. 
and on Algebra as the Science of Pure Time. 353 
Seay Perea ot (186.) 
a theorem which shows that we may transform the expression of an analogy (or pro- 
portion) between two pairs of effective steps in a manner which may be called alter- 
nation. (Compare the theorem (11.).) 
It is still more easy to perceive that we may invert an analogy or proportion 
between any two pairs of effective steps; or that the following theorem is true, 
c By je 
a, Syaput 
= mae (187.) 
Combining inversion with alternation, we see that 
bree 
d c 
it et (188.) 
(Compare the theorems (12.) and (13.).) 
In general, if any two pairs of effective steps a, b and c, a be analogous, we do 
not disturb this analogy by interchanging the extremes among themselves, or the 
means among themselves, or by substituting extremes for means and means for ex- 
tremes ; or by altering proportionally, that is, altering in one common ratio, or mul- 
tiplying by one common multiplier, whether positive or contra-positive, the two con- 
sequents, or the two antecedents, or the two steps of either pair: or, finally, by 
altering in inverse proportion, that is, multiplying respectively by any two reciprocal 
multipliers, the two extremes, or the two means. The analogy (185.) may therefore 
be expressed, not only in the ways (186.), (187.), (188.), but also in the following : 
axd axb d b axa b 
en Te Ser: ae 189. 
c a oa xee DUST Frias oh ae ( 9.) 
uwaxd eee, eee axbd 
Bac ae uae alain cis = sane (190.) 
a denoting any ratio of one effective step to another, and u a denoting the reciprocal 
ratio, of the latter step to the former. 
23. We may also consider it as evident that if any effective step ¢ be com- 
pounded of any others a and », this relation of compound and components will not 
be disturbed by altering the magnitudes of all in any common ratio of magnitude, 
that is by doubling or halving it, or multiplying all by any common positive multi- 
plier ; and we saw, in the 12th article, that the same relation of compound and com- 
ponents is not disturbed by reversing the directions of all: we may therefore mul- 
