and on Algebra as the Science of Pure Time. 367 
é and & being any two positive whole numbers assumed at pleasure : with this choice, 
nn 
therefore, of the numbers m and 7’, some one (at least) such as —, among the 
squares of fractions (249.), that is, some one at least among the following squares of 
fractions, 
1 1 Q Q 3 3 Qitk | @iik 
Sek Dik Lik Pe tk Dik ek OTe Qik 
of which the last is =72, must lie between any two proposed unequal positive ratios 
6 and 6”, of which the greater 6” does not exceed that last square 77, and of which 
i 1 aj) ‘ 
the difference 9 6'+6" is not less than BP and positive whole numbers i and & can 
always be so chosen as to satisfy these last conditions, however great the proposed 
ratio 6" may be, and however little may be its excess © b'+b" over the other pro- 
posed ratio U’. 
27. With these preparations it is easy to prove, in a new and formal way, the ex- 
istence of one determined positive square root 6 for every proposed positive ratio 
6, whether that ratio 6 be or be not the square of any whole or of any fractional 
number ; for we can now prove this Theorem : 
The square aa of the determined positive ratio a, of which ratio the existence 
was shown in the IIId. Lemma, is equal to the proposed positive ratio } in the same 
Lemma; that is, 
‘ / t 
: n n'n | 
if @> —, whenever ——; < B, 
mM mm 
n'! n' ni! 259. 
and a < all whenever ailonl. b, ¢ ) 
then aa=b, a= Vb, j 
mn’ m' n' being any positive whole numbers which satisfy the conditions here men- 
tioned, and } being any determined positive ratio. 
For if the square a a of the positive ratio a, determined by these conditions, could 
be greater than the proposed positive ratio 4, it would be possible, by the [Vth Lemma, 
to insert between them some positive fraction which would be the square of another 
positive fraction = ; that is, we could choose m and 7 so that 
ah, pee <a * (253.) 
mm” ? mm 
