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

8

= a 1+ l)+ - ). - -1)2 + n(n- 1)(-c2).(a 1) +&c.
X (a-l)+. 1.2 ' (a 1.2.3
2 3
by finding the terms of the coefficient of x, and putting for
the coefficients of x2,;, C&c.,.B, C, D, which involve numerical
qulantities and powers of a only.
And by putting A for (a-l)- -1i ( _ -1- &c.
2 3
a =1 + A + A + + C3 + DX4 + &c.
Now this expression for aX is true, whatever value may be assigned
to., let x become x + z.
Then + —= 1 + A (x + ) + B (+ + Z)+ C (' + )3 + &c.
but ax+ =ax. a'= {1l + Ax + -Bx + CX3 + &c.}
Equating the coefficients of x in these identical values of aT'+^
A + 2 Bz+3 Cz2+4_Dz +&c.Aa =A = A {l+A + iBz2C+ CS3+&c.}
And, lastly, equating the coefficients of the same powers of z in
these identical series,
21 =A2, 3C =-4B, 41= AC, &c.
A2 AB _A3 A C- A4
Whence B= -., C = 2 - - 7 &c..-. = ax = 1+ 4x+ ~.2.3+ 123 4 + +& 2 c.
in which = (a - 1) - -1) (a &-.
2 3
Con. If e denote that value of a in the equation ua —a which
males the series for A2 equal to unity, then u- ea" and log, t u x, and
the series for u or e" becomes
X2.'
e - 1 + x + 1 — +.2 - 3 + c&., which is the number u expressed in terms of x the logarithm of e; and making x- 1
1 1
1 - 1 + 1 + —1 + 3 + &c. = 2.7182818....which is the numerical value of the base e of Napier's system of logarithms.1
12. PrOP. SiewZv that the base of the Narpicriani system of logaritJtms
is inconlcensurable.
Here e- 1 + 1 +- -.2. + 2 —4 +
1 1 1.. e - 2 - + 1.23 + 1.2 3- 4 * +.. a series which is
1 1 1 1
cvidently less than the series -- +-.
2+ +2+ +

1 It is seldom the value of the base of Napier's system of logarithms is required
in any calculations beyond seven places of decimals. The following is the value of e
calculated to twenty places of decimals: e=2-718281,828459,045235,36.....