From these series, may logarithmic sines and cosines, independently of the values of the natural sines, be computed to 15 places; and, this inconvenience is avoided; if the natural sines had been taken, consisting of more than 7 places, no Tables in common use would give their logarithms. The logarithms indeed of the numbers m, 2n−m, 2n+m, &c. are supposed to be taken to 15

places, and these can be had, since the numbers will not consist

m

of more than 6 figures for

n

cannot exceed *; therefore, since

n = 90.60.60 = 324000, n + m, 2n+m, &c. cannot exceed 1000000.

As an instance to the preceding formula, suppose the logarithmic sine of 9o to be required: here m=1, n=10.

This is the log. sin. 9° to 10 places: and the decimal part is the logarithm of 15643446 the natural sine of 9°, found, p. 73, &c.

Rule, with its proof, for finding the logarithmic sine and tangent of very small arcs, (see pp. 96, &c. and Introduction to Taylor's Logarithms, p. 17, &c.)

For the sine; to the logarithm of the arc reduced into seconds with the decimal annexed, add the constant quantity 4.6855749,

m

If ">, the series for the cosine would be used for computing the sine, since sin. (45° + A) = cos. (45° — A): it is obvious, that the logarithms of m, n+m, &c. may be dispensed with entirely, by ex

panding log. (1.- m2) ); but then, to attain the same exactness, we

n

2

must make the series consist of more terms. It is also plain, that instead of fifteen places in the numerical coefficients of the series, any number may be used. See Callet's Logarithms, p. 48.

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.

For the tangent; to the log. arc and above constant quantity, add two-thirds of the Arithmetical complement of the log. cosine, the sum is the log. tangent of the given arc.

hence, log. sin. x = log. v, + § log. (1 − 2)

= log. x + log. cos. x,

and introducing the tabular radius 1010, the logarithm of which 10, log. sin. x = log. (10-log. cos. x).

x

Now r is in parts of the radius; let n be its value reduced to seconds, then 2 r x 3.14159, &c. : x :: 360.60.60 : n;

consequently,

log.xlog.n+log. 2r+log. 3.14159, &c.-2 log. 36 - log. 1000 log. n + 4.6855749,

and log. sin. x = log. n+4.6855749-(10-log. cos. x) which, since 10-log. cos. x is the Arithmetical complement of the logarithmic cosines, proves the rule.

1

d

hence, as before, n being the number of seconds in the arc x, log. n + 4.6855749 + arith'. comp'. log. cos. x ;

log. tan.

20—(log. n +4.6855749 +÷ arith'. comp'. log. cos. x).

These are the proofs of the two rules for finding the sine and tangent from the arc; but, there are also two rules for the reverse operation of finding the arc from the log. sine and log. tangent. These are subjoined. See the Introduction to Taylor's Logarithms, p. 22.

Rule to find the Arc from the Sine.

To the given log. sine of a small arc add 5.3144251 and § of the arithmetical complement of log. cosine; subtract 10 from the index of the sum, and you will have the logarithm of the number of seconds with the decimal fraction of the given arc.

Rule to find the Arc from the Tangent.

To the log. tangent add 5.3144251, and from the sum subtract of the arithmetical complement of log. cosine, and subtract 10 from the index, and you will have the logarithm of the number of seconds with the decimal fraction of the given arc.

Proof of the First Rule.

If y=sine, and x = arc, then d x =

dy

√(1 − y2)................(p. 106);

= ƒdy (1−y3) ̄* = ƒdy (1+2) nearly, since y is small,

x=

..z ➡y++ (nearly) = y(1 + 23) = y (1–2)−†

2 3

... log. x = log. y — log. (1 327)

=log. y-log. cos. ;

or to tabular radius, log. x=log. y+§ (10—log. cos. x), but, p. 255, log. x = log. n + 4.6855749 = log. n + (10−5.3144251);

consequently,

log. n = log. y + 5.3144251 + arith'. comp'. log. cos. x 10 which is the rule symbolically expressed.

hence, expanding, by the Binomial Theorem, the value of dar as far as two terms, and integrating, there results

Hence, log. xlog.t-log. sec. xlog.t-log.

supplying the Tabular radius, and putting for x its preceding value,

log.n=log. t + 5.3144251

(10- log. cos. x) — 10

—

log. t +5.3144251-arith'. comp'. log. cos. x 10,

which is the rule symbolically expressed.

K K