and on Algebra as the Science of Pure Time. 377 
1 n 3 n \ 3 3 
zs ) over the cube ‘= ) ,is 
{ 
i 
for example, when m=3, the excess of the cube ( 
n> Man\?_1 ie Dp, The byiyss ere ae 
e(>) +( m, } = 4 = | nay m, eh) m, ) 2 (285.) 
In general, the number of the terms (or added parts) in the expression (282.), 1 
ol+m : : 
m, and they are all unequal, the least being we ey) , and the greatest being 
: el+m y 4 
| ; their sum, therefore, is less than the hes multiple of this greatest term, 
that is, 
el+m 
pa a ee (284.) 
7 
and therefore the excess (281.) is subject to the corresponding condition 
1 1 el-+m 2 
o(- oy 44th <5 =*) ; (285.) 
aC m, ) + +C Oy ™, 4g =f) : (286.) 
However this excess (281.) increases constantly with x, when m, remains unaltered, 
because p so increases; so that the 1 +7 fractions of the series 
) m Sym a 
i Ce Ge Co (287.) 
increase by increasing differences, (or advance by increasing intervals,) which are each 
for example, 
oel+m . 
less than — c= , and therefore than z? if we choose m, and 7 so as 
™, 7] C 
to satisfy the conditions 
r _o9l+m kmin 
l+tn=im, m=kmxi ——— 
. (288.) 
i and & being any two positive whole numbers assumed at pleasure ; with this choice, 
therefore, of the numbers m,and 7, some one (at least), such as es jd of the 
series of powers of fractions (287.), of which the last is =7”, will fall between any 
two proposed unequal positive ratios 3' and 4”, if the greater 4” does not exceed that 
last power 7”, and if the difference © J’ + 4” is not less than >; and these condi- 
k 
tions can be always satisfied by a suitable choice of the whole numbers 7 and k, how- 
