feries converges, and its value, that is, m is found 2.30258509, &c. Having thus ob

=

tained a value of m, it is evident that a or log. 1+x, x being any small fraction, is to be found from the feries (x X -- = x2 +

x++, &c.).

10. When, by help of this feries, or others of the fame kind, the logarithms of the prime numbers are computed, thofe of all other numbers may be deduced from them. Thus, if the logarithms of 2 and 3 are known, we have from thence the logarithms of an infinity of other numbers; log. 42 log. 2; log. 6= log. 2 + log. 3; log. 5 log. 1o.log. 2. 1-log. 2; log. 83 log. 2; log. 9=2 log. 3; log. 12 =2 log. 2 + log. 3, &c.; also log. 20 = 1 + log. 2, log. 25 = 2 log. 5, log. 30

=

APPEN

APPENDIX.

I.

No. VI.

Arithmetic of Sines, &c.

THE

HE application of algebra to trigonometry has given rife to a calculus of a particular kind, which is of great ufe in the higher geometry. The following theorem is the foundation of it.

I. Let a and b be any two arches of a circle, of which the radius =1, fin. (a+b) fin. a X cos. b + cos. a X fin. b.

CD

CB

AC: AB :: fin. b: fin. a+b; and fin. b

CD

ABX CD

(AD+DB)CD ADXCD DBX CD

AC: AB:: CB: fin. a+b= ACXCB

=

ACX CB

ACXCB+ ACX CB'

Now,

fin. a+b=

bfin. a X cos. b + cos. a X

fin. b. Q. E. D.

P

SCHOLIUM.

In this theorem, and in those which follow, it is fuppofed that the fine and cofine of an arch lefs than 90° are both affirmative. These two fuppofitions are arbitrary; but, when they are once laid down, it follows neceffarily that the fine of any arch lefs than 180° is affirmative, but that the fine of an arch greater than 180° is negative; alfo, that the cofine of an arch between 90° and 270° is negative, but that of an arch between 270° and 360°, the co

fine is affirmative. Laftly, that, when an arch changes from + to, or from

to

+, its fine undergoes a like change, but its cofine retains the fame fign.

x

II. If, in the preceding theorem, we fuppose the arch b to become negative, then fin. b will also be negative, and therefore fin. a x cos. b - cos. a X

fin. a

b

fin. b.

III. Cos. a+b= cos. a x cos. b- fin. a fin. b. For, cos. a+b = sin. (90°—a

But fin. c-b= fin. c X cos. bcos. cx

fin. b; now, fin. c = cos. a, and cos. c =

IV. If, in the laft theorem, we make b negative, fin. b will alfo become negative, and therefore cos. a b = cos. a x cos. b fin. a X fin. b.

This I have ascertained inaaches terminating in each of the a quadrants.

V. Since fin. a+b= fin. a x cos. b + cos. a x fin. b, and fin. ab = fin. a x cos. b cos. a fin. b; therefore fin. a +b+ fin. a ➡b b2 fin. a X cos. b.

VI. By fubtracting the fame two equations,

Sin. a+b fin. a-b= 2 cos, a × fin. b.

VII. Also, fince by theor. 3d, cos. a+b = cos. a X cos. ¿ fin. a × fin. b, and, by theor. 4. cos. a — - b = cos. a X cos. b fin. a X fin. b, therefore cos. a+b +cos. a→ b = 2 cos. a X cos. b.

VIII. Again, by fubtracting, cos. a+b

3. By help of these theorems, the products and powers of the fines and cofines of arches may be expreffed in terms of the fums or differences of the fines or cofines of certain multiples of thofe arches. Thus, fince 2 fin. a X fin. bcos. a ·b⋅ -COS. a+ b, if b = a, 2 fin. 2a = 1 - COS. 2 a. Again,