COR. 1. Since A is a constant quantity, entirely dependent on the value of a, let e be the quantity whose value

1

ले

is such that ε = a;

(iii.)

We shall hereafter find the value of which is a constant quantity 2.7182818.... Logarithms to the base e are called Napierian, from Napier, the inventor of logarithms, who adopted this base because logarithms to it are more easily calculated than those to any other base; as is evident from comparing the series (ii.) and (iii.)

COR. 2. 1α=1{1 + (a− 1)}, which by (iii.) is

calculated to the base e are multiplied to give the logarithms

La' by which the logarithms

15. We shall next investigate some rapidly converging formulæ for the calculation of logarithms.

If a be a little greater than 1, this series converges very

rapidly.

= m . {(x2 – 1) − ↓ . (a" − 1)2+...}, by (iv.) {(xTM −

1

and, (a being greater than 1), by assuming m of sufficient magnitude, am may be made to differ from 1 by any definite quantity, however small the quantity may be; in which case, the succeeding terms of the series may be neglected as being of inconsiderable magnitude with respect to the first, and we have

1

m

1x = m . (xTM –

1).....

.(vii.)

16. Having given 1x, to find 1 (x + z), z being small when compared with x.

which is useful in computing le (1 + x) from lex, particularly

17. Having given the Napierian logarithms of two successive numbers, x-1 and x, to find that of the number next following.

18. To expand a* in a series ascending by powers of x; i. e. to expand the number in a series ascending by powers of

the logarithm.

...9

a3,... be

let the coefficient of x, which is (a−1)− 1. (a−1)2+}. (a − 1 )3 — ....., be represented by P1, and let the coefficients of x2, represented by P2, P3, ...;-then we have

or 1+ P1 · (x + ≈) + P2 · (x + x)2 + ... + P2 · (x + x)” + ...

Pn

and equating the coefficients of the terms involving xx, x2x,

x2-1%, we have

...

Now p1 = (a1) - § . (a − 1)2 + § . ( a − 1)3 — ...