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

9

which is a geometrical series whose sum is 1.
Hence e is greater than 2 but less than 3.
Again, the value of e cannot be expressed by any finite fraction
-, for then we should have
mn 2 - 1 1. 1
2 =J + -..- -
n - 1.2 1.2.3 1.2.3.i 1.2..
1.2.... ( + 1)
Multiplying both sides by 1. 2. 3... n..'. 2.1. 3....( - 1)-1 3.22... = 3.... 4. 5...-,.,.... + ~1 + n+i + (z + l)(n + 2) +
Now the left side of this equation is an integer; the right side
must also be an integer.
1 1
But the series - 1 + ( - 1)( + - *. is less than the snm of
1 1 1
the series - + + &c. *. which is equal to 1
n - 1 (n + 1)-2. The right side cannot be a whole number.
Hence there is no integer or finite fraction by which the exact
value of e can be expressed.
13. PFor. To find the logarithim of any numzber in a series in terms of
the base and of the nzumber itself; or to find x from the equation zt = a
in terms of u and a.1
The following is from La Grange's Calcul de Fonctions: —
Here uc - as and x = logra.
Now 1 + (u - ) = { + (a- 1)}-.'. { + ~(a -~}-={ +(C- l)}t
Expanding each side by the binomial theorem,
t (t ) _) t (t.- 1) (t 2)
+1 +t (4 C - -1) + 1.(-1)2+ 1.2.3 ( - 1)' + &c.
tx (tx - l).(a - 1)2 +- tx (t - 1) (tx - 2).(a &
= 1 + t (a-)+ - 1.2 1.2.3
subtracting 1 from each side and dividing every term by t,
+(c ( -) + -:t(t + (t - 1)(t - 2).(_ -1) + &c.
~. " (- 2) + —1 - 1[2.3
(Cb + J(tr 1 ) (( -1 1) 2 + (toX - ) (tX- ) (
172 (a 1.2.3
This equation being true lwhatever be the value of t. Let t = o
2 + 2
(. -i) (u-i)2 (_-l)3 x{(a-1) (a-l)2 (a-I3 -~. orlog. = -- {(al-1) (a-1) — 2 ])
(_ + - 1) - ( + 3 -- &c..'. z 01'o log (a ) ( 1) 3
(a- 1) - -2 + -- - &c.
which is the logarithm of the number u in terms of qu and the base a,