and on Algebra as the Scienee of Pure Time. 345 
The properties (114.) and (115.) of algebraic products of whole numbers extend to 
products of fractional numbers also ; that is, we may change in any manner the order 
of the fractional factors ; and if we resolve any one of those factors into two or 
more algebraic parts by the rules of algebraic addition and subtraction, we may com- 
bine each part separately as a partial factor with the other factors proposed, so as to 
form by algebraic multiplication a partial fractional product, and then add together 
those partial products algebraically to obtain the total product: or, in written 
symbols, 
aKa Sx Gy ey (154.) 
and 
zx (2+5)=€ «5)4 (E* 2): &e., (155.) 
because 
x(t v)=(Zxw)+(2xv), (156.) 
whatever steps may be denoted by wv and vw” and whatever fractional (or whole) 
number by ~ . We may also remark that 
be 
> = 
y x B=y xa, according as B = a, if y> 0, (157.) 
< < 
but that 
< ‘ a . 
y x B=y xa, according as B = a, if y <0, (158.) 
= < 
a B y denoting any three fractional (or whole) numbers. 
The deduction of one of two fractional factors from the other and from the product, 
_ may be called (by analogy to arithmetic) the algebraic division of the given fractional 
product as a dividend, by the given fractional factor as a divisor ; and the result, 
which may be called the quotient, may always be found by algebraically multiplying 
the proposed dividend by the reciprocal of the proposed divisor. This more general 
conception of quotient, agrees with the process of the 15th article, for the division of 
one whole number by another, when that process gives an accurate quotient in whole 
numbers ; and when no such integral and accurate quotient can be found, we may 
still, by our present extended definitions, conceive the numerator of any fraction to 
be divided by the denominator, and the quotient of this division will be the fractional 
number itself. In this last case, the fractional number is not exactly equal to any 
