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Five copper-plates at the end of the book.

EXPLANATION OF THE CHARACTERS OR MARKS

USED IN THE FOLLOWING WORK.

+, 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 3.

—, Minus or less, the sign of subtraction, and shows that the second quantity is to be taken from the

first; as CBGB shows that the line CB is to be diminished by the line GB.

×, Into or by, the sign of multiplication; as EDX DC signifies the rectangle formed by the lines ED and DC, and axb expresses the product of the quantity a by the quantity b. Also ab or ab signifies the same thing.

, Divide by, as PB÷Cs, or signifies that PB is to be

divided by cs.

PB
CS

AB2, AB3, 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.

√ora,3√a or a3, express the square and cube root of A.

A

A

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 quantities together, as A+B×m, or (A+B). M,

signifies that A and B are first to be added

as is to S

is to

• Therefore,

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 usually read A is to в as c is to D.

Angle, as A, signifies the angle a.

Greater than, as AB, shows that A is greater than B.
Less than, as AB, 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

TRIGONOMETRY.

BOOK I.

CHAPTER I.

THE NATURE AND PROPERTIES OF LOGARITHMS.

(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.rl

.p2.23.p.4.7.5.6, &c. be such a series, increasing

1 1 1 1 1 1

from 1; or 1.

r

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-, &c. decreasing from 1; which last series, agreeably to the established notation in algebra, may be thus expressed, 1. r-1.r-2. p-3, p−4.

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5 -6, &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

B

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