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• 368

Page Prop. 4. To find the excess of the three angles of a spherical triangle, above two right angles

366 5. To reduce the angles of a spherical triangle (whose

sides are very small arcs) to those of a rectilineal tri-
angle, having its sides of equal length with the sides

of the spherical triangle
6. Given two sides of a spherical triangle, and the angle

comprehended between them; to find the angle con-
tained between the chords of these sides, supposing
the chords not to differ materially from the arcs which
they subtend

371 7. The angles of elevation of two distant objects being

given, together with the oblique angle contained between the objects, to find the horizontal angle 373

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BOOK IV.

THE THEORY OF NAVIGATION.

CHAPTER I. Definitions and Plane sailing
CHAP, 11. Parallel and Middle Latitude sailing
CHAP. III. Mercator's sailing

376 to 380 381 to 383 383 to 392

TABLES.

I. A Table of the LOGARITHMS of numbers, from an
unit to ten thousand

393 to 409 II. A Table of NATURAL SINES to every degrec and minute of the quadrant

409 to 417 III. A Table of LOGARITHMICAL SINES and TANGENTS to

every degree and minute of the quadrant 418 to 440 IV. A Table of the REFRACTION in altitude of the heavenly bodies

441 V. A Table of the depression or Dip of the horizon of

441 VI. A Table of the sun's PARALLAX in altitude

441 VII. A Table of the augmentation of the moon's semidiameter

441 VIII. A Table of the right ascensions and declinations

of thirty-six principal fixed.stars, corrected to the
beginning of the year 1822

442

the sea

.

Five copper-plates at the end of the book.

EXPLANATION OF THE CHARACTERS OR MARKS

USED IN THE FOLLOWING WORK.

PB

+, Plus or more, the sign of addition ; as AD+DC, signifies

that the line ad is to be increased by the line Dc; and 4+3 signifies that the number

4 is to be increased by the number B. -, Minus or less, the sign of subtraction, and shows that the

second quantity is to be taken from the first; as CB – GB shows that the line CB is to

be diminished by the line GB. X, Into or by, the sign of multiplication ; as ED X DC sig

nifies the rectangle formed by the lines ED and Dc, and a x b expresses the product of the quantity a by the quantity b. Also a:b

or ab signifies the same thing. • , Divide by; as PB+Cs, or signifies that PB is to be

divided by cs. AB”, AB, signify the square and the cube of AB; also

14) signifies that 14 is to be involved to the third power, and then the fourth root

is to be extracted. or at, sva or A}, express the square and cube root of a.

=, Equal to, as AB=CD, shews that AB is equal to CD.
~, Difference, as A~ B, shews that the difference between

A and B is to be taken.
A vinculum or parenthesis, serves to link two or more quan-

tities together, as A+BXm, or (a +B).m,

CS

{i

together, and then to be multiplied by the
quantity m.
Proportion, A : B::C: D signifies that a has

to B the same ratio which c has to D, and is to as is to

is usually read a is to B as c is to D. .: Therefore, Ľ Angle, as į A, signifies the angle A.

Greater than, as AB, shows that A is greater than B. [Less than, as ACB, shows A to be less than B.

The other characters are explained among the definitions in the work. : N.B. The letters within the parentheses, at the beginning of the different paragraphs of the work, are for references. Thus, (C. 2.) refers to the article marked (C) at page 2. ; (H. 25.) refers to the article marked (H) at page 25, and so on.,

ERRATA.

Page 93 and 94, in the note, for Chap. XI. read Chap. XIV.
Page 305, line 10. for 5th of October, read 5th of August.

AN

INTRODUCTION

TO

PLANE AND SPHERICAL

TRIGON O M E T R Y.

BOOK I.

CHAPTER I.

THE NATURE AND PROPERTIES OF LOGARITHMS.

1

(A) Definition. LOGARITHMS are a series of numbers contrived to facilitate arithmetical calculations; so that by them the work of multiplication is performed by addition, division by subtraction, involution by multiplication, and the extraction of roots by division.

They may therefore be considered as indices to a series of numbers in geometrical progression, where the first term is an unit. Let 1.gd .ge .gl .gt .go5.86 , &c. be such a series, increasing

1 1 1 1 1 from 1; or 1.

-&c. decreasing from

22 go3 z4 z5 zo 1; which last series, agreeably to the established notation in algebra, may be thus expressed, 1.r-1.8-2.7-3. r-4. go5 gomb, &c. Here the common ratio is r, and the indices 1.2.3, &c. or -1.-2.-3, &c. are logarithms. Hence it is obvious, that if a series of numbers be in geometrical progression, their logarithms will constitute a series in arithmetical progresssion. And, where the series is increasing, the terms of the geometrical progression are obtained by multiplication, and those of the arithmetical progression, or logarithms, by addition; on the contrary, if the series be decreasing, the

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