334 Professor Hamiiron on Conjugate Functions, 
to the base, as such or such a particular multiple thereof, or as mentally generated 
from that base by such or such a particular act of multipling; and that every such 
particular relation, and every such particular act of multipling, may be distinguished 
from all such other relations, and from all such other acts, in the entire series or 
system of these relations, and in the entire system of these acts of multipling, by its 
own special or determining whole number, whether ordinal or cardinal, and whether 
positive, or contra-positive, or null. Now ‘every such relation or act must be con- 
ceived to have a certain inverse or reciprocal, by which we may, in thought, connect 
the base with the multiple, and return to the former from the latter: and, generally, 
the conception of passing (in thought) from a base or unit-step to any one of its 
multiples, or of returning from the multiple to the base, is included in the more com- 
prehensive conception of passing from any one such multiple to any other ; that is, 
from any one step to any other step commensurable therewith, two steps being said to 
be commensurable with each other when they are multiples of one common base or 
unit-step, because they have then that common base or unit for their common mea- 
surers The base, when thus compared with one of its own multiples, may be called 
a sub-multiple thereof ; and, more particularly, we may call it the ‘second positive 
sub-multiple” of its own second positive multiple, the “first contra-positive sub- 
multiple”? of its own first contra-positive multiple, and so forth; retaining always, to 
distinguish any one sub-multiple, the determining ordinal of the multiple to which it 
corresponds: and the act of returning from a multiple to the base, may be called an 
act of swb-multipling or (more fully) of sub-multipling by the same determining 
cardinal number by which the base had been multipled before; for example, we may 
return to the base from its second contra-positive multiple, by an act of thought 
which may be called sub-multipling by contra-positive two. Some particular sub- 
multiples, and acts of sub-multipling, have particular and familiar names; thus, the 
second positive sub-multiple of any given step, and the act of sub-multipling a given 
step by positive two, may be more familiarly described as the half of that given step, 
and as the act of halving it. And the more comprehensive conception above men- 
tioned, of the act of passing from any one step b to any other step ¢ commensurable 
therewith, or from any one to any other multiple of one common measure, or base, 
or unit-step a, may evidently be resolved into the foregoing conceptions of the acts 
of multipling and sub-multipling; since we can always pass first by an act of sub- 
multipling from the given step b, considered as a multiple of the base a, to that 
base a itself, as an auxiliary or intermediate thought, and then proceed, by an act of 
multipling, from this auxiliary thought or step, to its other multiple c. Any one 
step ¢ may therefore be considered as a multiple of a sub-multiple of any other 
