TUI

497

TEI

®>ne of its supplement; the supplement being an arc of such“•■aensions as will, together with the original arc, make anarc of 180°, or, in other words, a semicircle : thus, an arc of*20° is the supplement of an arc of 60°, and vice versd, because+ 120° = 180°. Similarly, the complement of an arc isatl arc of such dimensions as, together with the original arc,make an arc of 90°; thus, an arc of 50° is the complementan arc of 40°, and vice versd.

Definition 3.—The tangent of an arc is a straight line,drawn from one extremity of the radius, a C, perpen-jhcular to the same, A c, terminated at the other extremity°y a continued radius drawn through the other extremity, b,ot the arc a b.

Definition 4.—The secant of an arc is a straight line, d c,Passing through one extremity of the arc and the centre, and

tel V»-rr *V> /-» + *-l. — ~--U *1. *■» ~«-l-

e itri

minated by the tangent, which passes through the otherl ~emity.

■The versed sine is that part of the radius which is inter-oPted between the foot of the sine and the beginning of

arc.

The co-tangent and co-secant are respectively the tangenttl a secant of the complement.

,, Definition 5.—The sine, tangent, or secant of an angle, isne same as that of the arc intercepted.

As trigonometry is chiefly performed by proportion, and as11 proportion consists of multiplication or division, or both,j 4 since multiplication and division is best performed byu §arithms, we shall here give a short sketch of the natureconstruction of logarithms.

^ The sum of the logarithms of any two numbers, x and z,logarithm of their product. Let x be the logarithm of

x, and z the logarithm of z; then taking the exponents of thepower of a constant root for the logarithm, we have x = a x ,

and z = a z >

Therefore xz = a x a z = a x + z ; therefore x + z is thelogarithm of xz; that is, the sum of the logarithms of anytwo numbers is the logarithm of their product.

Corollary. —Hence, if x = z the logarithms of x 2 is 2 x orof 2 ? = 2 z.

In the same manner the logarithm x y z will be foundto be x + y + z, and consequently the logarithm of x 3= 3x; therefore, universally, the logarithm of a power isthe logarithm of the root multiplied by the index of thepower.

The difference of the logarithms of any two numbers, xand 2 , is equal to the logarithm of their quotient; since xz=a xx a x

and 2 = a z , therefore —= — = a x —z ; therefore, x—z isz a z

X

the logarithm of —.

The logarithm of the wth root of any number, x, is thelogarithm of that number divided by n.

Since a; = a x ; by taking the wth root of both sides, we1 x

have x n = an-, but — is the logarithm of x". And in thesame manner we find the logarithm of the wth power of anynumber, x, is w times the logarithm of that number, forx = « x ; raise both sides to the wth power, then willa;" = a" x ; therefore w x is the logarithm of x".

U T° determine two such functions of two quantities, x + 1 and z + 1, so that if the same operation be separately performed* + 1 and 2+1, and upon their product, the sum of the functions of x + 1 and z + 1, may be equal to the function of

e,r product:

I. That is, 0 {x + 1) + <j> (z + 1) = 6 [(a; + 1) (z + 1)] = <j> (1 + x + z + x z\

II. Assume ^(l + a:) = a + 5ar + ex' + dx 3 +, &c.

III. Therefore, <jt (l + z) = a + b z + C 2 S + tfz 3 +, &c. And by the same operation we obtain,

b x

c x 1

dx 2

IY. 0 [1 + (x + z + x z)] = a + b z + (b + 2 c) xz + (2 c + 3 d) x*z +, &C.

c z 5 (2 c + 3 d) xz 3

and by adding No. II. and No. III. together, we obtain,

V. 0 (1 + x) + * (1 + 2) = 2 a + b X + - dx \ +, &C.

b z cz* dz 2

Then comparing the co-efficients of IV. and V. we obtain,

a = o, b = b, c = — = e — — &c., whence we obtain,

2 3 4

3?

YI. 0(1 +z) = 5(z — o+Y-+ &c ") wll >ch is the logarithm of 1 + 2 .

2 3 4

Let M = b, substitute — z for x, in the series VI. and we obtain,

VII. L'(l-z) = M(-z-l 2 _J 3 _ C-, &c.)

Subtract No. VII. from No. VI. and we obtain,

VIII. l’ (1 + z) — l’ (1 — z) = 2 m (z* + ~ + *L+ ~ +, &c.) or,

6 L