Elementary arithmetic, with brief notices of its history... by Robert Potts.

10

Since u = a,, in which x = loguc.
2XP' An3X A4X4
And af== 1. + 2x +3 1.2.3.4 & c., is true whatever values may be assigned to a and x.
Let x, a, become respectively z, t, and A1 the new value of A4
A2'2 " A, 3 z3 A_4 I4
Then u=Z 1 + 1. + 1 —23 + 12. + &c.
in wich A, (. 1) 2 (~-2 (u- 1)' (u- 1)'
in which A,1 = (.)- ) - - &c.
2 ' 4
But since u =a,. '. =( () a= 1 + xz + 12 +. 1.2.3 &.
But it has been shown above that
A12 2 -1 3 z3
ez=I + -4,Z + 1.2 + 12 3 + &c.
Hence, equating the coefficients of S in these identical series,
A = AX,
16 _ 1)2 (ZG-1 )B (?G- )
A -, 2 q 3 4 -.
^ 1)- -2 +- - - 4 + &c.
and x or logan il (2 3 --- —- 4 --- —---
which is the logarithm of the number u, to base a, expressed in terms
of the number itself and the base.
It is obvious that this expression is not adapted for the calculationa
of the logarithms of natural numbers either to the base 10, or to the
base of Napier's system, because the series do not converge for
numbers greater than 2.
14. PRor. To find a converging series for calctcating 2apicer's logaritAms.
(.- _ 1)( ( -- &)
It has been proved, that log, =
(- () (a- i) (a- i) &
2 3-.
whatever may be the assumed values of a and u.
Now in the Napierian system the base is denoted by e; writing e fora.
But in Napier's system e is of such a value that the series
1)^( ^ - + - 0. —1
2 3
iHence the expression becomes
loA D )'- 1 (- I 2 1 &
loge_- (u-1)-(-i21)' + ( — 1)_ &c., which is the Napierian
o in t s of
logarithmu of u expressed in terms of the number u only.