and on Algebra as the Science of Pure Time. 335 
step b, if those two steps be commensurable ; and the act of passing from the one to 
the other is an act compounded of sub-multipling and multipling. 
Now, all acts thus compounded, besides the acts of multipling and sub-multipling 
themselves, (and other acts, to be considered afterwards, which may be regarded as 
of the same kind with these, being connected with them by certain intimate relations, 
and by one common character,) may be classed in algebra under the general name of 
multiplying acts, or acts of algebraic multiplication ; the object on which any such 
act operates being called the muléiplicand, and the result being called the product ; 
while the distinctive thought or sign of such an act is called the algebraic multiplier, 
or multiplying number : whatever this distinctive thought or sign may be, that is, what- 
ever conceived, or spoken, or written specific rule it may involve, for specifying one 
particular act of multiplication, and for distinguishing it from every other. ‘The 
relation of an algebraic product to its algebraic multiplicand may be called, in general, 
ratio, or algebraic ratio ; but the particular ratio of any one particular product to 
its own particular multiplicand, depends on the particular act of multiplication by 
which the one may be generated from the other: the mwmber which specifies the act 
of multiplication, serves therefore also to specify the resulting ratio, and every 
number may be viewed either as the mark of a ratio, or as the mark of a multiplica- 
tion, according as we conceive ourselves to be analytically examining a product 
already formed, or synthetically generating that product. 
We have already considered that series or system of algebraic integers, or whole 
numbers, (positive, contra-positive, or null,) which mark the several possible ratios of 
all multiple steps to their base, and the several acts of multiplication by which the 
former may be generated from the latter; namely all those several acts which we 
haye included under the common head of multipling. The inverse or reciprocal acts 
of sub-multipling, which we must now consider, and which we have agreed to regard 
as comprehended under the more general head of multiplication, conduct to a new 
class of multiplying numbers, which we may call reciprocals of whole numbers, or, 
more concisely, reciprocal numbers ; and to a corresponding class of ratios, which we 
may call reciprocals of integer ratios. And the more comprehensive conception of 
the act of passing from one to another of any two commensurable steps, conducts to 
a correspondingly extensive class of multiplying acts, and therefore also of multiplying 
numbers, and of ratios, which we may’ call acts of fractioning, and fractional 
numbers, or fractional ratios ; while the product of any such ‘act of fractioning, or 
of multiplying by any such fractional number, that is, the generated step which is any 
multiple of any sub-multiple of any proposed step or multiplicand, may be called a 
Fraction of that step, or of that multiplicand. A fractional number may therefore 
