362 Professor Hamitton on Conjugate Functions, 
ratio 5 of smallness, <1, 56> 0. But instead of thus comparing, with the progression 
of the positive ratio 2, the connected but opposite progression of the connected posi- 
tive ratio ux, and showing that these progressions meet each other in a certain 
intermediate state or positive ratio a, we might have compared the two connected 
and not opposite progressions of the two connected positive ratios 7 and 2x2, of 
which the latter is the square of the former ; and might have shown that the square 
(=a x v= x) increases constantly and continuously with the root (=x), from the 
state zero, so as to pass successively through every state of positive ratio 4. To 
develope this last conception, and to draw from it a more formal (if not a more con- 
vineing) proof than that already given, of the necessary existence of a conceivable 
positive square-root for every conceivable positive number, we shall here lay downa 
fow Lemmas, or preliminary and auxiliary propositions. 
Lemma I. If 2’ =a, and x>0, a'>0, then aa’ 
ax; (227.) 
<< 
Allv 
that is, the square a’x’ of any one positive number or ratio 2’, is greater than, or 
equal to, or less than the square wz of any other positive number or ratio xz, ac- 
cording as the number ’ itself is greater than, or equal to, or less than the number 
x 3; one number 2’ being said to be greater or less than another number x, when it is 
on the positive or on the contra-positive side of that other, in the general progression 
of numbers considered in the 21st article. This Lemma may be easily proved from 
the conceptions of ratios and of squares ; it follows also without difficulty from the 
theorem of multiplication (200.). And hence we may obviously deduce as a corollary 
of the foregoing Lemma, this converse proposition : 
if 2’ 
ALLY 
va, and x>0, a>0, then 2’ 
ALLY 
z5 (228.) 
that is, if any two proposed positive numbers have positive square-roots, the root of 
the one number is greater than, or equal to, or less than the root of the other 
number, according as the former proposed number itself is greater than, or equal to, 
or less than the latter proposed number. 
The foregoing Lemma shows that the square constantly increases with the root, 
from zero up to states indefinitely greater and greater. But to show that this in- 
crease is continuous as well as constant, and to make more distinct the conception of 
such continuous increase, these other Lemmas may be added. 
Lemma Il. If @ and a” be any two unequal ratios, we can and must conceive the 
