358 Professor Haminton on Conjugate Functions, 
We may also call the numbers vb and @ yb by the common name of roots, or 
(more fully) sqguare-roots of the positive number}; distinguishing them from each 
other by the separate names of the positive square-root and the contra-positive 
square-root of that number b, which may be called their common square: though we 
may sometimes speak simply of the square-root of a (positive) number, meaning then 
the positive root, which is simpler and more important than the other. 
25. The idea of the contizxwity of the progression from moment to moment in 
time involves the idea of a similarly continuous progression in magnitude from any 
one effective step or interval between two different moments, to any other unequal 
effective step or other unequal interval ; and also the idea of a continuous progres- 
sion in ratio, from any one degree of inequality, in the way of relative largeness or 
smallness, as a relation between two steps, to any other degree. Pursuing this train 
of thought, we find ourselves compelled to conceive the existence (assumed in the 
last article) of a determined magnitude », exactly intermediate im the way of ratio 
between any two given unequal magnitudes a and ”’, that is, larger or smaller than 
the one, in exactly the same proportion in which it is smaller or larger than the 
other ; and therefore also the existence of a determined number or ratio a which is 
the exact square-root of any proposed (positive) number or ratio 6. ‘To show this 
more fully, let A B D be any three given distinct moments, connected by the relations 
De 
B—A 
= i, iS ib (214.) 
which require that the moment 8B should be situated between a and p ; and let c be 
any fourth moment, lying between B and p, but capable of being chosen as near to B 
or as near to D as we may desire, in the continuous progression of time. Then the 
two ratios 
D—A 
Cas 
and 
Besa C—A 
will both be positive ratios, and both will be ratios of largeness, (that is, each will be 
a relation of a larger to a smaller step,) which we may denote for abridgement as 
follows, 
—A D—A 
c =— 
Sy Sax (215.) 
B-—A cC—*& 
but by choosing the moment c sufficiently near to B we may make the ratio x ap- 
proach as near as we desire to the ratio of equality denoted by 1, while the ratio y 
