352 Professor Hamitton on Conjugate Functions, 
of number, or of ratio, from contra-positive to positive, corresponds in all respects to 
the thought from which it was deduced, of the progression of time itself, from mo- 
ments indefinitely early to moments indefinitely late. 
22. It is manifest, on a little attention, that the ratio of one effective step b to 
another a, is a relation which is entirely determined when those steps are given, but 
which is not altered by multiplying both those steps by any common multiplier, 
whether positive or contra-positive ; for the relative largeness of the two steps is not 
altered by doubling or halving both, or by enlarging or diminishing the magnitudes of 
both in any other common ratio of magnitude, that is, by multiplying both by any 
common positive multiplier: nor is their relative direction altered, by reversing the 
directions of both. We have then, generally, 
pS Ses (182.) 
= (183.) 
Hence, by (167.), the two steps= x b and © x eare related in one common ratio, 
namely the ratio = to the common step ce, and therefore are equivalent to each 
other ; that is, we have the equation, 
c b 
abe acs (184.) 
whatever three effective steps may be denoted by a b ec. 
In general, when any four effective steps a b ¢ @ are connected by the relation 
d b 
= = as (185.) 
that is, when the ratio of the step a to c is the same as the ratio of the step b to a, 
these two pairs of steps a, b and c, a may be said to be analogous or proportional 
pairs ; the steps a and ¢ being called the antecedents of the analogy, (or of the 
proportion) and the steps b and a being called the consequents, while a and d are the 
extremes and b and ¢ the means. And since the last of these four steps, or the 
second consequent 4, may, by (168.), be expressed by the symbol 2 x ec, we see, by 
(184.), that it bears to the first consequent b the ratio < of the second antecedent 
¢ to the first antecedent a; that is, 
