368 Professor Hamitton on Conjugate Functions, 
and then, by the Corollary to the Ist Lemma, and by the conditions (252.), we should 
be conducted to the two following incompatible relations, 
= a 254 
m SiGe 4 mm (254.) 
A similar absurdity would result, if we were to suppose aa less than b; a a must 
therefore be equal to 4, that is, the theorem is true. It has, indeed, been here as- 
sumed as evident, that every determined positive ratio a has a determined positive 
square a a3 which is included in this more general but equally evident principle, that 
any two determined positive ratios or numbers have a determined positive product. 
We find it, therefore, proved, by the most minute and rigorous examination, that 
if we conceive any positive ratio 2 or a to increase constantly and continuously from 
0, we must conceive its square a x or aa to increase constantly and continuously with 
it, so as to pass successively but only once through every state of positive ratio 6: 
and therefore that every determined positive ratio 6 has one determined positive 
square root yb, which will be commensurable or incommensurable, according as 6 can 
or cannot be expressed as the square of a fraction. When ) cannot be so expressed, 
it is still possible to approximate in fractions to the incommensurable square root v }, 
by choosing successively larger and larger positive denominators, and then seeking 
for every such denominator m’ the corresponding positive numerator m’ which satisfies 
the two conditions (244.); for although every fraction thus found will be less than 
the sought root v 6, yet the error, or the positive correction which must be added to 
it in order to produce the accurate root v 6, is less than the reciprocal of the deno- 
minator m’, and therefore may be made as little different as we please from 0, (though 
it can never be made exactly = 0,) by choosing that denominator large enough. 
This process of approximation to an incommensurable root v 6 is capable, therefore, 
of an indefinitely great, though never of a perfect accuracy ; and using the notation 
already given for limits, we may write 
P rn ., nn }+n l+n’ ; 
SE RN yh ELAR SN (255.) 
mn mim m m 
and may think of the incommensurable root as the limit of the varying fractional 
number. 
The only additional remark which need be made, at present, on the subject of the 
progression of the square x .r, or aa, as depending on the progression of the root x, 
