If be very small, the first member sin xx and the terms in the parenthesis after the first vanish. Hence x A x .. A= 1, and

Now, the length of an arc of one degree is so small that if x be equal to this, the third term of the above series will contain no significant figure in the first ten places of decimals.

Retaining therefore only the first two terms of (1) we have, when x is small,

that is, the quantity within the brackets being = cos x, by (2)

sin xx cos x;

therefore introducing radius to render the expression homogeneous, we have

introducing radius and applying logarithms

log. x = log. n + log. 3.14159, &c. +10- log. 180 × 602

log. sin x=log. n + 4.68558 - ar. comp. log. cos x.

(4)

Hence this rule. To the logarithm of the arc reduced into seconds add the constant 4.68558, and from the sum subtract one-third of the arithmetical complement of the log. cosine; the remainder will be the logarithmic sine of the given arc. 9. To find the logarithmic tangent of a very small arc.

The second member of this equation may be formed from the second member of (3) in the last article, by adding the arithmetical complement of the log. cos x; therefore from (4)

log. tan x= = log. n + 4.68558+ arith. comp. log, cos x (5)

hence this rule. Add the logarithm of the arc reduced to seconds, the constant 4.68558, and two-thirds of the arithmetical complement of the log. cosine, the sum is the log. tangent required.

10. To find a small arc from its log. sine.

From (4) Art. 8, Appendix I.,

log. n log. sin x- 4.68558arith. comp. Iog, cos x
=log. sin x + 5.31442+ arith comp. log. cos x- 10

Hence the rule is this. Add the log. sine of the arc, the constant 5.31442, and of the arithmetical complement of the log. cosine; subtract 10 from the index of the sum, and the remainder will be the logarithm of the number of seconds in the

are.

To find a small arc from its log. tangent.

From (5) last art.

log. n= log. tan x-4.68558-arith. comp. log. cos x
log. tan x + 5.31442-arith. comp. log. cos x

RULE. Add the log. tangent of the arc, the constant 5.31442, and subtract } of the arithmetical complement of the log. cosine, reject 10 from the index, and the result will be the logarithm of the arc in seconds.

* If we substitute for cos x in the last fraction its value

1

we may

11. The values of a, which satisfy the equation sin a=0 are 0,± -,* ± 2′′, ± 3′′

....

Consequently the series which is the development of sin a must be divisible by a, (Alg. Art. 238, Prop. II.) a—, a +, a −2x, a +2π.

If therefore k be a constant whose value is afterwards to be determined, we have

* being 180°, or the seini-circumference whose radius is 1.

Its successive factors approxi

The above form serves to compute a logarithmic sine at once, without first com

puting the natural sine. †

From (2) in a similar manner may be derived the following formula for calculating logarithmic cosines.

м is the modulus of the system, equal for the common to 43429448.

12. Adding and subtracting (3) and (7) of Art. 70, and adding and subtracting (6) and (8) of the same article, the following formulas are obtained.