310. 1864

6

5.1

3.1

66

56.6 34.

6

108. 64.9

6

159.4 954

6

310.9 186.7

7

6.

3.6

67

57.4 344

7

108.9 65.4

7

1604 96.3

7

311.7 137.3

8

6.9

4.1

68

58.3 35.

8

109.7 65.9

8

161.1 96.8

8

313.6 187.7

9

7.7

4.6

69

59.1 35.5

9

110.6 66.4

9

163. 97.3

9

813.4 1384

10

8.6

5.3

70

60. 36.1

130

111.4 67.

100

168.9 97.9

950

314.3 188.8

11

9.4

5.7

71

60.9 36.6

1

1134 67.5

1

163.7 98.4

1

815.1 1394

IS

10.3

6.3

73

617 37.1

3

113.1 68.

3

1644 98.9

3

316. 139.8

13

11.1

6.7

73

63.6 37.6

3

114. 684

3

165.4 99.4

3

316.9 130.3

14

13.

7.8

74

63.4 38.1

4

114.9 69.

4

166.3 99.9

4

817.7 130.8

15

13.9

7.7

75

644 38.6

5

115.7 694

5

167.1 100.4

5

318.6 131.3

16

13.7

8.8

76

65.1 39.1

6

116.6 70.

6

168. 100.9

6

819.4 1314

17

14.6

8.8

77

66. 39.7

7

117.4 70.6

7

168.9 101.5

7

830.3 132.4

18

154

9.3

78

66.9 40.3

8

1184 71.1

8

169.7 108.

8

331.1 138.9

19

164

94

79

67.7 40.7

9

119.1 71.6

9

170.6 103.5

9

888. 133.4

90

17.1

10.3

80

68.6 41.3

140

180. 78.1

900

171.4 103.

860

833.9 133.9

31

18.

10.8

81

69.4 41.7

1

1.K>.9 786

1

1784 103.5

1

883.7 134.4

28

18.9

114

83

70.3 48.3

3

181.7 73.1

3

173.1 104.

3

884.6 134.9

33

19.7

114

83

71.1 48.7

3

183.6 73.7

3

174. 104.6

3

835.4 1354

34

30.6

18.4

84

73. 43.3

4

183.4 74.3

4

174.9 105.1

4

836.3 136.

35

31.4

18.9

85

78.9 434

5

1344 74.7

5

175.7 105.6

5

887.1 1364

38

83.3

13.4

86

73.7 44.3

6

135.1 75.3

6

176.6 108.1

6

838. 137.

37

33.1

13.9

87

74.6 44.8

7

186. 75.7

7

177.4 106.6

7

888.9 1375

38

34.

14.4

88

75.4 45.3

8

136.9 76.3

8

1784 107.1

8

839.7 138.

39

34.9

14-9

89

764 45.8

9

187.7 76.7

9

179.1 107.6

9

830.6 1384

30

35.7

154

00

77.1 46.4

ISO

138.6 77.3

810

180. 108.8

970

331.4 139.1

31

36.6

16.

91

78. 46.9

1

139.4 774

1

180.9 108.7

1

3334 139.6

33

374

16.5

93

78.9 47.4

3

130.3 78.3

3

181.7 109.3

3

333.1 140.1

33

38.3

17.

93

79.7 47.9

3

131.1 78.8

3

188.6 109.7

3

334. 140.6

34

39.1

17.5

94

80.6 48.4

4

138. 794

4

183.4 1104

4

834.9 1411

35

30.

18.

95

81.4 48.9

5

138.9 79.8

5

1844 110.7

5

835.7 141.6

36

30.9

184

98

884 49.4

6

133.7 80.3

6

185.1 111.9

6

836.6 148.3

37

31.7

19.1

97

83.1 50.

7

134.6 80.9

7

186. 111.8

7

337.4 143.7

38

33.6

19.6

98

84. 504

8

135.4 81.4

8

186.9 113.3

8

338.3 143.3

39

33.4

801

99

84.9 51.

9

136.3 81.9

9

187.7 1184

9

339.1 143.7

40

34.3

30.6

100

85.7 51.5

100

137.1 88.4

990

188.6 1134

380

340. 1444

41

35.1

31.1

1

86.6 58.

1

138. 88.9

1

189.4 113.8

I

340.9 144.7

43

36.

31.6

9

87.4 58.5

3

138.9 83.4

3

1904 1144

3

341.7 145.3

43

38.9

88.1

3

88.3 53.

3

139.7 84.

3

191.1 1144

3

343.6 145.8

44

37.7

88.7

4

89.1 53.6

4

140.6 84.5

4

193. 115.4

4

343.4 146.3

45

38.6

33.3

5

90. 54.1

5

141.4 85.

5

193,9 115.9

5

3444 1464

48

39.4

23.7

6

90.9 54.6

6

1434 854

6

193.7 116.4

6

345.1 147.3

47

40.3

34.3

7

91.7 55.1

7

143.1 86.

7

194.6 116.9

7

346. 1474

48

41.1

34.7

8

98.6 55.6

8

144. 864

8

195.4 117.4

8

346.9 1484

49

43.

85.3

9

93.4 56.1

9

1444 67.

9

198.3 117.9

9

347.7 1484

50

43.9

854

110

944 56.7

170

145.7 67.6

8330

197.1 1184

300

348.0 149.4

51

43.7

86.3

1

95.1 574

1

146.6 88.1

1

19a 119.

1

349.4 149.9

53

44.6

33.8

3

98. 57.7

3

147.4 88.6

9

J98.9 1194

3

3503 150.4

53

45.4

87.3

3

96.9 58.8

3

1484 89.1

3

199.7 130.

3

351.3 150.9

54

46.3

87.8

4

97.7 58.7

4

149.1 89.6

4

800.6 180.5

4

358. 151.4

55

47.1

38.3

5

98.6 59.3

5

150. 90.1

5

801.4 131.

5

3584 151.9

56

48.

38.8

6

99.4 59.7

6

150.9 90.6

6

3084 1814

6

853.7 158.5

57

48.9

89.4

7

1004 604

7

151.7 91.3

7

303.1 138.1

7

3544 153.

58

49.7

39.9

8

101.1 60.8

8

158.6 917

8

304. 188.6

8

355.4 1534

59

50.6

30.4

9

103. 614

9

153.4 984

9

304.9 183.1

9

3564 154.

60

51.4

30.9

130

103.9 614

180

1544 98.7

340

305.7 133.6

300

357.1 1544

dirt.

dep.

<LUt.

dirt.

dep. d.lat

dirt.

dep. d.Ut.

dirt

dep. d. lat.

dirt.

dep. d.lat.

Dirtanee, Departure and Diff Latitude.
Courts 500.

Digitized by VjOOQIC

142

TABLE V.

Cearee 39«.

Distance, Dtff Latitude and Departure.

dist.

d. lat.

dep.

dist.

d. lat. dep.

dial

d. lat.

dep.

diet.

d.lat.

**

diat.

d.lat.

de*

1

0.8

0-5

01

51.7 3*3

131

1084

64.1

181

1534

053

341

994-4

197.7

9

1.7

1.1

09

53.6 33.9

3

1034

64.7

9

1544

96.4

9

3053

1993

3

8.5

1.6

63

53.4 33.4

3

1044

65.9

3

1553

97.

3

908.1

1384

4

3.4

8.1

64

54.3 33.9

4

105*9

65.7

4

156.

974

4

9063

1904

5

4.3

3.6

65

55.1 34.4

5

106.

66.9

5

1563

08.

5

9074

1904

5.1

33

66

50. 35.

6

106.9

66.8

6

157.7

084

9084

130.4

7

5.9

3.7

67

56.8 354

7

107.7

674

7

1584

89.1

7

9094

1333

8

6.8

4.3

68

57.7 38.

8

1084

67.8

8

159.4

89.6

8

9104

131.4

9

7.6

4.8

69

564 36.6

S

109.4

68.4

9

1604

1003

•

9113

1313

10

84

54

70

59.4 37.1

130

1103

683

100

161.1

100.7

3M

919.

1334

11

9.3

5.8

71

60.* 376

1

111.1

69.4

1

Mi.

1013

1

9193

133.

1*

10.3

6.4

73

61.1 38.3

9

111.9

69.9

3

1684

101.7

9

914.7

1334

13

11.

6.9

73

61.9 38.7

3

1194

704

a

163.7

1034

9

9144

134.1

14

11.9

7.4

74

63.8 39.3

4

1134

71.

4

1644

1094

4

3154

1344

15

1*7

7.9

75

63.6 39.7

5

1144

714

5

165.4

1034

5

9164

13S-1

16

13.6

(15

76

64.5 40.3

6

1154

79.1

6

166.9

103.9

6

917.1

133.7

17

14.4

9.

77

65.3 40.8

7

116.9

734

7

167.1

104.4

7

9173

1363

18

15.3

•4

78

66.1 413

8

117.

73.1

8

1673

104.9

8

9184

136.7

19

16.1

10.1

79

67. 41.9

9

1173

73.7

9

1684

1054

•

3194

1373

90

17.

10.6

80

67.8 43.4

140

118.7

74.9

8JO0

1694

106.

300

9904

1373

n

17.8

11.1

81

68.7 49.9

1

119.6

74.7

1

1704

1064

1

«U

1384

93

m

11.7

83

694 43.5

9

180.4

75.9

3

1714

107.

9

29*3

1384

S3

19.5

13.3

83

70.4 44.

3

1914

754

3

173.3

1074

3

333.

1394

94

39.4

13.7

84

715 444

4

183.1

764

4

173.

106.1

4

3933

1393

35

31.9

13.3

85

79.1 45.

5

183.

764

5

1734

108.6

5

9*4.7

19M

46

83.

13.8

88

73.9 45.6

6

1934

77.4

6

174.7

1093

6

9954

141.

87

3*9

14.3

87

73.8 46.1

7

134.7

773

7

1754

109.7

7

896.4

1414

S8

83.7

14.8

88

74.6 46.6

6

1354

78.4

8

176.4

1103

8

3374

149.

89

•.'4.6

15.4

89

754 47.9

1*6.4

79.

1773

1104

338.1

1494

30

85.4

15.9

00

763 47.7

190

1873

794

910

1784

1114

970

399.

143.1

31

36.3

16.4

91

77.3 48.3

1

128.1

80.

1

1783

11L8

1

8994

1434

39

37.1

17.

93

78. 48.8

9

188.0

804

8

1794

1194

9

890.7

144.1

33

38.

174

93

78.9 49.3

3

1894

81.1

3

1806

1193

3

8314

144.7

34

88.9

18.

94

79.7 49.8

4

130.6

61.6

4

1814

11X4

4

8394

1453

35

8i).7

18.5

05

80.6 50.3

5

131.4

83.1

5

183.3

113.9

5

8333

145.7

36

30.5

19.1

96

61.4 50.9

6

133.3

83.7

6

1833

1144

6

834.1

1444

37

31.4

19.6

97

8*3 51.4

7

133.1

83.3

7

184.

115.

7

9343

146*

38

33.3

80.1

98

83.1 51.9

6

134.

83.7

8

1843

1154

8

9354

1474

39

33.1

80.7

90

84. 534

9

1344

84.3

185.7

118.1

9364

1474

40

333

91.3

1O0

84.8 53.

100

135.7

84.8

3930

1864

116.6

330

9374

148.4

41

34.8

81.7

1

85.7 534

1

1364

85.3

1

1874

117.1

1

9384

1483

49

35.6

83.3

3

864 54.1

3

137.4

854

9

1884

117.6

9

939.1

1494

43

38.5

83.8

3

87.3 54.6

3

1383

86.4

3

189.1

1183

3

940.

159.

44

37.3

83.3

4

883 55.1

4

139.1

86.0

4

190.

118.7

4

9404

1594

45

38.3

33*

6

89. 55.6

5

1393

87.4

5

1904

1199

5

941.7

151.

46

39.

34.4

6

89.9 56.3

6

1404

88.

6

191.7

1194

6

9494

1514

47

399

94.9

7

90.7 58.7

7

141.6

68.5

7

1914

1904

7

9434

159.1

48

40.7

35.4

8

•1.6 57.3

8

1434

89.

8

103.4

1904

8

9443

1594

49

41.6

86.

9

•9.4 574

9

1434

89.6

9

1943

191.4

9

945.1

15X1

SO

43.4

864

110

93.3 584

170

1443

00.1

330

105.1

191.9

390

9453

15X7

51

43.3

87.

1

04.1 584

1

145.

00.6

1

1953

199.4

1

9464

1543

5)

44.1

87.6

3

95. 59.4

8

145.9

01.1

8

106.7

199.9

9

9474

154.7

53

44.9

88.1

3

•5.8 59.9

3

146.7

•L7

3

1974

1934

3

9484

1554

54

45.8

83.6

4

•6.7 60.4

4

147.6

•93

4

198.4

m.

4

9494

1554

55

46.6

89.1

5

•74 60.9

5

148.4

09.7

5

1994

1944

5

9503

1564

56

475

89.7

6

08.4 614

6

140.3

•3.3

6

900.1

135.1

6

951.

1963

57

48.3

39.9

7

•9.9 69.

7

150.1

034

7

901.

195.6

7

9513

1574

56

49.3

30.7

8

109.1 694

8

151.

•44

8

9914

196.1

8

959.7

1573

59

50.

31.3

9

109.9 63.1

9

151.8

04.0

B

309.7

196.7

9

9534

UM

09

50.9

31.8

130

101.9 63.6

180

1594

•5.4

940

8034

1973

300

9544

1.53

dt*t

dep.

d.tat

diet.

dep. d. lat.

diet.

dep.

d.lat.

diet.

dep.

1UL

diet

dep diat.

pittance, Departure and Diff Latitude
Course 58°.

Digitized by VjOOQIC

TABLET.

Course 33°.
Dfrtaace, Diff Latitude and Departure.

143

dirt.

d. lat.

dap.

dirt.

d. lat

dep.

dirt

d. lat.

dep.

dirt'

d. hit.

dep.

dirt.

d. lat.

dep.

1

0*

0*

01

51*

33.9

1*1

101*

65.9

191

151*

98.6

341

909.1

131*

9

1.7

11

69

59.

33*

9

169.3

66.4

9

159.6

90.1

9

903.

131*

3

15

1.6

63

59*

34.3

3

103*

67.

3

153.5

99.7

3

903*

133*

4

3.4

9.9

64

53.7

34.9

4

104.

67*

4

154.3

1009

4

904.6

139*

5

4.9

9.7

65

54*

3S.4

5

104*

68.1

5

155*

100*

5

905*

133.4

6

5.

3*

66

55.4

35.9

6

105.7

68.6

6

156.

101.3

6

3UK.3

134.

7

5.9

3.8

67

50*

36*

7

1065

69*

7

156.8

101.8

7

907.9

134.5

8

6.7

4.4

68

57.

37.

8

107.3

69.7

8

157.7

109.4

8

304.

135.1

9

7.5

4.9

69

57.9

37.6

9

108*

70*

9

158*

MM*

9

908*

135.6

10

8.4

5.4

70

58.7

38.1

130

109.

70*

100

159*

103.5

350

909.7

136.9

11

9.9

6.

71

59*

38.7

1

109.9

71.3

1

160*

104.

I

910*

136.7

19

10.1

6*

79

69.4

39*

9

110.7

71.9

9

161.

104*

9

911.3

137*

13

10.9

7.1

73

61.9

39*

3

HI*

79.4

3

161.9

105.1

3

919.3

137*

14

11.7

7.6

74

69.1

403

4

119.4

73l

4

169.7

105.7

4

913.

138*

15

19.6

8.9

75

69.9

40*

5

113*

735

5

163*

10G.9

5

913.9

138.9

16

13.4

8.7

76

637

41.4

6

114.1

74.1

6

164.4

108.7

6

914.7

139.4

17

14.3

9.3

77

64.6

41.9

7

114.0

74.6

7

165*

107*

7

915.5

140.

18

15.1

9*

78

65.4

49*

8

115.7

75.9

8

166.1

107*

8

9104

140*

19

15.9

10*

79

66*

43.

9

116.6

75.7

9

166.9

106.4

9

917*

141.1

30

M*

10.9

80

67.1

43.6

140

117.4

76*

300

167.7

108.9

300

918.1

141*

SI

17.6

11.4

81

67.9

44.1

1

118.3

76.8

1

168.6

109*

1

918.9

149*

99

18.5

19.

' 89

68*

44.7

9

119.1

77.3

9

169.4

110.

9

319.7

149.7

93

19.3

19*

83

69.6

45.9

3

119.9

77.9

3

170*

1106

3

999.6

143*

94

99.1

13.1

84

70.4

45.7

4

190*

78.4

4

171.1

111.1

4

391.4

1438

95

91.

13.6

85

71.3

46.3

5

191*

79.

5

171.9

111.7

5

394.9

144*

98

91*

14.9

86

79.1

46.8

6

199.4

79*

6

179.8

119.3

6

993.1

144*

97

99.6

14.7

87

73.

47.4

7

1933

80.1

7

173.6

119.7

7

993.9

145.4

98

93*

15.9

88

73*

47.9

8

194.1

80.6

8

174.4

ra*

8

334*

146.

99

94*

15*

89

74.6

48*

9

195.

81.9

9

1753

113*

9

935.6

146*

30

95*

16*

00

755

49.

150

195*

81.7

310

176.1

114.4

370

S9M

147.1

31

96.

16.9

91

76.3

49.6

1

196*

89*

1

177.

114.9

1

3-J7*

147.6

39

96*

17.4

99

77.9

59.1

9

197*

89*

9

177*

115.5

9

238.1

148.1

33

97.7

18.

93

78.

50.7

3

198.3

83*

3

178.6

116.

3

339.

148.7

34

98*

18*

94

78*

51.9

4

199*

83.9

4

179*

116.6

4

399*

149*

35

99.4

19.1

95

79.7

51.7

5

130.

84.4

5

180.3

117.1

5

930.6

149*

36

303

19.6

96

895

59*

6

130*

85.

6

181*

117.6

6

931*

150.3

37

31.

99.9

97

81.4

59*

7

131.7

85*

7

189.

118.3

7

339*

150 9

38

31.9

99.7

98

89*

53.4

6

139*

86.1

8

189*

118.7

8

333*

151.4

39

39.7

9J.9

99

83.

53*

9

133.3

86.6

9

183.7

119.3

9

334.

153.

40

33*

91.8

100

839

54*

100

134.9

87.1

330

184*

119*

300

934*

159*

41

34.4

99*

1

84.7

55.

1

135.

87.7

1

185.3

190.4

1

835.7

153.

49

35.9

99.9

9

85*

55.6

9

135.9

68.9

9

186*

190.9

9

836*

153*

43

36.1

93.4

3

86.4

56.1

3

136.7

88*

3

187.

191*

3

337*

154.1

44

36.9

94.

4

87.9

56.6

4

137.5

89*

4

187.9

199.

4

938*

154.7

45

37.7

94*

5

88.1

57.9

5

138.4

89.9

5

1887

129*

5

939.

155*

46

38.6

95.1

6

88.9

57.7

6

139*

90.4

6

189*

1931

6

939.9

155*

47

39.4

95.6

7

89.7

58.3

7

140.1

91.

7

190.4

193.6

7

940.7

156*

48

40.3

96.1

8

90.6

58*

8

1409

91.5

8

101*

194.9

8

941.5

156*

49

41.1

96.7

9

91.4

59.4

9

141.7

99.

9

199.1

194.7

9

943.4

157.4

50

41.9

97*

110

99*

59.9

170

149*

99.6

330

1999

195.3

300

943.9

157*

51

49*

97*

1

93.1

00*

1

143.4

93.1

1

193.7

195*

1

944.1

158*

59

43.6

98*

9

93.9

61.

9

144.3

93.7

9

194.6

196.4

9

944.9

159.

53

44.4

98*

3

94*

61.5

3

145.1

94*

3

195.4

196.9

3

945.7

159.6

54

45.3

99.4

4

95.6

69.1

4

145.9

94*

4

198.9

197.4

4

946.6

160.1

55

46.1

39.

5

96.4

69.6

5

146*

95.3

5

197.1

198.

5

347.4

160.7

59

47.

39*

6

97.3

63.9

6

147.6

95.9

6

197.9

198*

6

948.9

161.9

57

47.8

31.

7

98.1

63.7

7

148.4

98.4

7

198.8

199.1

7

949.1

161*

58

48.6

31.6

8

99.

64*

8

149*

98.9

8

199.6

199.6

8

949.9

169*

59

49*

39.1

9

99*

64.8

9

159.1

97*

9

900.4

130.3

9

959*

169*

69

50*

39.7

199

100.6

65.4

180

151.

98.

940

9013

130.7

300

951.6

16M

dirt.

dap.

d.lat

dirt.

dap.

d. lat.

dirt.

dap.

d.lat

dirt.

dep.

d. lat.

dirt.

dep.

d.lat

Dirtaaee, Departare and DiC Latitude.
Course 570.

Digitized by VjOOQIC

144

TABLE V.

Course 340.
Dirtance, DUE Latitude and Departure.

dirt. d. lat.

dep.

dirt.

d.Iat

dep.' dirt.

d. lat.

dep.

dirt.

d.lat

dep.

dirt.

d.lat.

**.

1

0.8

0.0

61

50.0

34.1

131

1003

67.7

181

150.1

1013

341

1993

1343

2

1.7

1.1

62

51.4

34.7

2

101.1

68.2

2

150.9

101.8

3

8003

1353

3

8.5

1.7

63

52,2

353

3

103.

88.8

3

151.7

1023

3

2013

135.9

4

3.3

2.2

64

53.1

353

4

102.8

69.3

4

1523

102.9

4

8023

136.4

5

4.1

2.8

65

53.9

363

5

103.6

69.9

5

153.4

1033

5

803.1

137.

6

5.

3.4

66

54.7

36.9

6

104.5

703

6

1543

104.

6

8033

1373

7

5.8

3.9

67

553

373

7

1053

71.

7

155.

104.6

7

8043

138.1

8

6.0

43

68

56.4

33.

8

106.1

71.6

8

155.0

105.1

8

9053

138.7

9

7.5

5.

69

57.2

38.6

9

106.9

78.1

9

156.7

105.7

9

SOM

1393:

10

83

5.0

70

58.

39.1

130

107.8

78.7

100

1573

1083

350

8073

1393

11

9.1

6.2

71

58.9

39.7

1

108.6

733

1

1583

1063

I

808.1

140.4

IS

9.9

8.7

79

59.7

403

2

109.4

733

8

1593

107.4

8

8083

1403

13

10.8

73

73

003

40.8

3

110.3

74.4

3

100.

107.9

3

809.7

1413

14

1J.0

73

74

613

41.4

4

111.1

743

4

1603

1083

4

8103

148.

15

12.4

8.4

75

68.2

41.9

5

111.9

753

5

161.7

109.

5

3114

1483

10

13.3

83

76

63.

42.5

6

112.7

76.1

6

1683

100.6

6

3123

1433

17

14.1

93

77

63.8

43.1

7

113.6

766

7

1633

110.2

7

213.1

143.7

18

14.9

10.1

78

64.7

43.6

8

114.4

77.2

8

164.1

110.7

8

2133

1443

1ft

15.8

10.6

79

653

443

9

1153

77.7

9

165.

1113

9

314.7

1443

30

10.0

11.2

80

863

44.7

140

116.1

783

300

1653

1113

300

8153

1453

SI

17.4

11.7

81

67.2

453

1

116.9

78.8

1

168.6

112.4

1

3183

1453

83

18.8

12.3

82

68.

45.9

2

117.7

79.4

3

1673

113.

3

2173

1483

S3

19.1

12.9

83

683

46.4

3

118.6

80.

3

1083

1133

3

818.

147.1

84

19.9

13.4

84

693

47.

4

119.4

803

4

169.1

114.1

4

8183

1473

35

80.7

14.

85

703

473

5

1203

81.1

5

170.

114.6

5

319.7

1483

38

81.8

143

86

71.3

48.1

6

121.

81.6

6

170.8

115.V

6

2-J03

148.7

27

23.4

15.1

87

72.1

48.6

7

121.9

88.2

7

171.6

115.8

7

231.4

1493

88

833

15.7

88

73.

49.2

8

122.7

88.8

8

173.4

116.3

8

222.3

1493

89

84.

163

89

733

493

9

1233

63.3

9

1733

116.9

9

823.

150.4

30

84.9

183

00

74.6

503

150

124.4

83.9

310

174.1

117.4

370

8333

151.

31

85.7

173

91

75.4

50.0

1

125.2

84.4

1

174.9

118.

1

834.7

1513

33

36.5

17.9

93

76.3

51.4

2

126.

85.

8

175.8

1183

9

8353

158.1

33

87.4

18.5

93

77.1

52.

3

126.8

85.6

3

176.6

119.1

3

896.3

158.7

34

88.8

19.

94

77.9

53.0

4

127.7

86.1

4

177.4

119.7

4

8373

1J33

35

89.

19.6

95

78.8

53.1

5

128.5

86.7

5

178.3

14)3

5

928.

1533

38

89.8

80.1

96

79.6

53.7

6

129.3

87.2

8

179.1

120.8

6

2383

1543

37

30.7

80.7

97

80.4

543

7

1303

87.8

7

179.9

121.3

7

329.6

1543

38

31.5

213

98

81.3

54.8

8

131.

88.4

8

180.7

121.9

8

2303

1555

39

323

813

99

82.1

55.4

9

131.8

88.9

9

181.6

1223

9

8313

156.

40

33.8

82.4

1O0

82.9

55.9

100

132.6

893

330

182.4

133.

380

232.1

1583

41

34.

82.9

1

83.7

56.5

1

1333

90.

1

183.2

183.6

1

833.

157.1

48

34.8

215

2

84.6

57.

2

1343

90.6

8

184-

124.1

3

8333

157.7

43

35.0

84.

3

85.4

57.6

3

135.1

91.1

3

I84.O

124.7

3

234.6

1583

44

36.5

84.6

4

86.2

58.8

4

136.

91.7

4

185.7

1253

4

235.4

1583

45

37.3

35.2

5

87.

58.7

5

136.8

92.3

5

1863

125.8

5

2363

15M

46

38.1

25.7

6

87.9

59.3

6

137.6

923

6

187.4

126.4

6

837.1

159.9

47

39.

26.3

7

88.7

59.8

7

138.4

93.4

7

188.2

128.9

7

337.9

1093

48

39.8

86.8

8

89.5

60.4

8

1393

93.9

8

189.

127.5

8

8383

161.

49

40.0

87.4

9

90.4

61.

9

140.1

943

9

1893

128.1

9

839.6

1613

50

41.5

89.

110

913

613

1T0

1403

95.1

330

190.7

1883

300

840.4

1893

51

423

88.5

1

92.

62.1

1

1413

95.6

1

101.5

129.2

1

841.8

16*7

52

43.1

89.1

2

92.9

62.6

9

142.6

96.2

8

192.3

129.7

9

842.1

1633

53

43.9

89.6

3

93.7

63.2

3

143.4

98.7

3

193.2

130.3

3

343.9

1833

54

44.8

30.2

4

943

63.7

4

1443

97.3

4

194.

130.9

4

843.7

184.4

55

45.0

30.8

5

95.3

64.3

5

145.1

97.9

5

1943

131.4

5

844.6

165.

»

40.4

31.3

96.9

64.9

6

145.9

98.4

6

195.7

132.

6

845.4

1653

57

47.3

31.9

7

97.

65.4

7

146.7

99.

7

1983

132.5

7

8463

186.1

58

48.1

33.4

8

97.8

66.

8

1473

993

8

1973

133.1

8

947.1

186.6

59

48.9

33.

9

98.7

063

9

148.4

100.1

9

198.1

133.6

9

847.9

1873

00

49.7

33.6

189

993

67.1

180

149.2

100.7

840

199.

134.2

309

948.7

187.8

dirt.

dej.

d. lit

dial.

dep.

d. lat.' dist.

dep. d. lat.

dirt.

dep.

d. lat.

diet.

dep.

d.lat-

Dirtance, Departure and Diff. Latitude.
CeurM56o.

Digitized by VjOOQIC

TABLE V.

Crarie 35©.
Distance, DUE Latitude and Departure.

145

diet

d. lat.

dep.

diet

d.lat

dep.

diet.

d. lat.

dep.

diet.

d. lat.

dep.

diet.

d.lat

dep.

1

03

0.6

61

50.

35.

131

99.1

69.4

181

1483

103.6

941

197.4

J383

9

L6

1.1

69

50.8

354

9

99.9

70.

2

14&1

104.4

2

1983

1383

3

9.5

1.7

63

51.6

36.1

3

100.8

705

3

149.9

105.

3

199.1

1394

4

33

93

64

59.4

36.7

4

101.6

7M

4

150.7

1054

4

1993

140.

5

4.1

9.9

65

533

373

5

102.4

71.7

5

151.5

106.1

5

900.7

1404

6

4.9

3.4

66

54.1

37.9

6

1033

79.3

6

152.4

106.7

6

9014

141.1

7

5.7

4.

67

54.9

3&4

7

104.

79*

7

1533

1073

7

202-3

141.7

8

6.6

4.6

68

55.7

39.

8

104.9

73.4

6

154.

107*

6

903.1

1423

9

7.4

S3

69

584

39.6

9'

105.7

74.

9

1543

108.4

9

204.

1423

10

83

5.7

70

57.3

403

130

1064

74.6

100

155.6

109.

950

9043

1434

11

9.

63

71

583

40.7

1

1073

75.1

1

1564

109.6

1

905.6

144.

19

9.8

6.9

79

59.

41.3

9

108.1

75.7

9

157.3

1J0.1

9

906.4

1444

13

10.6

74

73

59.8

41.9

3

108.9

763

3

158.1

1.0.7

3

9073

1454

14

11.5

a

74

60.6

42.4

4

109.8

76.9

4

158.9

1113

4

908.1

145.7

15

19.3

a6

75

61.4

43.

5

110.6

77.4

5

159.7

1113

5

9083
909.7

1463

16

13.1

9.2

76

69.3

43.6

6

111.4

m

6

1U0.6

112.4

6

1463

17

13.9

03

77

63.1

443

7

1123

7&G

7

16 '.4

113.

7

9104

147.4

18

14.7

103

78

63.9

44.7

8

113.

793

8

1693

113.6

8

211.3

148.

19

15.6

JOJ

79

64.7

453

9

113.9

79.7

9

163.

114.)

9

2133

M8.6

90

16.4

114

60

654

45.9

140

114.7

803

900

1633

114.7

960

213.

149.1

91

173

19.

81

66.4

464

1

1154

60.9

1

164.6

1153

1

9133

149.7

93

18.

196

89

673

47.

9

1163

81.4

9

1654

115.9

9

214.6

1503

93

ia8

133

83

68.

47.6

3

117.1

89.

3

1663

116.4

3

215.4

1503

94

19.7

134

84

68.8

48.9

4

na

82.6

4

167.1

117.

4

2163

151.4

95

204

14.3

85

60.6

48.8

5

1183

83.2

5

1673

117.6

5

217.1

152.

96

21.3

143

86

70.4

49.3

6

1194

83.7

6

168.7

1183

6

2173

1524

97

29.1

153

87

713

49.9

7

190.4

84.3

7

169.6

118.7

7

2ia7

153.1

98

22.9

16.1

88

79.1

504

8

1213

84.lt

8

170.4

1193

8

2194

153.7

99

9X8

164

89

79.9

5J.

9

199.1

854

9

171.9

119.9

9

220.4

1543

30

94.6

17.2

00

73.7

51.6

150

192.9

86.

910

179.

1204

970

2213

154.9

31

25.4

173

91

74.5

523

1

123.7

86.6

1

1723

191.

1

222:

1554

39

26.9

ia4

99

75.4

52.8

9

1244

87.2

9

173.7

121.6

9

2223

156.

33

27.

18.9

93

763

53.3

3

1253

87*

3

1744

192.2

3

223.6

1564

34

27 3

19.5

94

77.

53.9

4

126.1

88.3

4

1753

192.7

4

224.4

1673

35

28.7

90.1

95

77.8

544

5

127.

88.0

5

176.1

123.3

5

2253

157.7

36

294

906

96

78.6

55.1

6

1273

894

6

176.9

1933

6

22611

1583

37

30.3

913

97

79.5

56.6

7

128.6

90.1

7

1773

1944

7

226.9

1583

38

31.1

91.8

98

80.3

563

8

129.4

90.6

8

1764

125.

8

227.7

1594

39

313

99.4

99

81.1

563

9

1303

913

9

179.4

125.6

9

2284

160.

40

393

99.9

100

81.9

57.4

160

131.1

91.8

990

1803

196.2

980

229.4

160.6

41

334

234

1

82.7

57.9

1

131.9

993

1

181.

126.6

1

2303

1613

49

34.4

24.1

9

83.6

584

9

132.7

92.W

9

181.9

1273

9

231.

1 S- 7

43

353

94.7

3

84.4

59.1

3

1334

934

3

182.7

127.9

3

2313

1693

44

36.

953

4

853

50.7

4

134.3

94.1

4

1834

1984

4

939.6

162.9

45

36.9

25.8

5

86.

603

5

1353

94.6

5

1843

199.1

5

2334

1634

46

37.7

96.4

6

86.8

603

6

136.

953

6

185.1

199.6

6

234.3

5S

47

38.5

97.

7

87.6

61.4

7

136.8

95.8

7

185.9

1303

7

235.1

1643

48

393

974

8

884

619

8

137.6

96.4

8

1863

1303

8

235.9

1653

49

40.1

98.1

9

89.3

694

9

1»4

96.9

9

187.6

1313

9

936.7

1653

50

4L

98.7

110

90.1

63.1

170

1393

974

930

188.4

131.9

900

9374

1663

51

413

99.3

1

90.9

63.7

1

140.1

9ai

1

1893

1394

1

23a4

1663

59

42.6

293

9

91.7

64.9

9

140.9

oa7

9

1«0.

133.1

9

2393

1674

53

43.4

30.4

3

92.6

643

3

141.7

99.2

3

190.9

133.6

3

240.

168.1

54

44.9

31.

4

93.4

65.4

4

142.5

99.8

4

191.7

1342

4

2403

lf.8.6

55

45.1

313

5

943

66.

5

143.4

100.4

5

1924

134.8

5

241.6

1693

56

45.9

39.1

6

95.

664

6

144.9

100.9

6

193.3

135.4

6

9424

1693

57

46.7

32.7

7

953

67.1

7

145.

101.5

7

194.1

135.9

7

9433

170.4

58

474

703

8

96.7

67.7

8

1453

102.1

8

195.

1364

8

944.1

170.9

59

483

333

9

974

683

9

146.6

109,7

9

1953

137.1

9

244.9

1714

60

49.1

34.4

190

983

6&8

180

147.4

103.2

940

196.6

137.7

300

945.7

172.1

diet.

dep.

d.lat

diet

dap. d.lat

diet

dep. d. lat

diet.

dep.

d.lat

diet I dep, d. lat

Dietanee, Departure and DiC Latitude.

Cearie H5&.

T

Digitized by VjOOQIC

146

TABLE V.

Course 3SO.

Distance, Diff. Latitude and Departure.

diBt.

d. lat.

dep.

diet.

d. lat.

dep.

diet.

d.lat

dep.

diet.

d. lat.

dep.

diet.

d.lat

dep.

1

0.8

0.6

61

49.4

35.9

191

97.9

71.1

181

146.4

106.4

941

195.

141.7

S

1.6

1.8

63

50.3

36.4

3

98.7

71.7

2

1475

107.

2

1955

1485

3

8.4

1.8

63

51.

37.

3

99.5

72.3

3

148.1

107.6

3

1965

1435

4

3.8

3.4

64

51.8

376

4

1005

72.9

4

148.9

1085

4

1914

143.4

5

4.

8.9

65

53.6

385

5

101.1

73.5

5

149.7

108.7

5

198.3

144.

G

4.9

3.5

66

53.4

38.8

6

101.0

74.1

6

150.5

1095

6

199.

1445

7

5.7

4.1

67

545

39.4

7

102.7

74.6

7

151.3

109,9

7

1995

145.3

8

6.5

4.7

68

55.

40.

8

103.6

75.2

8

158.1

110.5

8

8005

1455

9

7.3

53

69

55.8

40.6

9

104.4

75.8

9

1535

111.1

9

301.4

M65

10

8.1

5.9

70

565

41.1

130

105.3

76.4

100

153.7

111.7

960

8035

1465

11

8.9

6.5

71

57.4

41.7

1

106.

77.

1

154.5

1185

1

303.1

1475

IS

9.7

7.1

73

585

48.3

3

1065

77.6

3

1555

118.9

3

803.9

148 J 1

13

10.5

7.6

73

59.1

439

3

107.6

78.2

3

156.1

113.4

3

304.7

148.7

14

114

8.3

74

59.9

43.5

4

108.4

78.8

4

156.9

114.

4

2055

1495

15

18.1

85

75

60.7

44.1

5

1095

79.4

5

1575

114.6

5

8085

1495

16

J3.9

9-4

76

615

44.7

6

110.

79.9

6

158.6

115.3

6

807.1

1505

17

13.8

10.

77

63.3

45.3

7

1105

80.5

7

159.4

115.8

7

307.9

151.1

18

14.6

10.6

78

63.1

45.8

8

111.6

81.1

8

160.3

116.4

8

308.7

1515

19

15.4

115

79

63.9

46.4

9

1135

81.7

9

161.

117.

9

3095

1585

90

16.3

115

80

64.7

47.

140

1135

83.3

900

1615

117.6

960

8105

1585

21

17.

18.3

81

655

47.6

1

114.1

835

1

168.6

1181

1

3115

153.4

SS

175

18.9

83

665

48.8

3

114.9

83.5

3

163.4

na7

3

81%

154.

S3

18.6

13.5

83

67.1

48.6

3

J15.7

84.1

3

1645

1195

3

813.8

1545

84

19.4

14.1

84

68.

49.4

4

116.5

84.6

4

165.

1195

4

8135

1555

85

30.8

14.7

85

685

50.

5

117.3

85.2

5

1655

180.5

5

314.4

1555

96

31.

15.3

86

69.6

50.5

6

118.1

85.8

6

166.7

181.1

6

3155

158.4

87

31.8

15.9

87

70.4

51.1

7

118.9

86.4

7

1675

181.7

7

316.

1565

88

83.7

165

88

715

51.7

8

119.7

87.

8

168.3

1323

8

3165

1575

89

83.5

17.

89

73.

535

9

14)5

87.6

9

169.1

183.8

9

817.6

158.1

ao

845

17.8

00

73.8

53.9

150

131.4

88.3

910

169.9

183.4

970

318.4

158.7

31

85.1

18.3

91

73.6

53.5

1

1335

88.8

1

170.7

184.

1

319.3

1595

38

85.9

18.8

93

74.4

54.1

3

133.

895

8

171.5

134.6

3

830.1

1595

33

86.7

19.4

93

755

547

3

1235

89.9

3

172.3

1855

3

8305

1605

34

875

80.

94

76.

555

4

124.6

90.5

4

173.1

135.8

4

S3 1.7

161.1

35

88.3

80.6

95

76.9

55.8

5

135.4

91.1

5

173.9

126.4

5

3835

1615

38

89.1

815

96

77.7

56.4

6

1365

91.7

6

174.7

187.

6

833.3

1635

37

89.9

81.7

97

78.5

57.

7

187.

92.3

7

175.6

1375

7

334.1

1685

38

30.7

88.3

98

795

57.6

8

1875

92.9

8

176.4

138.1

8

8845

163.4

39

31.6

88.9

99

80.1

585

9

138.6

935

9

1775

188.7

9

835.7

164.

*&

38.4

834

100

80.9

585

160

139.4

94.

990

178.

1895

960

3365

164.6

335

84.1

1

81.7

59.4

1

130.3

94.6

1

178.8

139.9

I

8275

1655

43

34.

84.7

3

835,

60.

3

131.1

955

3

179.6

130.5

8

838.1

1655

43

34.8

85.3

3

83.3

60.5

3

131.9

95.8

3

180.4

131.1

3

839.

1665

44

35.6

85.9

4

84.1

61.1

4

133.7

96.4

4

181.3

131.7

4

8395

166.9

43

36.4

865

5

84.9

61.7

5

133.5

97.

5

183.

132.3

5

330.6

167.5

46

3T.8

87.

' 6

855

68.3

6

1345

97.6

6

182.8

132.8

6

831.4

168.1

47

38.

876

7

86.6

68.9

7

135.1

98.3

7

183.6

133.4

7

3335

168.7

48

38.8

38.3

8

87.4

63.5

8

135.9

98.7

8

184.5

134.

8

233.

1695

49

39.6

88.8

9

885

64.1

9

136.7

99.3

9

185.3

1346

9

8335

1695

50

405

89.4

110

89.

64.7

170

1375

995

930

186.1

135.2

990

834.6

1705

51

41.3

30.

1

89.8

655

1

138.3

100.5

1

1865

1355

1

835.4

171.

58

48.1

30.8

3

90.6

65.8

S

139.3

101.1

8

187.7

136.4

8

836.2

171.6

53

43.9

31.3

3

91.4

66.4

3

140.

101.7

3

188.5

137.

3

837.

1725

54

43.7

31.7

4

93.8

67.

4

140.8

103.3

4

189.3

1375

4

2375

1785

55

445

335

5

93.

67.6

5

1415

103-9

5

190.1

138.1

5

838.7

173.4

56

45.3

38.9

6

93.8

685

6

143.4

103.5

6

190.9

138.7

6

8395

174.

57

46.1

33.5

7

94.7

68.8

7

143.3

104.

7

191.7

1395

7

3405

1745

58

46.9

34.1

8

955

69.4

8

144.

104.6

8

1935

139.9

8

841.1

1755

59

47.7

34.7

9

965

69.9

9

144.8

105.2

9

193.4

1405

9

3415

175.7

60

48.5

355

180

97.1

70.5

180

145.6

105.8

340

1945

141.1

300

848.7

1765

diet.

den.

d. lat.

diet.

dep.

d. let.

diet.

dep. d. lat.

diet

dep.

d. lat.

diet.

dep.

d. lat.

Distance, Departure and DUE Latitude.
Course 54o.

Digitized by VjOOQIC

TABLX ▼•

Couuree 3T°.
tMrtance, DUE Latitude and Departure

147

diet.

d.lat

dep.

diet.

d. lat.

dep.

dirt.

d. lat.

dep.

diet.

d.lat

dep.

dirt.

d. lat. dep.

1

0.8

0.6

61

48.7

36.7

191

98.6

79.8

181

144.6

108.9

941

1993 145.

9

1.8

13

69

49.5

373

9

97.4

73.4

2

145.4

1093

9

193.3 1453

3

9.4

13

63

50.3

37.9

3

983

74.

3

146.9

110.1

3

194.1 1463

4

3.9

9.4

64

51.1

38.5

4

99.

74.6

4

146.9

110.7

4

1943 146.8

5

4.

3.

65

51.9

39.1

5

99.8

753

5

147.7

1113

5

195.7 147.4

6

4.8

3.6

66

59.7

39.7

6

100.6

75.8

6

148.5

111.9

6

1963 148.

7

5.6

49

67

53.5

403

7

101.4

76.4

7

1493

119.5

7

1973 148.6

8

6.4

4.8

68

54.3

40.9

8

1093

77.

8

150.1

113.1

8

198.1 1493

•

7.9

5.4

69

55.1

415

9

103.

773

9

1503

113.7

9

196.9 149.9

10

8.

6.

TO

55.9

49.1

130

103.8

783

100

151.7

1143

950

199.7 1503

11

8.8

6.6

71

56.7

49.7

1

104.6

783

1

1593

114.9

1

9003 151.1

19

9.6

7.9

79

57.5

43.3

9

105.4

79.4

9

1533

1153

9

9013 151.7

13

10.4

7.8

73

58.3

43.0

3

1083-

80.

3

154.1

1163

3

909.1 1523

14

11.9

8.4

74

59.1

443

4

107.

80.6

4

154.9

1163

4

902.9 152.9

15

19.

0.

75

59.9

45.1

5

107.8

81.2

5

155.7

117.4

5

903.7 1533

16

193

9.6

76

60.7

45.7

6

108.6

81.9

6

1563

118.

6

904.5 154.1

17

13.6

10.9

77

61.5

46.3

7

109.4

89.4

7

1573

118.6

7

205.2 154.7

18

14.4

10.8

78

69.3

46.9

e

1103

831

8

158.1

1193

8

206. 155.3

19

15.9

11.4

79

63.1

473

9

111.

83.7

9

158.9

1193

9

2063 155.9

9)0

16.

19.

60

63.9

48.1

140

1118

84.3

900

159.7

190.4

960

907.6 1563

91

163

19.6

81

64.7

4*7

1

119.6

84.9

1

160.5

121.

1

308.4 157.1

99

17.6

139

89

6*5

493

9

113.4

85.5

9

161.3

191.6

9

9093 157.7

93

18.4

13.8

83

663

50.

3

1143

86.1

3

169.1

1223

3

910. 1583

94

19.9

14.4

84

671

50.6

4

115.

86.7

4

169.9

192.8

4

9103 158.9

45

90.

15.

85

67.9

51.9

5

1153

87.3

5

163.7

123.4

5

911.6 1593

98

90.8

15.6

86

68.7

51.8

6

116.6

87.9

6

1643

194.

6

919.4 160.1

97

91.6

163

87

69.5

59.4

7

117.4

883

7

165.3

194.6

7

913.9 160.7

98

99.4

16.9

88

70.3

53.

8

1183

89.1

8

166.1

195.2

8

914. 161 3

99

93.9

17.5

89

71.1

53.6

9

119.

89.7

9

1663

1953

9

9143 161.9

30

94.

18.1

00

71.9

54.9

150

119.8

90.3

910

167.7

196.4

970

915.6 1625

31

94.8

18.7

91

79.7

54.8

1

120.6

90.9

1

1683

197.

1

916.4 1631

39

95.6

19.3

99

73.5

55.4

9

191.4

91.5

9

1693

197.6

9

917.9 163.7

33

96.4

19.9

33

74.3

56.

3

1993

99.1

3

170.1

1983

3

218. 1643

34

97.9

90.5

■K

75.1

56.6

4

193.

99.7

4

170.9

1983

4

9183 1843

35

98.

91.1

95

75.9

573

5

193.8

933

5

171.7

199.4

5

919.6 1653

38

96.8

91.7

96

76.7

57.8

6

194.6

939

6

1793

130.

6

990.4 166.1

37

993

99.3

97

77.5

58.4

7

125.4

943

7

173.3

130.6

7

9913 166.7

38

30.3

99.9

96

78.3

59.

8

1963

95.1

8

174.1

131.9

8

929. 1673

38

31.1

333

99

70.1

59.6

9

197.

95.7

9

1743

1313

9

992.8 167.9

40

31.0

94.1

100

79.0

60.9

100

1973

96.3

990

175.7

132.4

980

923.6 1683

41

39.7

94.7

1

80.7

60.8

1

198.6

96.9

1

1763

133.

1

224.4 169.1

49

333

95.3

9

81.5

61.4

9

129.4

973

9

177.3

133.6

9

2253 169.7

43

343

95.9

3

893

69.

3

1303

98.1

3

178.1

1343

3

226. 1703

44

351

963

4

83.1

69.6

4

131.

98.7

4

178.9

134.8

4

9263 1703

45

35.9

97.1

5

83.9

63.9

5

1313

99.3

5

179.7

135.4

5

997.6 1713

48

36-7

97.7

6

84.7

63.8

6

139.6

99.9

6

180.5

136.

6

228.4 179.1

47

37 3

98.3

7

853

64.4

7

133.4

1003

7

181.3

136.6

7

9293 179.7

48

384

983

8

863

65.

8

134.2

101.1

8

189.1

1373

8

230. 173.3

49

30-1

993

9

87.1

65.6

9

135.

101.7

9

189.9

137.8

9

9303 173.9

90

39.9

30.1

110

87.8

66.9

190

135.8

102.3

930

183.7

138.4

900

9313 174.5

51

40.7

30.7

1

88.6

66.8

1

136.6

102.9

1

1843

139.

1

932.4 175.1

59

41.5

31.3

9

89.4

67.4

9

137.4

1035

9

185.3

139.6

9

233.2 175.7

53

43.3

31.9

3

903

68.

3

1383

104.1

3

186.1

140.9

3

234. 176.3

54

43.1

39.5

4

91.

68.6

4

139.

104.7

4

186.9

1403

4

234.8 176.9

55

433

33.1

5

91.8

60.9

5

1393

105.3

5

187.7

1414

5

935.6 1773

58

44.7

33.7

6

99.6

69.8

6

140.6

105.9

6

188.5

149.

6

936.4 178.1

57

45.5

343

7

93.4

70.4

7

1414

106.5

7

1893

149.6

7

9373 178.7

58

463

34.9

8

94.9

71.

8

1493

107.1

8

190.1

143.2

8

938. 1793

59

47.1

353

9

95.

71.6

9

143.

107.7

9

190.9

1433

9

9383 179.91

80

47.9

.36.1

190

95.8

793

180 | 1433

1083

940

191.7

144.4

300

939.6 1803]

diet.

*l>

d.lat.

ditt.

Idep.

4. lat.

<ftst. Idep.

d.latldUt.

dap.

d. lat.

dirt.

dep. d. 1H.

Distance, Departure and Diff Latitude.
Comra* »3°.

Digitized by VjOOQIC

J48

TABLE r.

Cewraa 30°.
Dirtanae, Diff Latitnde and

dirt.

d.lat.

dep.

dirt.

d.lak

dep.

dirt.

d.lat.

dep.

dirt 7

d.lat.

dap.

dirt.

Hat.

4a*

1

04

0.6

61

48.1

374

181

95.3

744

181

1424

1114

841

1894

1484

s

14

ia

63

484

384

3

96.1

75.1

9

143 4

1124

3

1907

149.

3

2.4

14

63

494

384

3

964

75.7

3

1444

112.7

3

191.5

1494

!

3.3

24

64

50.4

39.4

4

97.7

764

4

145.

1134

4

192.3

1504

3.9

3.1

65

8?

40.

5

98.5

77.

5

1454

1134

5

1931

1504

6

4.7

3.7

66

40.6

6

994

774

6

146.6

1144

6

1934

1514

r

5.5

44

67

52.8

414

7

100.1

784

7

147.4

115.1

7

1944

152.1

6.3

4.9

68

534

414

8

1009

784

6

148.1

115 7

8

195.4

152.7

?

7.1

54

69

544

424

9

101.7

79-4

9

1484

1164

9

1964

1534

10

7.9

63

70

55.3

43.1

130

102.4

80.

190

149.7

117.

800

197.

1534

8.7

64

71

55.9

43.7

1

1034

80.7

1

150.5

117.6

1

1974

U44

TO

94

7.4

72

2j

444

2

104i

814

3

1514

1184

9

198.0

155.1

13

103

a

73

575

44.9

3

104.8

81.9

3

153.1

1184

3

190.4

1553

14

11.

8.6

74

584

45.6

4

1054

824

4

159.9

119.4

4

8094

U64

1$

11.8

94

75

59.1

464

5

1064

83.1

5

153.7

190.1

5

2004

157.

»?

194

9.9

76

59.9

464

6

1074

83.7

6

1544

180.7

6

801.7

1574

i?

13.4

104

77

00.7

47.4

7

108.

843

7

1554

1*14

7

802.5

1584

s

144

11.1

78

614

48.

8

103.7

85.

8

156.

191.9

6

8014

1584

15.

11.7

79

024

484

9

1094

85.6

9

1564

1924

294.1

1594

n

1*4

12.3

80

63.'

494

140

1104

864

200

157.6

193.1

123.7

804

3044

140.1

$6.5

12.9

81

634

49.9

1

111.1

864

1

158.4

1

805.7

M6.T

»

17.$

134
$4.2

.82

04.6

504

2

111.9

87.4

8

1594

194.4

8

989.5

1414

23

l&i

83

65.4

51.1

3

113.7

88.

3

160.

125.

8

8074

1619

34

18.9

144

84

664

51.7

4

113.5

88.7

4

1604

1254

d

908.

1884

SS

19.7

154

83

67.

524

5

114.3

89.3
89.9

5

1614

1264

6

2084

1684

99

904

16.

S

674

53.9

6

115.

6

1624

1264

6

8094

1484

87

914

16.6

68.8

534

7

115.8

904

7

103.1

1874

7

910.4

1444

28

*-M

17.2

88

69.3

$

8

116.6

911

8

163.9

198.1

8

2114

144.

29

22.9

$7.9

89

70.1

9

117.4

91.7

9

164.7

198.7

9

918.

1444

a©

23.6

184

w

70.9

S 4

150

1184

934

810

1654

1994

390

9184

1444

31

24.4

19.1

19.7

W

71.7

1

119.

93.

1

1663

1994

1

9134

1444

33

853

92

724

58.6
$74

9

1194

93.6

9

167.1

1304

|

9144

1474

33

26.

204

93

734

3

120.6

94.9

3

1674

131.1

i

9151

148.1

34

86.8

20.9

3

741

57.9

4

131.4

944

4

168.6

131.8

4

9154

148,7

3$

27.6

li- 5

749

58.5

5

122.1

95.4

5

169.4

133.4

6

9147

1804

36

28.4

22.2

96

75.6

$•<

1224

96.

6

1704

133.

6

8174

1494

37

29.2

e

S

76.4

59.7

7

133.7

96.7

7

17L

133.6

7

9184

1744

38

39.9

774

604

8

1344

97.3

8

171.8

134.9

8

919.1

1914

33

30.7

44.

99

78.

OL

9

1254

97.9

9

179.6

1344

9

9184

1714

40

314

946

10Q

788

614

160

196.1

98.5

89)0

173.4

135.4

380

220.6

1184

41

324

854

1

794

62.2

1

126.9

99.1

1

174.9

1

281.4

174

42

33.1

2

804

02.8

9

137.7

99.7

9

1744

136.7

9

92*4

173.6

J

33.9

364

s

814

63.4

3

138.4

100.4

3

1737

1374

3

»

1744

34.7

27.1

1

82.

64.

4

1294

101.

4

1764

137.9

4

9234

49

354

27.7

82.7

64.6

5

130.

101.6

5

1774

1384

5

9844

1744

46

364

m

6

835

65.3

6

1304

102.9

6

178.1

139.1

6

285.4

174.1

47

37.

i

844

65.9

7

131.6

102.8

7

1784

139.8

7

926.9

174.7

18

37.8

&

83.1

664

6

133.4

103.4

8

179.7

140.4

8

2864

1774

49

38.6

9

85.9

67.1

9

1334

104.

9

1804

141.

9

927.7

1774

K

•39.4
40i

304

110

86.7

67.7

170

134.

104.7

880

1814

1414

300

288.5

1744

31.4

874

684

1

134.7

1058

1

189.

1494
1494

1

8294

1784

33

41.

&

9

864

5*6

9

1354

1054

9

1828

9

S.i

1704

33

41.8

So

3

89.

3

136.3

1064

3

1836

143.4

3

1884

54

42.6

33.2

2

89.8

70.2

t

137.1

107.1
107.7

4

164.4

144.1

4

931.7

161.

55

43.3

33.9

5

90.6

704

137.9

5

1854

144.7

5

2324

1414

53

44.1

344

o

914

71.4

138.7

108.4

6

186.

1454

6

8334

1824

S

44.9

35.1

7

924

78.

J

139.5

109.

7

1864

145.9

7

834.

1884

45.7

35.7

01

7*4

1404

109.6

8

1874

1464

8

234.8

1814

33

464

364

3

934

734

9

1411

110.9

9

1884

147.1

9

SK

184.1

3D

47.3

36.9

120

94.6

734

186

1414

1104

940

189.1

1474

300

144.7

dfrt.

**

d.lat

dirt.

dep.

d. lat.

dirt.

dep.

Hat

dirt.

dep.

d. lat.

dirt.

dep.

d. lat.

pittance. Departure and Diit Latitude.

Cawaa»8q.

Digitized by VjOOQIC

TABUS ▼.

Course 39°

Distance, DUE Latitude and Departure.

U§

disk

d.lat.

dep.

dist.

d.let.

dep.

dist.

(List.

dep.

dist.

d.lat.

dep.

dist.

d. lat.

dep.

1

0.8

0.6

•1

47.4

38.4

1»

94.

76.1

181

140.7

113.9

»*1

187.3

151.7

S

1.6

1.3

69

484

39.

8

944

764

3

14L4

1144

3

188.1

158.3

9

13

19

63

49.

39.6

3

956

77.4

3

1434

1154

3

188.8

153.0

4

3.1

9.5

64

49.7

40.3

4

96.4

78.

4

143.

1154

4

189.6

153.6

5

3*

3.1

65

50.5

40.9

5

97.1

78.7

5

143.8

116.4

5

190.4

154.3

4.7

34

66

51.3

414

6

97.9

794

6

1444

117.1

6

1914

1544

7

5.4

4.4

67

53.1

484

7

98.7

79.9

7

145.3

117.7

7

199.

155.4

8

64

5.

68

59.8

434

8

994

80.6

8

146.1

118.3

8

199.7

156.1

9

7.

5.7

69

53.6

43.4

9

100.3

814

9

1464

118.9

9

1934

156.7

10

7.8

64

70

54.4

44.1

130

10L

814

190

147.7

1194

350

1944

157.3

11

84

«4

71

55.9

44.7

1

1014

83.4

1

148.4

190.3

1

195.1

158.

1*

9.3

7.6

79

56.

45.3

3

103.6

83.1

2

1494

1904

9

195.8

153.6

13

10.1

8.9

73

56.T

45.9

3

103.4

83.7

3

150.

1914

3

196.6

1594

14

10.9

e.8

74

574

46.6

4

104.1

844

4

150.8

199.1

4

197.4

1594

15

11.7

9.4

75

58.3

474

5

104.9

85.

5

1514

199.7

5

198.9

1604

16

1*4

10.1

76

59.1

474

6

105.7

85.6

6

1594

1934

6

196.9

161.1

17

13.8

10.7

77

59.8

484

7

106.5

86.3

7

153.1

194.

7

199.7

161.7

18

14.

11.3

78

60.6

49.1

8

1074

86.8

8

1534

194.6

8

9004

162.4

19

14.8

19.

79

61.4

49.7

9

106.

874

9

154.7

1254

9

801.3

163.

90

154

136

80

694

504

140

1084

88.1

aoo

155.4

135.9

I960

802.1

163.6

n

16.3

134

81

69.9

51.

1

109.6

88.7

i

1564

1964

1

8034

1644

S3

17.1

184

82

63.7

51.6

3

110.4

89.4

3

157.

197.1

9

803.6

164.9

93

17.9

144

83

644

59.3

3

111.1

90.

3

157.8

197.8

3

804.4

165.5

94

18.7

15.1

84

65.3

59.9

4

111.9

90.6

4

1584

198.4

4

8054

166.1

85

19,4

15.7

85

66.1

534

5

113.7

914

5

1594

199.

5

805.9

166.8

46

80.9

16.4

86

66.8

54.1

6

1134

91.9

6

160.1

199.6

6

306.7

167.4

87

81.

17.

87

67.6

54.8

7

1144

984

7

160.9

130.3

7

3074

168.

88

91.8

17.6

88

68.4

55.4

8

115.

93.1

8

1614

130.9

8

8084

168.7

39

92.5

18.3

89

694

56.

9

115.8

93.8

9

168.4

131.5

9

809.1

1694

30

93.3

18.9

00

69.9

564

150

116.8

94.4

aio

1634

1394

9370

809.8

169.9

3t

84.1

194

91

70.7

57.3

1

117.3

95.

i

164.

1328

1

310.6

1704

32

24.9

30.1

93

714

57.9

8

118.1

95.7

3

1644

133.4

9

311.4

1714

33

856

90.8

93

78.3

58.5

3

118.9

96.3

3

1654

134.

3

8194

171.8

34

96.4

914

94

73.1

59.3

4

119.7

96.9

4

1664

134.7

4

313.9

173.4

35

37.9

99.

95

73.8

59.8

5

1904

97.5

5

167.1

135.3

5

313.7

173.1

36

9*.

98.7

96

74.6

60.4

6

1314

984

6

167.9

135.9

6

3144

173.7

37

33.8

93.3

97

75.4

61.

7

183.

98.8

7

168.6

136.6

7

3154

1744

38

99.5

93.9

98

76.3

61.7

8

133.8

99.4

8

169.4

137.9

8

216.

175.

3J

304

94.5

99

76.9

69.3

9

133.6

100.1

9

1704

1374

9

316.8

175.6

40

311

95.9

100

77.7

69.9

160

134.3

100.7

aao

171.

138.5

»80

317.6

1764

41

31.9

95.8

1

78.5

63.6

1

185.1

101.3

i

171.7

139.1

1

918.4

176.8

43

33.6

964

3

79.3

64.9

3

125.9

101.9

3

179.5

139.7

9

3194

1774

43

33.4

97.1

3

80.

64.8

3

136.7

103.6

3

173.3

140.3

3

319.9

178.1

44

34.9

97.7

4

80.8

65.4

4

1974

1034

4

174.1

141.

4

330.7

178.7

45

35.

98.3

5

81.6

66.1

5

1284

103.8

5

174.9

1416

5

3314

179.4

46

35.7

98.9

6

8-2.4

66.7

6

139.

1044

6

175.6

149.3

6

3333

180.

47

36.5

916

7

83.3

67.3

7

139.8

105.1

7

176.4

143.9

7

333.

180.6

49

37.3

304

8

83.9

68.

8

130.6

105.7

8

177.2

143.5

8

333.8

1814

49

38.1

304

9

84.7

684

9

131.3

106.4

9

178.

144.1

9

224.6

181.9

50

38.9

314

110

854

69.9

170

1311

107.

830

178.7

144.7

390

235.4

183.5

51

396

39.1

1

86.3

69.9

1

139.9

107.6

1

1794

145.4

1

336.1

183.1

52

40.4

39.7

3

87.

70.5

3

133.7

1084

3

1803

146.

9

336.0

1834

53

414

33.4

3

87.8

71.1

3

134.4

106.9

3

181.1

1464

3

327.7

184.4

54

49.

34.

4

88.6

71.7

4

135.2

1094

4

181.9

147.3

4

388.5

185.

55

49.7

34.6

5

89.4

78.4

5

136.

110.1

5

189.6

147.9

5

8394

185.6

56

43.5

35.9

6

90.1

73.

6

136.8

110.8

6

183.4

1484

6

830.

186.3

57

44.3

35.9

7

90.9

736

7

137.6

111.4

7

184.9

149.1

7

3304

186.9

58

45.1

364

8

91.7

74.3

8

1384

112.

8

185.

149.8

8

331.6

1874

59

45.9

37.1

9

934

74.9

9

139.1

1126

185.7

150.4

9

3334

188.2

60
dist.

46.6

374

190

93.3

754

180

139.9

113.3

840

186.5

151.

300

233.1

188.8

rlep.

d.lat

dist.

dep.

d. lat.

dist.

dep.

d. 1st.

dist.

dep.

d. 1st.

dist.

dep.

d.lat.

Distance, Departure and Diff Latitude.
Coarse 51°.

Digitized by VjOOQIC

160

TABLE V.

Course 40°.

Distance, Diff. Latitude and Departure.

disk

d.lat.

dep.

dist.

d.lat.

dep.' dist.

d.Iat.

dep.

dist.

d.Iat. dep.

dist.

d.lat.

dep.

1

0.8

0.6

61

46.7

394

iai

987

77.8

181

138.7 1163

841

184.6

1544

9

1-5

1.3

03

473

39.9

3

935

78.4

9

139.4 117.

3

185.4

1553

3

93

1.9

63

48.3

403

3

944

79.1

3

1404 117.6

3

186.1

1564

4

3.1

3.6

64

49.

41.1

4

95.

79.7

4

141. na3

4

186.9

1563

5

33

34

65

49.8

41.8

5

95.8

80.3

5

141.7 118.9

5

187.7

1573

6

4.0

3.9

66

50.6

49.4

6

963

81.

6

1433 119.6

6

188.4

158.1

7

5.4

43

67

51.3

43.1

7

97.3

81.6

7

143.3 1304

7

1894

1583

8

6.1

5.1

68

53.1

43.7

8

98.1

82.3

8

144. 1303

8

190.

159.4

9

6.9

. 53

69

53.9'

44.4

9

98.8

82.9

9

1443 121.5

9

190.7

160.1

10

7.7

6.4

70

53.6

45.

130

99.6

83.6

100

1453 133.1

850

1913

160.7

11

8.4

7.1

71

54.4

45.6

1

100.4

843

1

1463 133.8

1

1933

1613

IS

9.3

7.7

79

554

463

9

101.1

84.8

3

147.1 133.4

8

193.

168.

13

10.

&4

73

55.9

46.9

3

101.9

85.5

3

1473 134-1

3

1933

1693

14

10.7

9.

74

56.7

47.6

4

103.6

86.1

4

148.6 134.7

4

1943

1633

15

113

9.6

75

573

484

5

103.4

86.8

5

149.4 1353

5

1953

163.9

16

133

103

76

584

48.9

6

1044

87.4

6

150.1 138.

198.1

1643

17

13.

10.9

77

59.

49.5

7

104.9

88.1

7

150.9 136.6

7

196.9

1654

18

13.8

11.6

78

594

50.1

8

105.7

88.7

8

151.7 137.3

8

197.6

1653

19

14.6

134

79

603

508

9

1063

89.3

9

153.4 137.9

9

198.4

1663

80

153

13.9

80

613

51.4

140

1074

90.

800

1534 138.6

860

1994

167.1

91

16.1

13.5

81

69.

53.1

1

ioa

90.6

1

154. 1394

1

199.9

1673

S3

16.9

14.1

89

69.8

53.7

9

1083

913

3

154.7 1293

8

800.7

168.4

93

17.6

14.8

83

63.6

53.4

3

1093

91.9

3

1553 1303

3

3013

169.1

94

18.4

15.4

84

64.3

54.

4

110.3

92.6

4

1563 131.1

4

3034

169.7

35

19.9

16.1

85

65.1

54.6

5

1111

93.2

5

157. LU8

5

903.

1703

96

19.9

16.7

86

65.9

55.3

6

1113

93.8

6

157.8 132.4

6

903.8

171.

97

90.7

17.4

87

66.6

55.9

7

113.6

943

7

158.6 133.1

7

904.5

1713

98

91.4

ia

88

67.4

56.6

8

113.4

95.1

8

1593 133.7

8

3053

1783

99

334

18.6

89

684

574

9

114.1

95.8

9

160.1 1343

9

906.1

1784

30

33.

19.3

00

63.9

57.9

150

114.9

96.4

»io

160.9 135.

870

8063

1733

31

33.7

19.9

91

69.7

58.5

1

115.7

97.1

1

161.6 135.6

1

307.6

1744

33

343

30.6

99

703

59.1

9

116.4

97.7

3

168.4 136.3

3

808.4

1743

33

35.3

914

93

714

59.8

3

1174

98.3

3

163.3 136.9

3

309.1

1753

34

36.

31.9

94

79.

60.4

4

118.

99.

4

163.9 137.6

4

309.9

176.1

35

363

333

95

79.8

61.1

5

I1&7

99.6

5

164.7 1384

5

810.7

1763

30

37.6

33.1

96

733

61.7

6

1193

100.3

6

1653 138.8

6

311.4

177.4

37

983

83.8

97

74.3

63.4

7

130.3

100.9

7

1664 139.5

7

3134

178.1

38

39.1

34.4

98

75.1

63.

8

131.

101.6

8

167. 140.1

8

813.

178.7

39

39.9

35.1

99

75.8

63.6

9

1313

1034

9

1673 140.8

9

813.7

1793

40

30.6

95.7

100

76.6

64.3

100

133.6

102.8

880

1683 1414

880

9143

180.

41

31.4

36.4

J

77.4

64.9

1

133.3

103.5

1

169.3 143.1

1

8153

180.6

49

334

87.

9

78.1

65.6

9

134.1

104.1

3

170.1 143.7

3

316.

1813

43

33.9

97.6

3

78.9

66.9

3

134.9

104.8

3

1703 1433

3

3163

1814

44

33.7

38.3

4

79.7

66.8

4

195.6

1054

4

171.6 144.

4

317.6

1883

45

343

98.9

5

80.4

673

5

136.4

106.1

5

173.4 144.6

5

3183

1834

46

354

99.6

6

814

68.1

6

1374

106.7

6

173.1 1453

6

319.1

1833

47

36.

304

7

89.

68.8

7

137.9

1073

7

173.9 145.9

7

319.9

1843

48

36.8

30.9

8

83.7

69.4

8

138.7

108.

8

174.7 1463

8

3303

185.1

49

373

31.5

9

83.5

70.1

9

139.5

108.6

9

175.4 1474

9

831.4

1853

50

38.3

33.1

110

84.3

70.7

1T0

1304

109.3

8330

176.9 147.8

800

8334

186.4

51

39.1

33.8

1

85.

71.3

1

131.

109.9

1

177. 1483

1

883.9

187.1

59

39.8

33.4

3

853

73.

9

131.8

110.6

3

177.7 149.1

3

883.7

187.7

53

40.6

34.1

3

86.6

73.6

3

1333

1114

3

1783 1493

3

3343

1883

54

41.4

34.7

4

87.3

73.3

4

1333

111.8

4

1793 150.4

4

8354

189.

55

43.1

35.4

5

88.1

73.9

5

134.1

112.5

5

180. 151.1

5

936.

1893

56

43.9

36.

6

88.9

74.6

6

1343

113.1

6

1803 151.7

6

986.7

1903

57

43.7

36.6

7

89.6

754

7

135.6

113.8

7

1916 153.3

7

9373

190.9

58

44.4

37.3

8

90.4

75.8

8

136.4

114.4

8

188.3 153.

8

8883

191.6

« »

454

37.9

9

914

763

9

137.1

115.1

9

183.1 153.6

9

899.

1984

60

46.

38.6

130

91.9

77.1

180

137.9

115.7

340

183.9 1543

300

8893

1983

4ist.

dep.

d. lat

dist.

dap.

d. lat.

1 dist.

dep.

d. lat.

dist. dep. d.lat.

dist.

dep

d.laL

Distance, Departure and Diff. Latitude.
Coarse 50°.

Digitized by VjOOQIC

TABLE ¥•
Cooric «o.

Distance, DUE Latitude and Departure.

151

dirt.

d.lat

dep.

dirt.

d. lat. dep. j dirt.

d. lat.

dep.

dist.

d. lat.

dep.

dirt.

d. lat.

dep.

1

0.8

0.7

61

46. 40.

191

913

79.4

181

136.6

118.7

941

181.9

158.1

9

1.5

1.3

69

46.8 40.7

9

99.1

80

2

137.4

119.4

2

189.6

158.8

3

9.3

9.

63

47.5 41.3

3

993

80.7

3

138.1

190.1

3

183.4

159.4

4

a

9.6

64

483 49.

4

93.6

81.4

4

138.9

120.7

4

184.1

160.1

5

3.8

3.3

65

49.1 42.6

5

94.3

82.

5

139.6

191.4

5

184.9

160.7

6

4.5

3.9

66

39.8 433

6

95.1

82.7

6

140.4

122.

6

185.7

161.4

7

5.3

4.6

67

50.6 44.

7

95.8

83.3

7

141.1

122.7

7

186.4

162.

8

6.

52

68

51.3 44.6

8

96.6

84.

8

141.9

123.3

8

1873

169.7

9

64

5.9

69

52.1 45.3

9

97.4

84.6

9

149.6

124.

9

187.9

163.4

10

7.5

6.6

70

52.8 45.9

130

98.1

85.3

100

143.4

194.7

950

188.7

&

11

a3

7.2

71

53.6 46.6

1

98.9

85.0

1

144.1

125.3

1

189.4

IS

9.1

7.9

72

54.3 47.2

9

99.6

86.6

2

144.9

126.

2

1903

1653

13

9.8

83

73

55.1 47.9

3

100.4

87.3

3

145.7

196.6

3

1903

166.

14

10.6

9.2

74

553 48.5

4

101.1

87.9

4

146.4

1273

4

191.7

168.6

15

11.3

9.8

75

56.6 49.2

5

101.9

88.6

5

147.9

197.9

5

192.5

167.3

16

19.1*

10.5

76

57.4 49.9

6

102.6

89.2

6

147.9

198.6

6

193.9

168.

17

12.8

113

77

58.1 505

7

103.4

80.9

7

148.7

J29.2

7

194.

168.6

18

13.6

11.8

78

58.9 513

8

104.1

90.5

8

149.4

129.9

6

194.7

169.3

19

14.3

195

79

593 51.8

9

1043

913

9

1503

130.6

9

1953

1693

90

15.1

13.1

80

60.4 523

140

105.7

91.8

900

150.9

131.2

960

1963

1703

91

15.8

13.8

81

61.1 53.1

1

106.4

92.5

1

151.7

131.9

1

197.

1713

99

16.6

14.4

82

61.9 53.8

9

1073

93.2

9

152.5

132.5

2

197.7

1713

93

17.4

15.1

83

62.6 54.5

3

107.9

93.8

3

153.2

133.2

3

198.5

172.5

94

18.1

15.7

84

63.4 55.1

4

108.7

94.5

4

154.

133.8

4

1993

173.2

95

18.9

16 4

85

64.2 55.8

5

109.4

95.1

5

154.7

134.5

5

900.

173.9

96

19.6

17.1

86

64.9 56.4

6

1103

95.8

6

1553

135.1

6

900.8

1743

97

90.4

17.7

87

65.7 57.1

7

110.9

96.4

7

1563

135.8

7

2013

175.9

98

91.1

18.4

88

66.4 57.7

8

111.7

97.1

8

157.

136.5

8

2023

1753

99

91.9

19.

89

67.2 5&4

9

1193

97.8

9

157.7

137.1

9

903.

1763

30

99.6

19.7

00

67.9 59.

150

1133

98.4

910

158.5

137.8

970

903.8

177.1

31

93.4

20.3

91

68.7 59.7

1

114.

99.1

1

1503

138.4

1

904.5

177.8

S

24.9

21.

99

69.4 60.4

9

114.7

99.7

9

160.

139.1

2

205.3

178.4

94.9

21.6

93

70.2 61.

3

1153

100.4

3

160.8

139.7

3

206.

179.1

34

25.7

22.3

94

70.9 61.7

4

116.9

101.

4

161.5

140.4

4

906.8

1793

35

26.4

23.

95

71.7 623

5

117.

101.7

5

1623

141.1

5

9073

189.4

36

27.2

93.6

96

72.5 63.

6

117.7

109.3

6

163.

141.7

6

9083

181.1

37

27.9

24.3

97

733 63.6

7

118.5

103.

7

1633

14*4

7

909.1

181.7

38

28.7

24.9

98

74. 643

8

119.2

103.7

8

1643

143.

8

9093

182.4

39

29.4

25.6

99

74.7 64.9

9

120.

104.3

9

1653

143.7

9

9103

183.

40

30.9

96.9

100

75.5 65.6

150

1903

105.

990

166.

144.3

980

911.3

183.7

41

30.9

26.9

1

76.2 66.3

1

191.5

105.6

1

1063

145.

1

919.1

184.4

49

31.7

97.6

2

77. 66.9

9

122.3

106.3

9

1673

145.6

2

919.8

185.

43

32.5

28.2

3

77.7 67.6

3

123.

106.9

3

1683

146.3

3

913.6

185.7

44

33.9

28.9

4

78.5 68.2

4

1233

107.6

4

169.1

147.

4

9143

1863

45

34.

99.5

5

79.9 68.9

5

194.5

108.2

5

1693

147.6

5

915.1

187.

46

34.7

30.2

6

80. 693

6

125.3

108.9

6

170.6

148.3

6

915.8

187.6

47

35.5

30.8

7

80.8 70.2

7

196.

109.6

7

171.3

146.9

7

916.6

1883

48

36.9

31.5

8

81.5 70.9

8

196.8

110.2

8

172.1

149.6

8

917.4

188.9

49

37.

32.1

9

82.3 713

9

1973

110.9

9

1793

1503

9

918.1

1693

50

37.7

39.8

110

83. 72.9

170

1983

1113

930

173.6

150.9

900

9183

1903

51

38.5

33.5

1

83.8 72.8

1

129.1

112.2

1

1743

151.5

1

919.6

190.9

59

39.2

34.1

9

843 733

9

1293

112.8

2

175.1

152.2

9

920.4

1913

53

40.

34.8

3

853 74.1

3

130.6

113.5

3

1753

152.9

3

921.1

199.9

54

40.8

35.4

4

86. 743

4

131.3

114.2

4

176.6

153.5

4

9913

1993

55

415

36.1

5

86.8 75.4

5

132.1

114.8

5

177.4

1543

5

999.6

1933

56

42.3

36.7

6

17.5 76.1

6

1323

115.5

6

178.1

154.8

6

993.4

194.9

57

43.

37.4

7

88.3 76.8

7

1333

116.1

7

178.9

1553

7

994.1

1943

58

43.8

38.1

8

89.1 77.4

8

1343

116.8

8

1793

156.1

8

924.9

1953

59

44.5

38.7

89.8 78.1

9

135.1

117.4

9

180.4

156.8

9

995.7

196.9

60

453

39.4

190

90.6 78.7

180

135.8

118.1

940

181.1

1573

300

99&4

1983

diet.

dep.

d. lat.

dist.

dep. d.lat.

diit.

dep.

d. lat.

dirt.

dep.

d. lat.

dirt.

dep.

d. let

Pittance, Departure and Diff Latitude.

Digitized by VjOOQIC

152

T4BLX V.

Covraa 400.
Qiafteaee, Diffi Latitude and Departure.

diet.

d.lat.

dep.

diet.

d.!at*

dep.

diet.

d. lac

dep.

dial.

d.lat;

dep.

dial.

d. lat. dap.

1

0.7

0.7

61

45.3

404

iai

895

81.

181

1344

131.1

Ml

170.1 1615

S

1.5

1.3

63

46.1

41.5

3

90.7

81.6

9

1855

1314

9

1705 1615

3

94

9.

63

46.8

43.8

3

91.4

88.3

3

130.

1934

3

1804 169.6

4

3.

9.7

64

47.6

434

4

92.1

83.

4

136.7

193.1

4

1814 163.3

5

3.7

3.3

65

48.3

43.5

5

93.9

83.6

5

137.5

1234

5

189.1 1635

6

4.5

4.

66

40.

44.3

6

93.6

845

6

1385

1345

1834 1644

7

5.9

4.7

67

404

444

7

04.4

85.

7

139.

135.1

7

1034 1655

8

5.9

5.4

68

50.5

455

8

95.1

85.6

8

139.7

1354

S

1845 1655

ft

6.7

6.

69

513

465

9

855

864

9

140.5

1364

9

185. 1664

10

7.4

6.7

70

59.

464

130

98.6

87.

190

1415

197.1

aso

1855 1675

11

85

7.4

71

53.8

474

1

97.4

87.7

1

1414

1374

i

1865 168.

19

8.9

8.

73

53.5

485

3

98.1

88.3

3

149.7

1984

3

1874 1684

13

9.7

8.7

73

54.3

484

3

98.8

89.

3

143.4

130.1

3

189. 1605

14

10.4

94

74

55.

49.5

4

094

89.7

4

1445

)3t>4

4

1884 170.

15

Jl.l
11.9

10.

75

55.7

505

5

1005

90.3

5

144.9

1305

5

1895 1704

16

10.7

76

564

50.9

6

101.1

91.

6

145.7

131.1

6

1904 1715

17

19.6

11.4

77

57.9

515

7

101.8

91.7

7

146.4

1314

7

101. 179.

18

13.4

13.

78

58.

53.3

8

1034

99.3

8

147.1

1334

8

191.7 1794

10

14.1

19.7

79

58.7

595

9

1035

93.

9

1475

1335

1995 1735

*0

144

1X4

80

59 5

535

140

104.

93.7

ftOO

1464

1334

8)6©

1935 174.

91

15.6

14J

81

605

545

1

104 8

94.3

1

1494

1345

1

194. 1744

99

16.3

14.7

83

60.9

54.9

3

105.5

95.

9

150.1

135.4

9

194.7 1755

93

17.1

154

83

61.7

55.5

3

1065

05.7

3

150.9

1354
1365

3

195.4 176.

94

17.8

16.1

84

63.4

565

4

107.

06.4

4

1514

4

1964 176.7

95

18.6

16.7

85

63.3

56.9

5

107.8

97.

5

1595

1375

5

196.0 1775

96

19.3

17.4

86

63.9

57.5

6

106.5

97.7

6

153.1

1375

6

197.7 178.

97

90.1

18.1

87

64.7

585

7

1094

98.4

7

1534

138.5

7

196.4 178.7

98

90.8
91.0

18.7

88

654

58.9

8

110.

99.

8

1544

1394

8

199.9 1795

99

19.4

69

66.1

594

110.7

90.7

9

1555

1394

1905 18a

30

99.3

90.1

00

00.9

605

ISO

1115

100.4

910

150.1

1405

»T0

9005 180.7

31

93.

30.7

91

67.6

60.9

1

U85

101.

1

1504

1415

1

901.4 1815

33

93.8

91.4

93

68.4

61.6

9

113.

101.7

9

157.5

1414

3

903.1 189.

33

94.5

23.1

93

69.1

62.8

3

113.7

103.4

3

1585

1494

3

9095 189.7

34

95.3
96.

33.8

94

69.9

634

4

114.4

103.

4

159.

1435

4

9086 1835

35

93.4

95

70.6

63.6

5

115.2

103.7

5

1594

143.9

5

904.4 184.

36

96.8

S4.1

06

71.3

64.3

6

115.0

104.4

6

1605

1445

6

906.1 184.7

37

97.5
98.9
99.

94.8

97

73.1

644

7

116.7

105.1

7

1614

1455

7

9064 1855

38

95.4

98

79.8

65.6

8

117.4

105.7

8

163.

145.0

8

806.6 180,

39

9&1

99

no

665

9

1185

106.4

9

109.7

1404

9075 186.7

4VO

99.7

964

100

74.3

66.9

160

1185

107.1 89)0

1634

1475

9j80

906.1 187.4

41

30.5

97.4

1

75.1

67.6

1

119.0

107.7

1

1645

147.9

1

9084 188.

49

315

93.1

8

75.8

6B.3

9

130.4

108.4

9

165.

148.5

9

909.6 188.7

43

39.

98.8

3

76.5

68.9

3

181.1

109.1

3

165.7

149.3

3

9105 189.4

44

39.7

39.4

4

77.3

694

4

121.0

100.7

4

1664

149.9

• 4

811.1 190.

45

33.4

30.1

5

78.

70.3

5

133.6

110.4

5

167.9

150.6

5

311.8 190.7

46

34.3

30.8

6

78.8

70.9

6

183.4

111.1

6

168.

151.3

6

3185 191.4

47

34.9

31.4

7

79.5

71.6

7

124.1

111.7

7

168.7

151.9

7

3135 199.

48

35.7

39.1

8

80.3

735

8

1844

118.4

8

169.4

153.6

8

914. 199.7

49

36.4

334

9

8L

735

9

135.6

113.1

9

1705

153.9

9

9145 1934

§o

37.9

335

110

81.7

73.6

170

186.3

1134

930

170.0

1534

9J90

3155 194.

51

37.9

34.1

1

89.5

74.3

1

187.1

114.4

1

171.7

154.6

1

9165 194.7

59

38.6

344

9

835

74.9

9

187.8

115)

9

179,4

1555

9

917. 1954

53

39.4

35.5

3

84.

75.6

3

148.6

115 8

3

173.3

155.9

3

917.7 196.1

54

40.1

36.1

4

84.7

76.3

4

139.3

116.4

4

173.9

156.6

4

3185 196.7

.55

40.9

36.8

5

85.5

77.

5

130.1

117.1

5

174.6

1575

5

2195 197.4

56

41.6

374

6

86.9

77.6

6

130.8

117.8

6

175.4

1579

6

990. 198.1

57

40-4

38.1

7

86.9

78.3

7

1315

118.4

7

176.1

158.6

7

990.7 198.7

58

43.1

384

8

87.7

79.

8

1395

119.1

8

176.9

150.3

8

9914 199.4

59

43.8

30.5

9

88.4

70.6

9

133.

1194

9

1774

159.9

883.9 900.1

60

44.0

40.1

190

895

80.3

180

1334

180.4

940

178.4

100.6

300

9995 800.7

diet*

de*.

d.lat

diet.

dep. d. lat.

diet.

dep.

Hat.

diet.

dep.

Hat.

diet.

Idep. d.lat»

Diataaee,De|tartaieaBdIHCLaiit«de.

Digitized by VjOOQIC

TABLTY.

158

Dirtajue, DUE Latitade and Dtpaitam

dirt.

d-lat.

(fop.

dirt.

d.lat dep.

dirt.

d-lat

dap.

dirt.

d.laL

dep. | dirt.

d.lat dep.

1

6.7

0.7

61

444 41.6

iai

884

844

181

139.4

193.4 941

1764 1644

8

14

1.4

68

454 484

8

894

819

9

1311

194.1

3

177. 161

a

34

8.

63

46.1 43.

3

90.

819

3

1334

1944

3

177.7 1617

4

8.8

8.7

64

464 416

4

90.7

84.6

4

1344

1954

4

1784 166.4

«

3.7

3.4

65

474 444

6

91.4

854

5

1354

1*64

5

1794 167.1

•

4.4

4.1

66

484 45.

*6

919

85.9

6

136.

196.9

6

179.9 1674

7

5.1

44

67

49. 45.7

7

919

86.6

7

1364

197.5

7

180.6 168.5

8

5.8

54

68

49.7 46.4

8

916

874

8

1374

1964

8

1814 169.1

64

6.1

69

504 47.1

9

944

81

9

1313

1984

9

1611 J694

10

74

64

VO

514 47.7

ISO

95.1

88.7

100

138.

199.6

980

1818 1704

XX

&

74

71

514 48.4

1

954

894

1

1317

1304

1

1816 171.3

n

84

19

79

58.7 49.1

9

964

90.

9

140.4

1304

9

1844 171.9

13

94

8.9

73

514 494

3

974

917

3

1414

1314

3

185. 1715

14

10JI

9.5

74

54.1 504

4

91

91.4

4

141.9

1384

4

1854 1713

15

11.

104

75

544 51.1

6

98.7

911

5

1416

131

6

1864 1719

16

11.7

104)

76

554 514

6

994

918

6

1434

133.7

6

1874 174.6

17

19.4

11.6

77

564 584

7

1004

93.4

7

144.1

134.4

7

188. 1754

18

134

18-3

78

57. 518

8

100.9

94.1

8

1444

135.

8

1817 176.

19

134

13.

79

574 534

9

10X7

944

9

1454

1317

9

1884 1716

SO

14.6

13.6

80

584 544

146

1094

954

900

1464

1314

900

1964 1774

81

15.4

144

81

594 554

1

1011

98.8

1

147.

137.1

I

1964 171

»

161

15

88

60. 55.9

3

1019

98.8

3

147.7

137.8

9

191.6 1717

93

164

15.7

83

60.7 56.6

3

104.6

97.5

3

1484

138.4

3

1994 179.4

84

17.6

16.4

84

614 574

4

1054

984

4

1494

139.1

4

1911 180.

85

113

17.

85

684 58.

5

106.

989

5

149.9

1394

5

1934 lfc0.7

86

19.

17.7

86

68.9 58.7

6

1064

99.6

6

150.7

1404

6

1944 181.4

87

19.7

18.4

87

616 594

7

1074

1004

7

151.4

1414

7

1954 1811

88

30.5

19.1

88

64.4 68.

8

1084

100.9

8

1511

1414

8

191 1894

88

914

194

88

65.1 617

8

109.

101.6

9

1594

1415

9

1917 1834

80

81.9

904

00

654 61.4

190

169.7

1813

910

1516

1434

970

1974 184.1

31

88.7

31.1

91

66.6 611

1

110.4

101

1

1544

1434

1

1919 184.8

38

83.4

91.8

99

67.3 617

9

111.3

1017

9

155.

144.6

9

1919 185.5

33

84.1

88.5

93

68. 614

3

1114

104.3

3

1554

145.3

3

199.7 1864

34

84.9

818

94

617 64.1

4

1116

105.

4

1564

1419

4

9604 186.9

33

85.6

819

95

694 64.8

5

1114

195.7

5

1574

1416

5

901.1 1874

33

964

94.6

96

704 654

6

114.1

108.4

6

15ft

1474

6

901.9 1884

37

87.1

85.3

97

70.9 664

7

1144

107.1

7

158.7

141

7

9016 188.9

38

878

85.8

96

71.7 664

8

115.6

107.8

8

J59.4

1417

8

9034 189.6

38

aw

864

99

794 674

9

1164

1014

9

1604

149.4

9

904. 1904

40

893

874

100

711 684

100

117.

109.1

9390

M0.9

150.

980

9044 191.

41

30.

88.

1

734 68.9

1

117.7

1094

1

1614

150.7

1

8054 191.6

48

30.7

98.6

3

74.6 69.6

3

1184

110.5

3

1614

151.4

9

9084 1994

43

314

89.3

3

754 704

3

1194

1113

3

1611

1511

3

907. 191

44

319

30.

4

78.1 719

4

1194

111.6

4

1618

1594

4

907.7 1917

45

38.9

30.7

5

764 71.6

5

190.7

1115

5

164.6

1514

5

908.4 194.4

46

33.6

31.4

6

774 784

6

191.4

1)19

6

1654

1541

6

9094 195.1

47

34.4

311

7

784 71

7

199.1

1119

7

161

154.8

7

900.9 1917

48

35.1

317

8

79. 717

8

199.9

114 6

8

166.7

155.5

8

9104 1964

48

354

33.4

9

79.7 744

9

1934

1154

9

1674

1564

9

9114 197.1

BO

364

34.1

110

80.4 75.

170

1944

115.9

930

1684

1519

900

9111 1974

51

374

344

1

8J4 75.7

1

195.1

116.6

1

1684

1574

1

9194 1984

58

38.

354

9

81.9 78.4

9

1954

1174

*

169.7

1519

9

8116 199.1

53

384

36.1

3

884 77.J

3

1964

118.

3

170.4

1519

3

914.3 1994

54

394

364

4

814 77.7

4

1974

1117

4

171.1

159.6

4

315. 9004

55

404

375

5

84.1 78.4

5

198.

1194

5

171.9

160.3

5

915.7 9014

56

41.

38.8

6

844 79.1

6

198.7

190.

6

1794

161.

6

9164 9014

57

41.7

38.9

7

816 794

7

199.4

190.7

7

1713

161.6

7

9174 9016

58

48.4

39.6

8

86.3 804

8

1304

191.4

8

1741

1613

8

9174 9034

58

43.1

40.9

9

87. 814

9

130.9

199.1

9

1744

163.

9

9117 9019

60

43.9

40.9

180

874 814

180

131.6

1918

940

1754

1617

300

9194 9044

dirt.

dep.

d. lat.

dirt.

dep. d. lat.

dirt. 1 dep.

d. lat.

dirt.

dep.

i. lat.

dirt.

dep. d. lat.

DieUBOT, Departure and Diff Latitude

<2emree4T0.

u

Digitized by VjOOQIC

164

TABLE V.

Oomrea *4°.
Distance, Difi Latitude mod Departure.

dirt.

(JUlmt.

dep.

dirt.

d.lat.

dep.

dirt

d.lat.

dep.

dirt.

d. *t.

dep.

dirt.

d.lat.

*»

1

0.7

0.7

61

43.0

424

191

87.

84.1

181

1303

135.7

941

173.4

187.4

3

1.4

1.4

63

44.6

43.1

3

87.8

84.7

3

130.9

186.4

8

174.1

168.1

3

3.3

8.1

63

45.3

433

3

885

85.4

3

131.6

187.1

3

1743

1883

4

8.9

3.8

64

46.

44.5

4

89.8

86.1

4

133.4

137.8

4

1755

1693

5

3.6

33

65

46.8

453

5

89.9

86.8

5

133.1

1483

5

1763

1703

6

4.3

4.3

66

47.5

45.8

6

90.6

873

6

133.8

1393

6

177.

1703

7

5.

4.9

67

48.3

46.5

7

91.4

88.3

7

1343

189.9

7

177.7

1713

8

5.8

5.6

68

483

47.9

8

93.1

88.9

8

1353

130.6

8

178.4

1733

9

63

6.3

69

49.6

47.9

9

933

89.6

9

136.

131.3

9

179a

173.

10

7.3

6.9

70

50.4

48.6

130

933

90.3

100

136.7

133.

990

1793

173.7

11

7.9

7.6

71

51.1

49.3

1

943

91.

1

137.4

133.7

1

1803

174.4

13

8.6

8.3

78

514

50.

3

95.

91.7

8

138.1

133.4

3

1813

179.1

13

9.4

9.

73

58.5

50.7

3

95.7

98.4

3

1383

134.1

3

183.

175.7

14

10.1

9.7

74

533

51.4

4

98.4

93.1

4

1393

1343

4

183.7

170.4

15

10.8

10.4

75

54.

53.1

5

97.1

933

5

1403

1353

5

183.4

177.1

16

11.5

11.1

76

54.7

53.8

6

973

945

6

141.

1363

6

1843

1773

17

J3.S

11.8

77

55.4

533

7

985

95.9

7

141.7

136.8

7

1843

1783

18

12.9

13.5

78

56.1

543

8

99.3

95.9

8

143.4

1373

8

1853

1793

10

13.7

13.3

79

583

54.9

9

100.

96.6

9

143.1

138.3

9

1863

1783

90

14.4

13.9

80

SIS

55.6

140

100.7

973

9JO0

143.9

138.9

SJ60

187.

180.6

si

15.1

14.6

81

583

563

1

101.4

97.9

1

144.6

1393

1

187.7

1813

93

153

15.3

83

59.

57.

8

103.1

98.6

3

1453

1403

3

1883

188.

33

16.5

16.

83

59.7

57.7

3

108.9

993

3

146.

141.

3

1893

183.7

34

17.3

16.7

64

60.4

58.4

4

103.6

100.

4

146.7

141.7

4

1893

1*3.4

35

18.

17.4

85

61.1

59.

5

104.3

100.7

5

1473

1434

5

190.6

184.1

36

18.7

18.1

* 86

613

59.7

6

105.

101.4

6

148.3

143.1

6

1913

1843

87

19.4

18.8

87

68.6

60.4

7

105.7

109.1

7

148.9

1433

7

193.1

1853

3d

30.1

•19.5

88

63.3

61.1

8

1065

103.8

8

149.6

1443

8

1938

1603

39

30.9

30.1

89

64.

613

9

1073

1033

9

150.3

1453

9

1933

1883

30

31.6

30.8

00

64.7

63.5

190

107.9

1043

910

151.1

145.9

970

1943

1873

31

84.3

315

91

65.5

63.9

1

108.6

104.9

1

151.8

146.6

1

194.9

1883

33

33.

833

93

663

63.9

3

1093

105.6

3

153.5

1473

8

195.7

168.9

33

33.7

83.9

93

66.9

64.6

3

110.1

W6.3

3

153.3

148.

3

196.4

1893

34

34.5

93.6

94

67.6

65.3

4

1103

107.

4

153.9

148.7

4

197.1

1903

35

353

843

95

683

66.

5

1113

107 7

5

154.7

149.4

5

1973

191.

36

853

85.

98

69.1

66.7

6

1133

108.4

6

155.4

150.

6

1983

1917

37

36.6

85.7

97

693

67.4

7

113.9

109.1

7

156.1

150.7

7

1993

198.4

38

37.3

86,4

98

704

68.1

8

113.7

109.8

8

1563

151.4

8

300.

193.1

39

88.1

87.1

99

713

68.8

9

114.4

110 5

9

1573

153.1

9

300.7

1933

40

383

87.3

100

71.9

693

160

115.1

lll.l

990

1583

1593

980

3014

1943

41

39.5

883

1

73.7

70.8

1

1153

111.8

i

159.

153.5

1

303.1

1953

43

30.3

89.3

3

73.4

70.9

3

1163

113.5

3

159.7

154.3

3

304.9

1953

43

30.9

89.9

3

74.1

713

3

117.3

113.3

3

160.4

154.9

3

303.6

1983

44

31.7

30.6

4

748

73.3

4

118.

113.9

4

161.1

155.6

4

8043

1973

45

33.4

31.3

5

755

73.9

5

118.7

114.6

5

161.9

156.3

5

805.

198

46

33.1

33.

6

76.3

73.6

A

119.4

115.3

6

163.6

157.

6

305.7

198,7

47

33.8

33.6

7

77.

74.3

7

180.1

116.

7

1633

157.7

7

3063

198 4

48

34.5

33.3

8

77.7

75.

8

1803

116.7

8

164.

158.4

8

8073

£00.1

49

35.3

34.

9

78.4

75.7

9

131.6

117.4

9

164.7

159.1

9

3073

9003

50

36.

34.7

110

79.1

76.4

190

1833

118.1

9)30

165.4

1593

900

3083

8013

51

36.7

35.4

1

793

77.1

1

133.

118.8

1

1663

160.5

1

309.3

9U9.I

53

37.4

36.1

3

80.6

77.8

3

133.7

119.5

8

166.9

161.3]

3

810.

8033

53

38.1

36.8

3

81.3

78.5

3

184.4

180.4

3

1673

161.9

3

310.8

8033

54

38.8

37.5

4

83.

79.3

4

1858

130.9

4

1683

163.6

4

3115

3043

55

39.6

38.3

5

83.7

7».9

5

135.9

131.6

5

169.

163.3

5

3183

8043

56

40.3

389

6

834

806

6

1363

133.3

6

169.8

163.9

6

3139

805.6

57

41.

39.6

7

84.3

81.3

7

1873

133.

7

1703

164.6
1653

7

813.6

8063

58

41.7

40.3

8

84.9

83.

8

138.

133.fi

8

1713

9

314.4

807.

59

43.4

41.

9

85.6

83.7

9

188.8

1343

9

171.9

166.

9

3151

807.7

60

43.3

41.7

190

663

83.4

180

1393

185.

840

173.6

166.7

300

8153

808.4

dirt.

dep.

d.lat

dirt.

dep.

d. let.

dirt.

dep.

d let

dirt.

dep.

d. let.

dirt.

dep.

llati

Dirtauee, Departure and Diff Latitude.
Ctwea 48*.

Digitized by VjOOQIC

TABLE ▼.

C««rM 49°.
Distance, Di€ Latitude tad Departure

155

dteL

d.lat.

dap.

dist.

d.lat.

dap.

dill

d.lat.

dap.

dial

d.lat.

dap.

diat.

d.lat.

dap.

1

0.7

0.7

61

43.1

43.1

101

85.6

85.6

181

138.

198.

041

1704

170.4

8

1.4

1.4

88

434

434

864

804

198.7

188.7

171.1

171.1

a

9.1

8.1

63

444

444

87.

87.

199.4

1804

1714

1714

4

84

84

64

454

454

87.7

87.7

130.1

130.1

178.5

1784

5

34

35

65

46.

46.

884

884

1304

1304

173.9

173.3

6

44

44

66

46.7

46.7

89.1

89.1

1314

1314

173.9

173.9

7

4.9

4.9

67

47.4

474

894

80.8

132.9

138.8

174.7

174.7

8

5.7

5.7

68

48.1

48.1

904

904

139.9

139.9

1754

175.4

9

8.4

8.4

69

484

484

914

914

1334

133.6

176.1

176.1

10

7.1

7.1

90

494

494

ISO

914

91.9

100

1344

134.4

090

1764

1764

u

74

74

71

504

594

99.6

984

135.1

135.1

1774

1774

18

84

84

78

504

504

913

934

1354

1354

1784

1784

13

94

94

73

514

514

94.

94.

1364

1364

1784

1784

14

9.9

94

74

584

594

944

944

1374

1379

179.6

179.6

15

10.6

104

75

53.

53.

954

954

137.9

137.9

1804

1804

16

114

114

76

53.7

53.7

96.9

96.9

138.6

138.6

181.

181.

17

19.

18.

77

54.4

54.4

964

96.9

1394

1394

131.7

181.7

10

19.7

18.7

78

554

55.9

97.6

974

140.

140.

188.4

188.4

19

13.1

134

79

554

554

984

984

1407

140.7

183.1

183.1

00

14.1

14.1

80

564

564

140

99.

99.

000

1414

1414

060

1834

1834

91

143

144

81

574

574

99.7

99.7

148.1

148.1

1844

184.6

it

15.6

154

89

58.

58.

1004

1004

1494

1484

1854

1854

»

164

164

83

58.7

58.7

101.1

101.1

1434

143.5

186.

186.

84

17.

17.

84

59.4

594

101.8

1014

1444

1444

186.7

186.7

S3

17.7

17.7

85

69.1

60.1

1094

1094

145.

145.

187.4

1874

96

18.4

18.4

85

604

604

1034

1034

145.7

145.7

188.1

168.1

fl7

19.1

19.1

87

614

614

103.9

103.9

1464

1464

1884

1A84

SB

19.8

194

88

69.9

684

104.7

104.7

147.1

147.1

1894

1894

89

90-5

804

89

684

684

105.4

105.4

1474

1474

1904

1904

oo

914

814

00

634

834

100

106.1

106.1

010

1484

1484

oyo

190.0

1904

ai

SL9

814

91

644

644

1064

1064

1494

1404

191.6

191.6

at

884

88.6

99

65.1

65.1

1074

1074

149.9

1494

1994

1994

33

834

834

93

654

654

1084

1084

150.6

150.6

193.

193.

34

94.

84.

94

664

864

1084

1084

1514

1514

193.7

193.7

as

94.7

84.7

95

674

674

109.6

109.6

159.

159.

1944

1944

34

955

954

98

674

67.9

1104

1104

•158.7

159.7

1954

1954

37

953

884

97

684

684

111.

111.

153.4

153.4

195.9

195.9

38

364

854

98

694

694

111.7

111.7

154.1

154.1

1964

1964

33

974

874

99

70.

70.

118.4

118.4

1544

1544

1974

1974

40

864

884

100

70.7

70.7

160

113.1

113.1

008

1554

155.6

OOO

198.

198.

41

99.

89.

1

71.4

71.4

1134

1134

1564

1564

198.7

19*7

43

99.7

99.7

9

78.1

79.1

114.6

1144

157.

157.

199.4

199.4

43

39.4

35.4

3

784

784

1154

115.3

157.7

157.7

890.1

900.1

44

31.1

31.1-

4

734

734

116.

116.

158.4

158.4

8004

8004

49

31J8

314

5

744

744

116.7

116.7

159.1

159.1

8014

8014

43

384

394

6

75.

75.

1174

1174

1594

1594

808.9

8084

47

334

334

7

75.7

75.7

118.1

118.1

1604

1604

808.9

8084

48

334

33.9

6

784

764

1184

1184

1614

1614

8034

8034

49

344

34.6

9

77.1

77.1

1194

119.5

1614

1614

8044

8944

00

354

35.4

119

774

774

170

1804

1804

008

1684

1684

000

805.1

805.1

51

38.1

36.1

1

784

785

180.9

1904

1634

1634

8054

8054

JO

354

364

8

794

794

1914

181.6

164.

164.

8054

8064

fl

374

374

3

794

794

1894

1894

1644

1644

8074

9974

54

384

394

4

894

894

183.

183.

1654

1655

8074

8979

55

334

35.9

5

814

814

183.7

183.7

1664

1664

8084

8684

58

394

394

6

88.

89.

1844

1844

1664

166.9

8094

9094

87

494

494

7

89.7

89.7

1854

1854

1674

1674

810.

810.

58

4L

41.

8

834

83.4

1854

1854

1884

1684

8M.7

910.7

59

417

417

9

84.1

84.1

1864

1864

169.

189.

8114

9114

88

484

484

ISO

844

844

180

1874

1874

840

169.7

169.7

300

818.1

818J

diat.

fr

d.lat

Idirt.

W

d.lat

diet.

dap,

d.lat

diat. dap.

d.lat

diat.

*8>

6.1st.

nataaaa.OapaitaraaadDtCLaUtada.

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE VI.

MERIDIONAL PARTS.

Digitized by VjOOQIC

Digitized by VjOOQIC

tabu n.

159

1

8©

1©

SO

So

40

50

60

70

80

OO

10O

lio

18©

ISO

MO

/

00

ISO

180

MO

300

861

481

483

549

603

664

785

787

848

1

1

61

81

81

41

01

69

99

83

43

04

65

86

88

50

1

9

9

68

99

89

49

09

63

93

84

44

05

68

87

89

51

8

3

3

63

83

83

43

03

64

94

85

45

06

67

88

90

59

3

4

4

64

84

84

44

04

65

85

86

46

07

68

89

91

53

4

5

5

65

95

85

45

' 05

66

86

87

47

08

69

30

98

54

5

6

8

66

88

86

46

06

67

87

88

48

09

70

31

93

55

6

7

7

67

87

87

47

07

68

88

89

49

10

71

38

94

58

7

8

8

68

88

88

48

08

09

SB

90

50

11

78

33

95

57

8

9

69

SB

89

49

09

70

30

91

51

IS

73

35

96

58

10

10

70

30

SO

50

10

71

31

99

59

13

74

36

97

59

10

1J

11

71

31

91

51

11

79

39

93

53

14

76

37,

98

60

11

IS

18

78

39

99

59

19

73

33

94

54

15

76

38

99

61

18

13

13

73

33

93

53

13

74

34

95

55

16

77

30

800

63

13

J4

14

74

34

94

54

14

75

35

96

58

17

78

40

01

63

14

15

15

75

35

95

55

15

76

30

97

57

18

79

41

OS

64

15

16

10

76

36

96

56

16

77

37

98

58

19

80

48

03

65

16

17

17

77

37

97

57

17

78

38

99

59

80

81

43

04

66

17

18

18

78

38

98

58

18

79

30

500

60

81

88

44

05

67

18

19

19

79

39

99

59

19

80

40

01

61

88

83

45

06

68

19

SO

80

80

40

900

60

90

81

41

09

69

83

84

46

07

69

80

81

81

81

41

01

61

91

89

49

03

63

84

85

47

08

70

81

a

SS

88

48

09

69

99

83

43

04

65

95

86

48

00

71

99

S3

83

83

43

03

63

83

84

44

05

68

SO

88

49

10

79

93

84

84

84

44

04

64

84

85

45

06

67

87

89

50

11

73

84

SS

SS

85

45

05

65

85

86

46

07

68

88

90

51

19

74

85

SO

98

86

46

08

66

96

87

47

08

69

99

01

58

13

75

86

87

87

87

47

07

67

87

88

48

09

70

31

98

53

14

76

97

88

88

88

48

08

68

88

88

49

10

71

39

93

54

16

77

98

89

88

89

49

09

69

99

90

50

11

79

33

94

55

17

78

29

30

30

90

50

10

70

30

01

51

19

73

34

95

56

18

79

30

31

31

91

51

11

71

39

98

59

13

74

33

96

57

19

80

31

38

39

98

59

19

79

33

93

53

14

75

36

97

58

80

89

38

33

33

93

53

13

73

34

94

54

15

76

37

98

50

31

83

33

34

34

94

54

14

74

35

95

55

16

77

3B

99

60

89

84

34

35

35

95

55

15

75

36

96

58

17

78

39

700

61

83

85

35

38

38

98

50

16

76

37

97

57

18

79

40

01

68

*4

86

36

37

37

97

57

17

77

38

98

58

IB

80

41

09

63

85

87

37

38

38-

98

58

18

78

SB

99

59

90

81

49

03

64

86

88

38

38

39

99

59

19

79

40

400

60

91

89

43

04

65

87

89

39

40

40

100

60

80

80

41

01

61

SS

83

44

OS

60

SB

90

40

41

41

01

61

81

81

49

09

69

83

84

45

06

67

SB

01

41

48

48

09

69

89

&i

43

03

63

94

85

46

07

68

30

99

49

43

43

03

63

S3

83

44

04

64

95

86

47

08

60

3i

93

43

44

44

04

64

84

84

45

05

65

96

87

48

09

70

39

94

44

45

45

05

65

S5

65

46

06

66

97

88

49

10

71

33

95

45

46

46

06

66

86

86

47

07

67

*8

89

50

11

78

34

96

46

47

47

07

67

87

87

48

08

68

99

90

51

19

73

35

87

47

48

48

08

68

98

88

40

09

60

30

01

59

13

74

36

98

48

49

49

09

60

99

89

50

10

70

31

09

53

14

75

37

99

49

50

50

10

70

30

90

51

11

71

39

93

54

15

76

38

900

50

51

51

11

71

31

01

59

19

79

33

94

55

16

78

3B

01

51

58

59

19

78

39

99

53

13

73

34

95

58

17

70

40

09

59

53

53

13

73

33

93

54

14

74

35

96

57

18

80

41

03

53

54

54

14

74

34

94

55

15

75

36

97

58

19

81

49

04

54

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80

06

54

44

36

89

85

88

88

85

90

37

35

36

13

96

81

68

56

46

37

31

86

84

84

86

31

39

36

37

14

98

83

69

57

47

38

33

88

36

88

SB

33

41

37

38

15

99

84

71

59

49

40

34

30

37

87

30

35

48

Si

39

17

3000

85

73

60

50

41

36

31

89

80

38

97

44

30

40

18

03

87

73

68

59

43

37

33

31

31

33

38

46

49

41

19

03

88

75

63

53

45

30

34

33

38

35

40

48

41

48

81

05

90

76

65

55

47

40

36

34

34

37

49

59

48

43

38

06

91

78

66

5ti

48

49

38

36

36

38

44

59

43

44

34

07

93

79

68

58

50

43

39

37

38

40

45

53

44

45

35

09

94

81

69

59

51

45

41

39

30

48

47

55

45

46

96

JO

95

83

71

61

53

47

43

41

41

44

49

57

49

47

88

13

97

84

73

63

54

48

44

43

43

45

51

50

47

48

89

13

98

85

74

64

56

50

46

44

44

47

59

61

49l

49

31

14

3100

87

75

65

57

51

47

46

46

49

54

68

• f

50

33

16

01

88

77

67

59

53

49

47

48

51

56

94

591

51

33

17

03

90

78

68

60

55

51

40

40

53

58

69

» .

53

35

19

04

91

80

70

68

56

58

50

51

54

CO

99

59

53

36

80

05

93

81

71

64

58

54

59

53

56

61

79

53

54

37

81

07

94

83

73

65

59

55

54

54

56

63

79

54

55

39

83

08

95

84

74

67

61

57

55

56

59

65

73

55

56

40

84

10

97

86

76

68

69

59

57

56

61

67

75

59

57

49

86

11

96

87

78

70

64

60

59

60

63

60

77

57

58

43

87

13

3900

89

7P

71

66

68

60

61

64

70

79

59

59

9944

3089

3114

01

39903381

3473

3967

3664

3763

3863

3966

4078

4iei

59

i

430

440

450

460

470 48©

400

500

510

580

53*

540

550

590

9

Digitized by VjOOQIC

TABLE VI.

103

Meridional Part*

1

570

580

590

OOO

610

4649

O30
4975

630
4905

640

650

66O

670

68O

690

700

1

♦183

4994

4409

4537

5039

5179

5384

6474

5631

5795

6966

1

84

96

11

39

51

77

07

43

81

9b

77

33

97

69

1

9

80

98

13

31

53

79

09

44

64

98

79

36

5600

73

8

3

8e

4300

15

33

55

81

13

40

86

31

83

39

03

75

3

4

90

03

17

35

57

84

14

49

88

33

84

43

06

78

4

5

93

04

19

37

60

86

16

51

91

36

87

44

09

81

5

6

94

06

31

39

63

88

18

53

93

38

89

47

11

84

6

7

96

Ob

93

41

64

90

80

55

96

41

98

50

14

86

7

8

97

OB

85

43

66

93

83

58

98

43

95

53

17

fc9

8

9

911

11

87

45

66

94

95

60

5900

46

97

55

90

93

9

10

4901

13

99

47

70

96

87

69

03

48

5500

58

93

95

10

1J

03

15

31

49

78

98

99

05

05

51

03

60

85

98

11

18

05

17

33

51

74

4801

31

67

07

53

05

63

38

0001

13

13

07

18

34

53

76

03

34

69

10

58

07

66

31

04

13

14

08

SO

36

55

78

05

36

71

13

58

10

68

34

07

14

15

10

93

38

57

80

07

38

74

14

61

13

71

37

10

15

its

19

34

40

59

83

09

40

76

17

63

15

74

39

13

16

17

14

36

43

03

84

11

43

78

19

66

18

76

43

16

17

18

16

93

44

04

87

14

45

81

83

68

90

79

45

19

18

19

18

30

46

06

89

16

47

83

84

71

83

89

48

33

10

90

90

33

48

68

91

18

49

85

86

73

86

85

51

85

30

91

SI

34

50

70

93

30

51

68

39

76

88

87

54

36

31

99

93

3D

53

73

95

S3

54

90

31

78

31

90

56

31

33

93

95

38

54

74

97

84

56

93

34

80

33

93

59

34

33

94

97

40

56

76

99

96

58

95

36

83

36

95

63

37

34

95

90

49

58

78

4701

89

60

97

38

85

39

98

65

40

85

96

31

44

60

80

03

31

63

99

41

88

41

5701

68

43

36

97

a

45

03

83

05

33

65

5108

43

90

44

04

71

46

37

98

34

47

64

84

07

35

07

04

46

93

46

06

74

49

38

99

36

49

06

86

10

37

09

06

48

95

40

09

76

59

39

30

38

51

68

88

19

39

78

08

50

98

59

19

79

55

30

31

40

53

70

90

14

43

74

11

53

5401

54

15

83

58

31

39

49

55

78

93

16

44

76

13

55

03

57

17

85

61

33

33

44

57

74

94

18

46

78

15

58

06

59

90

88

64

33

34

46

50

76

96

90

48

81

18

60

08

63

93

91

67

34

35

47

•1

78

98

93

50

83

90

63

11

65

95

94

70

35

30

49

63

80

4600

94

53

85

83

65

13

67

*J8

96

73

36

37

51

05

83

03

36

55

87

85

67

16

70

31

99

76

37

38

53

07

84

04

38

57

90

87

70

18

73

34

5903

79

38

39

55

09

86

06

31

59

93

99

73

81

75

36

05

83

30

40

57

70

88

08

33

01

94

33

75

83

78

39

08

85

40

41

59

79

90

10

35

03

96

34

77

86

80

43

11

88

41

49

09

74

0-2

19

37

65

99

36

80

88

83

45

14

91

44

43

09

70

94

14

39

68

5001

39

88

31

86

47

17

94

43

44

04

78

95

16

41

70

03

41

84

33

88

50

19

97

44

45

06

80

97

IP

43

78

05

43

87

36

91

53

83

6100

45

40

08

83

99

90

45

74

08

46

89

38

94

56

35

03

46

47

70

84

4501

93

47

76

10

48

98

41

96

58

88

08

47

48

79

86

03

95

50

79

13

51

94

43

99

61

31

09

48

49

74

88

05

97

59

81

14

53

97

46

5603

64

34

13

49

50

75

90

07

99

54

83

17

55

99

48

04

67

37

15

50

51

78

99

09

31

56

85

19

58

5301

51

07

70

40

18

51

59

80

94

11

33

.58

87

31

60

04

54

10

78

43

31

53

53

89

96

13

35

60

90

33

03

06

58

13

75

46

84

53

54

84

98

15

37

69

98

86

65

09

59

15

78

48

87

54

55

85

90

17

39

04

94

98

67

11

61

17

81

51

30

55

5ft

87

4401

10

41

06

96

30

69

14

64

30

83

54

33

56

57

89

03

91

43

09

98

33

78

16

66

33

86

57

36

57

5*

91

05

33

45

71

4901

35

74

19

69

35

89

60

40

58

59

4993

07

45*25

4647

4773

03

5037

5176

5331

5471

5638

5793

9963

6143

59

/

570

58©

590

OOO

610

690

H

640

650

66O

670

680

690

TOO

1

Digitized by VjOOQIC

164

TABU VI.

Meridional Part*

f

710

730

730

740

750

760

77°

780

790

80O

810

880

830

84P

6146

6335

6534

6746

6970

79J10 7467 7745

B0468375

9739

9145

9606

10137

6

1

49

38

36

49

74

14

73

49

51

81

45

53

14

147

1

s

53

41

4J

53

78

18

76

54

56

87

58

60

98

158

f

3

55

45

45

57

88

S3

81

59

61

93

58

67

31

186 1

4

58

48

4b

60

86

87

85

64

67

98

65

74

39

K?

4

5

61

51

53

64

90

31

90 68

73 8404

71

88

47

Iff

5

6

64

54

55

68

94

35

94,' 73

77

10

78

89

55

195

8

7

67

58

58

71

97

39

98! 78

83

16

84

96

64

7

8

70

61

68

75

7001

43

7503J 83

88

S3

91

9903

79

815

8

9

73

64

65

79

05

47

07 88

93

87

97

11

69

994

•

10

77

67

69

88

09

53

18' 93

99

33 8804

18

89

934

»l

11

80

71

73

86

13

56

16 98

8104

39

10

95

97

944

111

12

83

74

76

90

17

60

817803

09

45

17

33

9706

954

Mi

13

86

77

79

93

81

64

85 06

15

51

83

40

14

964

Ul

14

89

80

83

97

85

68

30, 19

80

57

30

48

93

974

14

15

93

84

86

6801

89

73

35: 17

85

63

30

55

31

994

»!

16

95

87

90

04

33

77

39 88

31

69

43

63

40

994

»

17

98

90

93

08

37

81

44 87

36

74

49

70

48

394

17 »

18

69)01

94

97

18

41

85

48 33

41

80

56

77

57

314

1<

19

05

97

6600

15

45

89

53 37

47

66

63

65

65

394

It

SO

08

6400

03

19

48

94

57 48

53

98

69

98

74

331

»l

SI

11

03

07

83

59

98

08, 47

58

98

76

9300

83

345

si!

S3

14

07

10

86

56

7303

661 53

63 8504

83

07

91

355

911

S3

17

10

14

30

60

06

71 57

68

10

89

15

9809

365

83

84

SO

13

17

34

64

11

76

68

74

16

96

88

09

SB

94.

35

83

17

81

38

68

15

80

67

79

98

8903

30

17

385

95'

96

86

90

94

41

78

19

85

73

85

88

09

37

96

398

»l

87

30

83

88

45

76

S3

89

77

90

34

16

45

35

488

97

98

33

87

31

49

80

98

94

83

96

40

93

53

44

418

9*

89

36

30

35

53

84

33

99

87

8*01

46

30

60

59

497

98

30

39

33

39

56

88

36

7603

98

07

53

36

69

61

437

»•

31

48

37

48

60

93

41

08

97

18

56

43

76

74

448

».

39

45

40

46

.64

96

45

13

7903

18

65

50

83

79

459

a 1

33

49

43

49

68

7100

49

17

07

S3

71

57

91

88

488

s

34

58

47

53

71

04

53

83

18

99

77

63

99

97

479

*.

35

55

50

56

75

68

58

86

17

34

83

70

9407

9901

498

35

36

58

53

60

79

18

63

31

98

40

69

77

14

15

591

38

37

61

57

63

83

16

68

36

87

45

95

84

88

94

511

37

38

64

60

67

86

80

71

40

33

51

8601

91

30

33

591

3?

39

66

63

70

90

84

75

45

37

56

07

96

38

49

539

39"

40

v71

67

74

94

88

79

50

43

69

14

9005

45

SI

543

41

41

74

70

77

98

33

64

54

48

67

SO

19

53

68

554

41

43

77

73

81

6901

36

88

59

53

73

96

18

61

69

585

41

43

80

77

85

05

40

93

64

58

79

39

85

69

78

sn

43

44

83

80

88

09

45

97

68

63

84

36

39

77

87

44

45

87

83

98

13

49

7401

73

68

90

44

39

85

96

597

45

46

90

87

95

17

53

06

78

73

95

51

46

93

10005

688

48

47

93

90

99

80

57

10

83

78

8301

57

53

9501

015

611

47

48

96

94

6703

84

61

14

87

83

07

63

60

69

084

•

49

99

97

06

88

65

19

93

89

IS

69

67

17

033

641

«

50

6303

6500

10

39

69

S3

97

94

18

78

74

85

043

881

51

51

06

04

13

36

73

87

7703

99

94

89

81

33

059

863

51

53

09

07

17

40

77

31

06

8004

89

88

88

41

061

674

51

53

19

11

80

43

61

36

11

09

35

95

96

40

071

861

53

54

15

14

84

47

85

41

16

14

41

8701

9103

57

*».

55

19

17

88

51

89

45

81

80

47

07

10

65

97

2!

56

S3

31

31

55

94

49

85

85

53

14

17

73

091

7W

38 t

57

85

84

35

59

98

54

30

30

58

SO

94

81

108

738

57'

58

88

88

3P

63

7903

58

35

35

64

88

31

89

Iffl

749

51.

59

6S33

6531

6743

696t

06

7463 7740

8O40

8369

8733

9138

9598

10197

1079

38

i

710

78P

73©

740

750

7«o

770

780

790

800

61*

88O

830

840

'

Digitized by VjOOQIC

TABLE VH.

AMPLITUDES

Digitized by VjOOQIC

166

TABLE VH.
Amplitude*

Declination of the Sun.

lat.

0O

10

30
SO o'

30

40

50

60

70

80 ,

90

l«o

IP .*.

03

P 0'

33 0'

40

50

OO 0'

70 0'

80 0'

90 c

J4P *

IP • *

1

•

• 1

2

•

• t

3

1

1

1

1

1 1

4

1

1

1

1

1

1

I

9 i

5

1

1

1

1

8

8

8

8

3 *

6

1

1

1

3

8

s

3

3

3

4 t

7

J

1

8

3

3

3

4

4

5

5 7

8

1

8

3

3

4

4

5

5

6

7 -

y

1

3

3

4

5

5

6

7

6

a •

10

8

3

4

5

6

7

7

8

t

» u

n

3

3

4

6

7

8

9

10

11

13 ii

l*

3

4

5

7

8

11

13

14

15 ii

13

3

5

6

8

10

11

13

14

16

10 U

14

4

6

7

9

11

13

15

17

19

99 i«

15

3

4

6

8

11

13

15

17

19

21

94 ii

Hi

o •

3

5

7

10

12

15

17

19

83

84

97 l.

17

3

5

8

li

14

17

19

83

85

88

31 IT

Id

3

6

9

13

15

19

82

85

88

3

14 .-

12

3

7

10

14

17

81

34

88

31

15

* is

40

4

8

13

15

19

83

$7

31

35

13

43 21

41

4

9

13

17

SI

36

30

34

39

43

4* I

22

5

9

14

19

84

88

33

38

43

48

53 a

33

5

10

16

81

86

31

36

43

47

53

50 a

24

6

11

17

83

3d

34

40

46

53

57

19 3 t*

25

6

13

19

85

31

37

44

50

56

11 3

9 si

2d

7

14

80

87

34

41

48

54

10 1

8

15 t

27

7

15

Si

82

37

44

53

59

7

14

92 r

2d

8

16

•84

33

40

48

56

9 4

13

SI

*> *•

29

17

86

34

43

58

8 1

9

18

87

3» tt

30

19

87

37

47

56

5

15

84

14

44 31

31

10

80

30

40

50

7

10

SI

31

41

59 Ji

32

11

83

33

43

54

5

16

87

38

49

IS • Ji

33

IS

83

35

46

58

10

31

33

45

57

9 J}

34

13

35

37

50

6 3

15

87

40

53

IS 5

W H

35

13

87

40

53

6

*0

33

47

11 1

14

9* li

36

14

88

43

57

11

85

40

54

94

99 »

37

15

30

45

5 1

16

30

47

10 8

18

14

49) 37

38

16

32

48

5

31

37

54

10

87

44

14 1 39

39

17

34

53

9

86

44

9 1

19

37

55

13 3tf

40

18

37

55

13

33

51

9

88

47

11 6

* " .

41

80

39

59

18

38

58

18

38

58

18

3* 41 >

42

81

48

4 3

83

44

8 5

86

48

18 9

31

51 49 •

43

32

44

6

SB

51

13

36

58

81

44

15 7 43

44

83

47

10

34

5*

SI

45

11 9

34

5R

99 4«

45

35

50

15

40

7 5

30

55

81

47

14 13

39 45

4:i

Si

53

19

46

13

&>

10 6

33

13 1 .

89

57 «t

47

88

56

84

53

81

49

18

46

16 1

45

16 IS 47

48

30

59

8J

59

89

59

30

IS

31

15 S

14 4-

4J

31

3 3

35

6 6

38

9 10

43

15

*l

91

54 4>

50

33

7

40

14

48

83

56

30

14 5 1

49

17 19 39

51

35

11

46

83

58

34

11 10

47

84

18 1

3»|31

53

37

15

53

30

8 8

47

85

13 4

43 1

93

18 13:

53

40

19

59

39

80

10

41

88

15 4 1

46

*• ,53

51

43

84

5 7

49

33

15

58

48

86

17 11

57 M

55

45

83

14

59

44

30

IS 16

14 3

50

37

19 9m a

56

47

35

83

7 10

58

46

35

85

16 15

18 5

57 >

57

50

40

31

88

9 13

11 4

56

48

49

16

39 39 57

59

53

47

40

34

88

83

13 18

15 14

17 10

19 8

91 • >

5 >

57

53

50

47

45

43

41

41

41

49

45 5

no

.

9,0 o

40

60

80 1

100 9

ISO 4

140 6

MP 10

180 14

990 19

*9o*> it

lat.

<p

P

SO

30

40

50

60

70

SO

•0

MO

110 tot

Do

elinatio

n of the

Ban.

1

Digitized by VjOOQIC

TABLE VII.

•

167

r

Amplitude**

Declination of the Sun.

tat.

1V>

130

MO

150

I60

170

ISO j

IO©

SOO

810

88O

230

let.
"io*

o°

.90 0'

130 p

140

150 0*

160

170 0'

ISO O'llbO

SO© O'iio

88° 0'

430

1

4)

1

3

1

1

I

1

1

I

1

1

1

s

3

1

1

1

1

1

3

3

8

8

• 8

8

3

4

9

8

9

9

8

3

3

3

3

3

3

4

4

5

3

3

3

4

4

4

4

5

5

5

5

5

ft

4

4

5

5

5

7

7

7

8

8

7

*

6

7

7

8

8

9

9

10

10

11

7

8

7

8

8

9

10

10

11

18

18

13

14

14

8

9

t

10

U

11

IS

13

14

15

10

16

17

18

9

io

11

19

13

14

15

10

17

18

19

SO

81

83

10

11

14

15

10

17

18

SO

SI

S3

S3

85

80

87

11

19

10

18

19

31

83

84

25

80

88

30

31

33

19

13

19

91

83

84

80

88

39

31

33

35

37

38

13

14

»

84

90

88

30

38

34

30

38

40

43

45

14

15

90

98

30

33

35

37

39

48

44

47

4V

58

15

16

89

38

35

37

40

48

45

48

51

53

50

5U

10

17

33

30

39

48

45

48

51

54

58 88

33

4

84

7

17

18

38

41

44

47

51

^ *

58

80

1

81

5

8

18

15

18

19

49

46

49

53

57

18 1

19 5

8

18

16

80

84

19

•20

47

51

55

59

17

3

8

18

10

80

85

30

34

90

il

58

57

15

1

16

10

15

80

85

89

34

40

44

81

23

57

14

8

7

13

18

93

38

34

30

44

50

55

39

•a

13

3

9

14

30

30

31

37

43

49

55

84

1

95

7

93

44

9

15

81

37

34

40

40

53

88

93

6

13

10

94

*5

16

88

89

30

43

49

50

81

3

10

18

35

33

35

3o

99

30

37

44

58

59

30 7

14

88

30

38

40

30

27

30

37

45

53

18

1

19 9

18

*0

34

43

53

90

1

97

id

37

40

54

17

3

11

80

SO

38

47

57

85

16

8H

99

45

54

10

3

13

98

33

41

51

83

1

84

11

88

33

3D

30

53 1

5

3

13

S3

34

44

54

88

5

10

37

38

49

30

31

14

8

13

84

34

45

57

81 8

19

31

43

55

97

7

31

32

11

83

34

40

58

80 10

88

35

47

95

30

13

80

33

33

SI

34

40

59

19

11

94

37

51

84

4

18

38

40

33

34

31

45

58

18

11

85

30

53

S3

7

88

37

58

88

7

34

35

48

50

17

11

35

40

55

S3 10

85

41

57

37

13

99

35

3G

54 J

9

84

39

55

81 11

87

44

-5

1

96

18

35

53

30

37

15

5

88

38

54

20

11

88

40

34

3

81

40

58

99

18

37

:w

Id

35

53

19

10

3d

47

83 5

34

43

97

3

38

33

44

38

39

31

50

18

8

37

40

88

90

40

80

7

88

49

30

11

39

40

45

17

5

85

45

.'1

5

98

47

95

9

31

54

89

17

40

40

41

59

90

48

SO

3

85

48

94 10

33

57

38

8J

40

31

11

41

42

16

15

37

19

S3

40

83 10

34

59

87

84

50

30

10

43

4*

43

31

55

19

44

98

8

34

85

30

80

53

39

SO

49

38

18

43

44

48

18

13

39

81

5

38

59

80

55

88

33

53

31

S3

54

44

45

17

6

33

80

38

57

84 35

55

37

SS

50

30

97

59

33

33

45

40

85

54

83

53

93

13

53

30 35

57

89

30

31

3

33

38

34

14

40

47

45

19

10

47 88

18

50

35 33

57

88

31

30

43

33

19

57

47

40

8

6

39

81

13 I

45

84

SO

55

37 30

39

7

44

38

S3

34

3

35

44

48

49

89

.*

3

38 .83

14

51

SO 88

88

45

31

85

33

7

49

36

33

49

50

58

89

88

7

45

35

94

87 3

44

30

96

39

9

53

35

39

37

98

50

51

19

17

57

30 ,84

17

59

41

89 85

31

9

55

34

43

30

32

38

83

51

53

44

11

86

83

8

53

90

30

88 81

30 8

56

33

45

35

30

37

89

39

94

53

53

90

13

57

43 35

88

37

10

89 4

54

38

45

34

38

36

33.

38

30

40

89

53

54

43

S

30

34

18 ,80

7

58

50

31 43

33

38

35

35

37

34

39

38

41

40

54

55

91

15

83

5

57

49

98

43

30 39

38 30

34

35

30

30

38

40

40

47

48

50

55

5<i

SO

43

95

38 '37

34

39

33

31 31

33 33

35

36

37

48

39

51

43

4

44

SO

50

57

39

86

84

84

90

88 88

33

30

84

38 38

34 34

30

43

38

54

41

9

43

87

45

50

57

5«

•23

6

85

7

97

10 39

14

31

31

33 89

35 40

37

54

40

18

48

33

44

58

17

30

58

5*»

49

54

980 | 3Q

10

39

31

34 35

36 58

39

19

41

37

44

40

40

49

SI

59

340 34

8«o 44

50 ,310 10

33037

35047

SB© 10

400 38

430 10

45047

480 31

510 84

to

r

1*>

I30 j 140

150 160

l?o

180

190

SOO

SI©

830

830

let.

I DeeUaetion of the San.

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE VIII.

TIME OF THIS SUN'S RISING AND SETTING.

Digitized by VjOOQIC

170

TABLE Vm.
Time of the Son's rising and setting .

App.ti«ae.t^2lr

when the UL and dec are of ff^y

I*

Declination of the Bon.

tat.

0O

I©

SO

30

40

50

60

70

80

90

lOo

lio

no

hu.

oo

6*

p 0-

6* 0-

6* 0-

6* 0-

6* 0-6* 0-

6» 0-

6* 0-

6* 0-

6* 0-

6* 0-

0* 0-

00

1

6

1

1

1

1

1

3

6

1

1

1

1

9

S

9

3

3

6

1

1

1

1

9

9

s

3

3

4

6

1

1

3

8

8

3

3

3

3

5

6

1

3

S

3

3

3

4

4

4

6

6

1

3

3

3

3

4

4

5

5

7

6

1

3

3

3

4

4

5

5

6

8

6

3

3

3

3

4

5

5

6

6

7

6

3

3

3

4

4

5

6

6

7

8

10

6

1

3

3

4

4

5

6

6

7

R

•

to

11

6

3

3

3

4

5

5

6

7

8

9

•

11

IS

3

3

3

4

5

6

7

8

9

9

IB

IB

13

8

3

4

5

6

6

7

8

•

10

11

n

14

3

3

4

i

6

7

8

9

10 1

11

19

M

15

3

3

4

6

8

9

10

11

18

13

15

16

3

3

5

6

7

8

9

10

19

13

14

M

17

3

4

5

6

7

9

10

11

18

14

15

17

18

3

4

5

7

8

9

10

13

13

14

If

to

19

3

4

6

7

8

10

11

13

14

15

17

19

90

3

4

6

7

9

10

18

13

15

M

10

80

SI

13

5

6

8

9

11

18

14

16

17

19

ta

S3

3

5

6

8

10

11

13

15

16

18

89

89

33

3

5

7

9

10

13

14

15

17

19

n

93

34

4

5

7

9

11

13

14

16

18

90

93

94

35

4

6

7

9

11

13

15

17

19

81

83

95

SO

4

6

8

10

IS

14

16

18

90

83

94

99

37

4

6

8

10

18

14

16

19

81

33

95

97

88

4

6

9

11

13

15

17

19

83

34

90

98

SB

4

, 7.

9

11

13

16

18

80

83

85

87

90

30

5

7

19

14

16

19

31

S3

M

98

39

31

5

7

10

18

14

17

19

S3

84

87

90

St

33

5

8

10

13

15

18

80

S3

85

88

31

39

33

5

8

10

13

16

18

31

84

86

89

33

33

34

3

5

8

11

14

16

19

S3

85

37

30

33

34

35

6

8

11

14

17

30

S3

85

SB

31

34

33

36

6

9

13

15

18

80

S3

86

SB

33

9*

39

37

3

6

9

13

15

18

81

84

87

31

34

37

37

38

3

6

9

13

16

19

83

85

86

g

35

SB

39

30

3

6

10

13

16

30

83

86

SB

36

40

39

40

3

7

10

13

17

80

34

87

31

34

38

41

49

41

3

7

10

14

17

31

35

88

33

35

30

43

41

43

7

11

14

18

33

85

89

33

37

40

44

49

43

7

11

15

19

S3

36

30

34

38

48

40

43

44

8

13

15

19

33

37

31

35

39

43

47

44

45

8

IS

16

30

34

38

33

36

41

45

40

45

46

8

13.

17

91

35

89

33

38

43

46

51

49

47

9

13

17

93

86

30

35

39

44

48

53

47

48

9

13

18

S3

87

31

36

41

45

50

55

48

49

9

14

18

S3

88

33

37

48

47

53

57

49

50

10

14

19

34

89

34

39

44

49

54

50

59

51

10

15

30

85

30

35

40

45

50

50

7* 1

51

53

10

15

91

86

31

36

41

47

59

58

3

59

53

11

16

31

87

38

38

43

49

54

7*

6

S3

54

11

17

33

88

33

39

45

50

56

3

8

54

55

11

17

33

89

35

40

46

53

58

4

11

SS

56

13

18

34

30

36

48

48

54

7* 1

7

13

59

57

6,

13

19

85

31

37

44

50

56

3

10

16

*7

58

13

19

96

38

39

45

53

59

6

13

SO

59

50

6.

13

30

37

33

40

47

54

7» 1

8

15

83

JO

60

6*

6» 7

6* 14"

6*81-

6*88-

6*35"

6*43-

6*49"

6*56-

4

11

19-

80

99

tat.

00

10

SP

30

40

50

60

TO I 8©

OO

100

lio

ISO

WU

» IS*— Ape. time at somas, gives the tins of setting.

Digitized by VjOOQIC

TABLE VIU.

Time of the Son'i rising and letting.

171

*»*■•« SS2T when the lat. and dec • « * ffJ^JS?

Dedication of the Sun.

lat.

13©

14©

15©

IfiO

17©

18©

19©

SO©

810

33©

S3©

83© 88'

lat

<K>

6» 0-

6* 0-

6* 0-

6» 0-

6» 0*

6* 0-

6» 0"

S» 0"

6k 0-

6» 0*

6» 0*

6* 0-

0©

1

1

1

1

1

1

1

1

1

S

3

8

S

1

3

3

8

8

3

9

3

3

3

3

3

3

3

3

3

3

3

3

3

4

4

4

4

5

5

5

&

3

4

4

4

4

5

5

5

6

6

6

6

7

7

4

5

5

5

5

6

6

7

7

7

8

8

9

9

5

6

6

6

6

7

7

8

8

9

9

10

10

10

6

7

6

7

8

8

9

9

10

10

11

11

• 19

19

7

8

7

8

9

9

10

10

11

13

IS

13

14

14

8

9

8

9

10

10

11

13

13

13

14

15

15

16

9

10

9

10

11

19

19

13

14

15

16

16

17

18

10

11

10

11

13

13

14

14

15

16

17

18

19

19

11

13

11

13

13

14

15

16

17

18

19

30

81

31

13

13

13

13

14

15

16

17

18

19

30

31

33

33

13

14

13

14

15

16

17

19

30

31

83

83

84

85

14

15

14

15

16

18

19

80

31

88

34

35

86

87

15

16

15

16

18

19

80

81

83

34

35

37

88

99

16

17

16

17

19

30

31

33

34

36

37

88

30

31

17

18

17

19

90

81

33

34

*6

37

89

30

38

39

18

19

18

SO

81

33

34

86

37

89

30

33

34

34

19

90

19

31

33

34

36

37

89

30

33

34

36

30

80

21

90

33

34

35

87

89

30

33

34

36

38

38

81

33

81

S3

35

37

88

30

33

34

36

36

39

40

83

S3

33

34

36

38

30

38

34

36

38

39

43

43*

33

34

34

35

37

89

31

33

35

37

39

41

44

45

84

35

35

37

89

31

33

35

37

39

41

43

46

47

85

36

36

38

30

38

34

36

39

41

43

45

48

49

80

37

37

89

31

34

36

38

40

43

45

48

50

51

87

38

38

30

33

35

37

40

48

45

47

50

53

53

38

39

39

33

34

37

39

43

44

47

49

53

54

56

39

30

31

33

36

38

41

43

46

49

51

54

57

58

30

31

33

34

37

40

48

45

48

51

53

56

59

7»

31

33

S3

36

39

41

44

47

50

53

56

58

7» 9

3

38

33

34

37

40

43

46

49

53

55

58

7» 1

4

5

33

34

36

39

43

45

48

51

54

57

7»

3

7

8

34

35

37

40

43

46

49

53

58

59

3

6

9

11

35

36

39

43

45

48

51

55

58

7» 1

5

8

13

14

36

37

40

43

47

50

53

57

7*

4

7

11

15

16

37

39

43

45

48

53

55

59

3

6

10

14

17

19

38

39

43

47

50

54

¥

7» 1

5

9

13

16

30

93

39

40

45

48

53

56

59

3

7

11

15

19

33

35

40

41

46

50

54

58

7» 9

6

10

14

18

88

37

39

41

43

48

53

56

7»

4

8

IS

17

31

35

39

33

44

43

50

54

58

3

6

11

15

19

34

39

33

36

43

1 44

53

56

7»

4

9

13

18

93

37

33

37

39

44

45

53

58

3

7

11

16

31

85

30

35

40

43

45

46

55

7»

4

9

14

19

84

89

34

39

44

47

46

47

57

3

7

IS

17

83

37

33

37

43

48

51

47

48

59-

4

9

14

19

95

30

35

41

47

53

55

48

49

7» 3

«

13

17

83

38

33

39

45

51

57

8»

49

50

4

9

14

30

85

31

37

43

49

55

8» 8

5

50

51

6

13

17

33

39

35

41

47

53

8»

7

10

51

53

9

14

30

36

33

38

45

51

58

5

18

15

59

53

11

17

33

89

36

43

49

56

8» 3

10

17

81

53

54

14

30

87

33

40

46

53

8*

8

15

33

37

54

55

17

33

30

37

44

51

58

5

13

81

3D

33

55

56

30

37

34

41

48

55

» 3

11

19

37

36

40

56

57

33

30

37

45

58

8*

8

16

35

34

43

48

57

58

37

34

43

49

57

5

14

S3

33

41

51

56

58

59

30

38

46

54

8» 3

11

80

89

39

49

9*

9* 5

59

60

34

43

51

59

8

17

38

36

47

58

9

15

60

lat.

130

140

15©

16©

17©

18©

190

SO©

31©

33©

33©

S3© 38

' lat.

1

* 13*— App, time at suiwrr, fives the time of rifinf

Digitized by VjOOQIC

Digitized by VjOOQIC

TABLE IX,

Digitized by VjOOQIC

174

TABLE DC.

TABLE X.

Atmospherical Refractions.

Barom. 30 in.
Fa. Therm. 50°

App. Alt.

Refrac-
tion.

Diffi
forlOO
Therm.

App. Alt.

Refrac-
tion.

Biff
for 20°
Therm.

10O

5' 20"

•7"

50O

1' 49"

•8"

11

4 51

6

'51

47

3

IS

88

5

53

45

3

13

8

5

53

44

3

14

3 50

5

54

43

3

15

34

4

55

41

3

16

SO

4

56

39

3

17

9

4

57

38

18

3 58

4

58

36

19

48

3

59

35

SO

39

3

60

f34

31

31

3

61

33

83

33

3

63

31

83

17

3 .

63

30

34

10

3

64

38

85

4

3

65

87

36

1 59

3

66

86

37

54

3

67

35

38

49

3

68

34

39

45

3

69

83

30

41

3

70

31

31

37,

3

71

80 1

33

33

3

73

19

33

30

3

73

18

34

86

3

74

17

35

83

3

75

16

36

89

3

78

14

37

17

3

77

13

38

14

78

IS

39

13

79

11

40

9

80

10

41

7

81

9

43

5

83

8

43

3

4

83

7

44

84

6

45

58

85

5

46

56

86

4

47

54

87

3

48

53

88

8

49

51

89

1

50

49

90

Augmentation of the

moon's semi-diamTr.

Altitude.

Augment.

10©

3"

15

4

30

5

35

7

30

8

35

9

40

10

45

11

50

IS

55

13

60

14

70

15

90

16

TABLE XL

Son's Alt

PTxinalL

150

9"

35

8

45

7

55

6

65

5

70

4

75

3

80

8

65

1

• When the Therm, is below 50° the correction in t Nwt
columns la addatiye to the refraction.

Digitized by VjOOQIC

Digitized by VjOOQIC

Digitized by VjOOQIC

MAURY'S NAVIGATION,

Jt! ELEMENTARY, PRACTICAL, AND THEORETICAL TEEAT1SB Off
NAVIGATION i WITH A MEW AND EASY FLAN FOE FIMB1MO D1FF.
LAT. t DEP., CQUB8E, AND DISTANCE 1Y PROJECTION. BY M. r
. MAURY, LIEUT., U. S. NAVY.

OPIHIO At OP MAYIQATORS AMU PftoriBSOKg.

Bos-row, April 21st, 1835.
A work of the kind you are preparing for
the pKN t containing the demonstrations of
the formulas of Nautical Astronomy, would
be very useful to those who have a tmite for
the ■object, and would like to eiamion tile
demoiMtfations of the rules.
Heajiectfully,

Your obedient sferv ant,

N. BOWDITCtf,

PniLAintPHiA, 14th April, 1835.
Dsa« 8n:

1 hate eiamined, with is mficli care as
th« nature of my engagements has permitted,

the book intended for the instruction of the

ionnfer ©fleers of the navy, which you
awe left with me. My opinion ia f that inch
a work will be valuable to them, and will
met with fawovr among them; aappl ying,
iw it does, the mathematical principle! in-
volved in the studies of their profession in
a sufficiently condensed form.

Coming from one of their own profession,
Especially, 1 nboold anticipate that the work
would be received among them, even with-
out the injunction of the authorities, who
would, however, 1 think, find it to the inte-
rest of the ■ertke to sinction it.

Allow me to say, that 1 consider the work
filly to aostaiii the high character for scion-
tilc acquirement, which I have always
heard attributed to its anther.
Respectfully yours,

A. 0. BAOHE, Prof, &e.

The nuiersigiied are of ©pinion that the
work of Lieutenant M. F. Maury, on navi-
gation, ii eminently useful as a school book
for nmutical students. It illustrates with
clearness and aimplieity, the principles on
whieh the oaknlmtions in nat igation are
founded. • ^

We felt the want of just such a oeok in
tad of our early studies, and cheerfully re-
eonunead it to all who desire to inform
thorns©!? es in this branch of education, with
a. vkw to the nautical profession.

f MANC1S H. GREGORY, Capt
KOBEMT F. STOCKTON, Cant
FEBBEWCK BUGLE, Com.
G. A. MAGRUDER, Com.

Mv Diam 8m:

. I have great pleasure in stating my be-
lief that it is of the utmost importance that
the midshipmen of the navy should have
some established work, containing within
itself all the information on mathematics
and Mmgfttion, including nautical astro-
nomy, which they arc required to know in
•tiler to pass am examination fur promo-

mentary, and embrace arithmetic, algebra,
geometry, and plane and spherical trigo-
nometry, io far, and so far only, as might
be necessary to the construction of all the
rules and fermnlie requisite to solve the
various problems in navigation and sur-
veying: When 1 Was preparing for exami-
nation, 1 felt an earnest desire to pssess
this information, and fo ascend step by step
to a complete understanding of the whole
subject, so as to hate the means of reach-
ing all the processes of which 1 availed
myself by my 6wn resources, without
taking any thing on trust. After my pro-
motion 1 devoted my whole time and atten-
tion, for a considerable period, in attaining
this object; and feeling how much my own
coorse hid been impeded by the want of
books containing the required knowledge,
I carefully preserved all that I had recourse
to for the purpose; such as La Croix, Be-
lont, Le'gendre, Lassafle, Borda, and Cal-
let, and determined at my earliest leisure
to compile from them a treatise, narrowed
down to. what was indispensable, so as to
spare others, ambitious of more complete
information on this branch of professional
knowledge than is usual, the great diffi-
culty 1 hail experienced in knowing where
to apply for information. The appearance
of your work, so exactly sifpplying what
Was needed, and, Hfom your infinitely
higher mathematical attainments, executed
in so superior a manner, to what it would
have been had the task been left to me,
took from me all motive and desire to go
on with the undertaking;. After having
exerted such commendable exertion, in-
genuity, and judgment in accomplishing
your task, 1 trust you may, at least, have
the satisfaction of seeing your work gene-
rally used by the midshipmen, if not as »
manual of practical navigation, which Bow-
ditch's admirable work so effectually surf*
plies, at least as an elementary treatise lor
the instruction of the young officers of the
navv, the more acceptable snd encouraging
for having been supplied by one, who was,
at the time, of their number.

Believe me, very truly,
And respectfully yonr%

ALEX. SL1BELL MACKEN BE,
Commander, U. S. Mavy.
Taftstrowff, 14th November, 1843.

1 should not hesitate to commend Maury's
Navigation for the use of the midshipmen
of our navy. . To those of them who lire
advanced ii tie elements of the science, it
sUppliea the prmdkal information nec->"-*"~~~
to nuke , them good navigators ;
1 others, who have commenced with 1

g OPINIONS OF NAVIGATORS AND PlOFJSMOfS.

of the principles are both ample, eisy , and i the work myself, I know liii valoe, andt f

well arranged. % therefore, hope it will become the aolho-

The work, I know, originated In the rized book of study for the midshipmen of

wants of the students of navigation on | our navy.

boafd ships, and I confidently belie? e that
It will supply those wants.

L. M. POWELL,
Commander, U. S. Navy.

Sib:

U. S. Nav? Yard, Goaf out, <

8th January, 1833. <

THOS. A. DOANIN.
Commander, U- S. Narfy.

1 am much pleased with your ''Treatise
on Nat igation." The mathematical inves-

ligations It affords of the principles of the
science, particularly of nautical astronomy,
place it upon different grounds from the
treatises upon the subject in common uae,
and adapt It much better to the purpose of
Instruction. 1 ana desirous of introducing
It, as far as may be practicable, among my
own classes; and recommend it to afltiie
younger officers in our na¥il service, who
desire to become acquainted with the theory,
as well as the practice of the mathematical
part of their profession. Its designation as
one of the books to be used In their exami-
nation, would, I think, conduce to elevate Sib:
the standard of mathematical attainments
among them.

With great retard,

Your obedient servant.
JOHH H. C. COFFIN.

Prof, of Mathematics, U. S. Navy.

Wasmhgtom, 20th Dec, 1842.
Diaji Sim:

1 have been much Interested in a hasty
examination of your work upon Naviga*
tion, more partlculatly with that part or it,
which relates to "Spherical Trigonometry,"
and " Nautical Astronomy," two branches
of navigation that hate oeen too superfi-
cially treated In the most popular works
npon the subject, to answer the Increasing
general Information of the practical alli-
gators of the present day.

During the last forty years, hut little im-
provement of this kind has been Introduced
into the standard works upon navigation,
and as a general remark, It may be safely
asserted that they are behind the wants of
those who use them.

Ever anilous for the general diffusion of
such Important knowledge in our profes-
sion, 1 trust you will be encouraged to
Introduce the work into the navy as a Text'
Booh 1 remain, respectfully, etc.,
JAMES GLYNN,
Commander, U. S. Navy.
Lt.M.F.MAOaT 9 U.S.N.

U. S. Ship Dam, 23d Oat, 1843.
1 feel great pleasure In recommending
Maury's Navigation as a work of real use-
fulness ami Importance to the young omV
cers of our navy.

It Imbodles whatever can be of utility to
the navigator, in a concise and perspicuous
*-anner; and Its explanation* and references
Mediated to excite, In youthful and in-
ns mmd*, a desire for the higher
atical attainments. Hating uaed

Fjlag Ship FamfsftvaiiiA, >
November 4th, 1843. |
Mr Beam Sm:

1 have, with great pleasure and care, ett>
mined your book on Navigation, and do de-
cidedly recommend and prefer It to any
other id use. The niathematlcal prinoipM
condensed in such a form, ia a sufficient
recommendation to eirery^ student of navi-
gation, and 1 recommend It to all mathema-
ticians and young officers belonginp; to, mml
at schools attached to the mrt y , and should
think that the Hon. Secretary of the Navy
would so order It.
I hate the pleasure to remain, with regard.
Your obedient servant,

E. F. KENNEDY,
Lt M. P. Maobf, U. S. N. P. C, Norfolk.

U. S. Ship Pbs»tlvavia 9 >

Nov. 3d, 1643. |

1 have been much gratified In the pcmaail
of your Treatise on Navigation; and tMnk
it veil adapted for use as a school booh, ami
one best calculated, of any that I have secn>-
to Induce a lote for the prosecution of the
stodj of navigation aa a science, sad not
merely as an art.

1 "am, wry respectfully,

Your obedient servant

A. G. PENDLETON,
Prof, of Mathematics. P. a Jtuvy.
Lt. M. F. Mawȴ, U. S. M.

U. S. Ship Ohio, Bosto*. >
Not. Mth, 1843, }

Sir:

I have eiamlned, with a food deal of at-
tention, your Treatise on Navigation, and
find It embraces all the elements necessity
to constitute a soientiSe navigator. II
would, I believe, be found a valuable aux-
iliary In our naval schools. *

The work of Dr. Bowditeb, although emi-
nently useful as a pmetiml one, faila almost
entirely In the development of the nrlnei*
pies from which its rules are derived.

It seems, therefore, desirable, that some
work explaining more fully the tAasrv of
navigation should be put into the hands of
our midshipmen, In order that they may
become, as we all desire, aotnlpe aa well
aa practical nat waters.

Hoping that tie pleasure of famishing
such a work for the younger officers of the
serv ice may be yours,

I remain, very respectfully,

Your obedient servant,

JOS. T. MOSTOH f
Prof, of Mathematios. U. a Maw.
Lt. M. F. Maw, U. 8. N.

U. S. Ship Boaww, Bosvov, 1
Nor.l5ut,1813; j

Bmm But:
1 am much plesfed to-ltwn, uutogfe. Hit

OPINIONS OF KAflGATOES AND PROFESSORS.

Army and Navy Chronicle, that i new edi-
tion of your valuable Navigator Is soon to
be published.

1 cannot doubt the success of a large se-
cond edition, iind I am cooident that it
will add to your reputation, and secure for
your book the celebrity which it deserves.
Itf merit is recognised in the navy, and 1
would recommend it in all schools of navi-
gation, particularly on account of the man-
ner in which many difficult and obscure
pointp are made eaay and plain.

Respectfully, etc.,_

'ENBEIGMAST.
Commander U. S. Navy.
lit. M. F. Maurt, U.S. Navy.

WiuciaoToa f Del. Nov. 5th, 1843.

1 eoniider Maury's Treatine on Na? iga-
tion, the very work that hai long_ been
wanting in our schools, and 1 hope it will
eventually be uaed in all of them (particu-
larly the naval onea) instead of Bow ditch's
Practical Na¥igator ; for 1 think it far su-
perior aa a book of instruction.
1. 8HUBR1CK,

Commander, U. S. P.

NoBFouf, Nov. 7th, 1843,
i think Maury's Navigation is admirably
adapted for the instruction of the young
©ffieers of the Navy. All the problems are
dedneed f mm theorems, in such a manner aa
to give the young seaman a correct idea of
the theory aa well as the practice of navi-
gation. The methods ate simple and accu-
rate, and the tables wel and carefully con-
structed. The work contains all that the
atudent of navigation can require; and, were
it made the authoriied text book of the
Navy, the standard of mathematical attain-
■entsamong midshipmen, would be greatly
etlerateal.

R. B. CUNNINGHAM.

Commander, Us 8. Navy.

Nofimim 8th, 1843.
1 peasess a copy of Maury's Navigation,
and consider it a valuable text-book for all
nautical students, whether in the United
Stales Naval Service, or in the commercial
marine.

The author, Lieutenant M. F. Maury, U.
8. Navy, is a man of science, and has pro-
duced this work under the advantage of
knowing from experience what the nantl
•at student and practical navigator require.
8. P. LEE,

U. U. S. Navy.

Sim:

U. 8. Natal Hospital,

New York, Nov. 10th, 1843.

Your volume seems to me well calcu-
lated to achieve the object for which it
seems designed: namely, to demonstrate the
formulas of Nautical Astronomy, anil ex-
plain the principles upon which the art of
navigation is founded. A better hook for
aehoola of navigation, than yonrs, i am ner-
amidcd dnea not exist in onr faunam Bat

after the expression of favourable opinions
of it, by such men as Bowditch, Alexander
Dallas Bache, P. J. Rodriguez, Edward C.
Ward, and John H.C. Coffin, all eminently
qualified to judge of such a work, few can
doubt its worth or set any value upon the
opinion of, Very respectfully,

* Your obedient servant, -

W. 8. W. 1USCBENBEBGER,
Surgeon, U. 8. N.
Lieut M. F. Maoey, U. S. Navy.

Data Sn:

U. S. Ship Cumberlahd, \
Boston, Nov. 14lh, 1843. J

From the cursory examination 1 have
given your treatise" on navigation, I, for
one, am proud that so useful and valuable
a work has been furnished by one of Onr
own corps.

With the improvements you propose to
add in the new edition you are preparing,
I doubt not it will possess advantages over
other works of the kind, and be found a
valuable auxiliary in our naval schools,
very truly yours,
JOSEPH SMITH,
Captain U. S. Navy.
Lieut M. F.JMauiiv, U. S. Navy.

Washihgtoh Citt, Nov. 23d f 1843.
Bkar Maumv,

1 take this occasion to express to yon the
pleasure 1 feel, in noticing the announce-
ment of a new edition of your treatise on
navigation. Its subject matter being strict-
ly professional, has called for the close scan
of* many of your brother officers, myself of
the numb'er. Ita worthiness to become the
text book of onr young naval officers, may,
with propriety, be judged of by those who
are called on to exhibit a certificate of
having been closely examined on, and found
to possess a thorough knowledge of the sub-
ject treated of in your book alluded to.

Without an exception, all of our brother
officers,and tfiey are many whom 1 hate heard
descant freely on the merita of your work
on navi|ration, pronounced it to be the best
text booli on that subject extant. In a foil
concurrence with that opinion,
1 am with much esteem,

Respectfully yonrs truly,
WM. W. BCJNTlp,
Lieut 0. S. Navy.
Lieut 11 F. Mauef, U. S. Navy.

Dear Sn:

U. S. Brio Pxmnv, |
Norfolk, ¥a., Not. 24th, 1843. |

i am pleased to leam yon are preparing
an improved edition of your Navigation,
more especially with the view of instruct-
ing the midshipmen in the theory of naviga-
tion. To carry the student beyond the
mechanical solution of nautical problems,
into a comprehension of those principles of
Mathematics and Aftronomy, upon whr
these problems are based, stems to fa
been waning in moat treatises on Nav;
tion.

OPINIONS OF »A¥IGATOfM AID PIOFESSOIS.

Even Dr. Bowdltch'a invaluable epitome,

fhat has laid every American, who has to

trice his way on the grreat deep, under a
lasting debt of g ratttnoe, is wholly practi-
cal ip Its method. But the value of a Claw
Book proposing to teach the young officer

the theory of one branch ctf his profession,
-which Imbues his mild with lome tincture
of science, and raises him above the blind
worker of mechanical problems, need not
be enlarged upon; though it cannot be bet-
ter appreciated than by those in the profes-
plon, who were deprived of the many ad-
vantage* now offered to our midshipmen,
find who were compelled to prepare for
their examination with a Practical Naviga-
tor for their sole instructor, aid a camp-
ftool between two guna, for their study
fooni.

Uniformity in Instruction too is a matter
pi great importance, and not doubting
that the iuperior authority, who alone can
prescribe the book to be taught, will fully
appreciate the honorable contribution af-
forded by four work, to the character anil
benefit of the Navy, and that it will be made
the standard at the examination of midship-
pen, I remain, my dear sir,

Very sincerely, &c.
8. P. DUPONT,
Commander, U. S. N.
feieiit M, F. Mauev, U. S. Nayy.

On. AvQvwnnm. S. Floaida, >
NoF.87tb t 184a5

My ©pinion can contribute nothing to Uhne
pstablisfaed reputation of your valuable work.
I ean only say, that when I firat looked

through it, 1 considered it as thoroughly
supplying the great desideratum of a mid-
phi pman's study of navigation, apd remarked
to those who were present, " How nnfortu-
nate were we, in not having inch a book
when we were etudenta or navigation."
My opinion has been confirmed by that of
jother and more competent judges, and 1
believe that throughout the entire sen ice,
there haa not been a disparaging voice
raised against it. 1 regard it as being to
Jiowiitcn what Bowditch was to Hamilton
Moor©.

Hespeetfuly, &c.
WE f . LYNCH,
Lieut U. 8. Navy,
punt M. F. if AUEY,

Yale College, Conn.

October 36th, 1843.
This laluaMft wot k «P Mm f gallon, theo-
retical and practical, it seems to me desira-
ble to -have placed in the hinds of every
naval officer, "who may have occaaiojj to
navigate a ship or to explain the principles
of Nautical Astronomy. These are well
brought out, and illustrated with examples
of their application, which render the trea-
tise clear and intelligible, and adapt it well
"~ *V purposes of a text book for learners,
"'bis book as a ^ulde In the rational*
ciples of navigation, and Bowditch
ipanioii In the practical compote-

tions, the young officer* of our Navy may
be well prepared for the important and ^re-
■ponaible duties of nafigalom Having

formerly used the work at sea, while en-
gaged as an instructor in the naval service
of the U. S., I cordially give it this recom-
mendation. •

/AMES HOOKEY, Jr.

Tutor in Nat. Philosophy,

I^te Prof, of Mali U. 8. N.

8prv»yi!!o 8c am. Gallativ. |

Philadelphia, Dec. 13th, 1843. |
Dear Sin:

1 have learnt with pleasure that a seooni
edition of jour work on Navigation wil
shortly be published.

1 have always considered this hook, from
its general arrangement, and the kinds of
solution employed, as decidedly the boot
w« k for students extant

Believe me, truly yours,

(Signed) GEO. 8. BLAKE.
Lt M. F. ftUuBT, 17. 8. N. Waibington. *

Opinion of Cmpt. M'Ihtosh, U. 8. N. |
Haw lonx, Oec. 12th, 1843. f

It affords me great pleasure to state, that I
consider your work on Navigation, aa one of
the very best now extant, and most cheer-
fully recommend it as most suable for •
school book for midshipmen.

Lieut Maury.

Opinio* of Copt. Pbhciyai., U. S. N. 1
U. 8. Ship CoMSTiruTioir, {

Goaport, Bee. 14th, 1843,

. In compliance with your feqiieat, 1 ham
perused Maury's "New Theoretical, and
Practical Treatise on Navigation," and hawe

found it, as far aa I am able to Judge, a eonffr*

nient reference, in illustrating the princi-
ples of Nautical Aatroiiomy. Its explana-
tions of the principles of Spherical Trigo-
nometry, and its application of them to the
solutions of the various astronomical pro-
blems, so essential in navigation, render it
a book, in my opinion, not only useful aa a>
supplement to our firat ( Bowditch *a) stan-
dard work on the subject; but valuable in
itself: an acquisition to the nautical stu-
dent, who, if ho is desirois of acquiring a
correct, and practical k now ledge of his pro-
fesiioDj may be largely aided by the ntudj
thereof!
Lt M. F. Maoev, U. S. N.

Qptaiom of Omwl Fommasr, U. 8. N . 1

. WasHiwovoa, Dec. 18th, 1843. J
I take much pleasure In recommending
your "New Theoretical and Practical Trea-
tise on Navigation-* ' The explanations and
illustrations are rendered clear and compre-
hensive, and 1 believe it to be joat each •
production as we require for the instruction
of the young ©ficera of our Navy, as well
aa others desirous of obtaining a well
grounded and accurate knowledge of tin
science.
LtM.r.MaoRT.U.&N.

OPINIONS OF MAViGATOlS AND FBOFE8SOB&

fiphiivn qf A§ U»& Mkwal Lymmm, >

Brooklyn, Mew York. §
Brooklt», Dec. 90th, 1843.

The undersigned, a committee to which
wu referred Lieut. Maury's " Treatise on

Navigation," report that they hi?© careful-
ly examined the same, and aw of opinion
that it is a work well adapted for the in-
•friction of the young officers of the Navy,
m mil the problems and formalB that mre
neceiiary in their profession, ire there
broug ht together in a oondenwd form f and
■o clearly, and concisely demonstrated, that
the student may easily inform himself of
the theory and principles on which his
practice Is founded; and the accompanying
tablea are no constructed aa to facilitate hit
calculations; all of which are systematically
wring ed, with a limplicity that has hereto-
fore Been generally wanting in works on
Navigation. They wonld therefore recom-
mend the same to be adopted for the use of
the Natal schools.

Signed,

J. H. STIIlfGHAM,
WM.D. NEWMAN,
ALEX. C. GIBSON.

The following ©pinions have already ap-
peared in print, but as they hire in all
probability escaped the notice of many
to whom these pages will be presented f
they are again inserted here.

« U. 8. M. &, Ate Work, Jmmmy 19, 1836.

'•Bear Sir,— 1 have bad mncfi pleasure
in the peraial of yonr " New Theoretical
and Practical Treatise on Navigation;" the
plan and arrangement* of which are origi-
nal; itcontainalittleor nothing superfluous,
and every part of it appears to be ai clear
and intelligible as the nature of the subject
will admit. Such a work has long been
wanted in our Naval Schools, and on board
opr vessels of war. I intend to make use
of it in the Naval School on this station; and
1 recommend it to be med by all the profes-
sors of Maiheipatica and Nautical Science,
in the Nawy of the United States.
«*Yoori Respectfully,
"EDW. C. WA1D,
M Proi:Math.U.aNa.vy."
"Passed Midshipman M. F. Maury,
« U.S. Navy."

" K S. JVbvy fur J, Gosport, March 7, 1836.

" 1 have examined a Treatise on Naviga-
tion' written by M. F. Maury of the U. S.
Navy; mnd have no hesitation in recom-
mending it to the atodents of that science.

The explanations are clear, the roles are
illustrated by many examples, and the new
arrangement of some of the tables simpli-
fy the calculations of the navigator. Mr.
Maury is deserving of great credit for the
work, and 1 wish him every success.

f . J. ROPlieUEZ.