and on Algebra as the Science of Pure Time. 365 
square of any whole or of any fractional number, it is possible to find, or to conceive 
as found, one positive ratio a, and only one, which shall satisfy all the conditions of 
the following forms : 
eae ha hs (239.) 
mn’ m' n" denoting here any positive whole numbers whatever, which can be chosen 
so as to satisfy these relations, 
geen ee 
nn n'n 
b 
By Ti v 
ve om 
———s 19 
m' m (240.) 
For if the proposed ratio 4 be not the square of any whole or fractional number, then 
the existence of such a ratio a may be proved from the two preceding Lemmas, or 
from their Corollaries, by observing that the relations (240.) give 
min’ nn 
n! 
—=— >. -_ alla, therefore —. >.—- 
mm” mim? ~ m ~ mm’? 
(241.) 
so that no two conditions of the forms (239.) are incompatible with each other, and there 
must be at least one positive ratio a which satisfies them all. And to prove in the same — 
case that there is ovly one such ratio, or that if any one positive ratio a satisfy all the 
conditions (239.), no greater ratio ¢ (> a@) can possibly satisfy all those conditions, we 
may observe that however little may be the excess 0 a+ c of the ratio c over a, this 
excess may be multiplied by a positive whole number m’ so large that the product 
shall be greater than unity, in such a manner that 
m (8 a+c)>1, (242.) 
and therefore 
1 1 
@ate>=—, and ¢c>—+a; (243.) 
m m 
and that then another positive (or null) whole number 7’ can be so chosen that 
, 
n'n 
Ll+n’ 1l+n’ 
7 = — b, x 5 z 
UL Li m m 
> b, (244) 
with which selection we shall have, by (239.) (240.) (243.), 
sve > : 2 
iri ea sy Cra ere * (245.) 
whereas, if c satisfied the conditions (239.) it ought to be less than this fraction 
aa , because the square of this positive fraction iS greater by (244.) than the pro- 
