and on Algebra as the Science of Pure Time. 343 
it is sufficient to add the numerators, because then the proposed fractional steps are all 
multiples of one common sub-multiple of the common unit-step b, namely of that sub- 
multiple which is determined by the common denominator ; it is therefore sufficient, in 
other cases, to prepare the fractions so as to satisfy this condition of haying a common 
denominator, and afterwards to add the numerators so prepared, and to combine their 
sum as the new or resulting numerator of the resulting fractional sum, with the 
common denominator of the added fractions as the denominator of the same frac- 
tional sum ; which may, however, be sometimes simplified by the omission of common 
factors, according to the principle (135.). ‘Thus 
v (’ x pw) + (uw xv) p y Bs vat py 5 
== er 5 .0r more concisely = -> = = =— , &e.; (150.) 
MX pe lM he 
Be 
i= 
for, as a general rule of algebraic notation, we may omit at pleasure the mark of 
tmaultiplication between any two simple symbols of factors, (except the arithmetical 
signs 1, 2, 3, &c.,) without causing any confusion ; and when a product thus denoted, 
by the mere juxta-position of its factors, (without the mark x ,) is to be combined 
with other symbols in the way of addition, by the mark +, it is not necessary to en- 
close that symbol of a product in parentheses : although in this Elementary Essay we 
have often used, and shall often use again, these combining and enclosing marks, for 
greater clearness and fulness. It is evident that the addition of fractions may be 
performed in any arbitrary order, because the order of composition of the fractional 
steps is arbitrary. 
The algebraical subtraction of one given fractional number — from another un- 
. . . fe “7° 
equal fractional number ~, is an operation suggested by the decomposition of a 
. . v . . . 
given compound fractional step — xb into a given component fractional step —, x b 
” Bh 
J 
and a sought component fractional step “x b, (these three steps being here sup- 
posed to be all effective :) and it may be performed by compounding the opposite of 
the given component step with the given compound step, or by algebraically adding 
the opposite O~ of the given fractional number ~ to the other given fractional 
le 
number ~, according to the rule (150.). When we thus subtract one fractional 
number from another with which it does not coincide, the result is positive or contra- 
positive according as the fraction from which we subtract is on the positive or contra- 
positive side of the other; and thus we have another general method, besides the 
tule (146.), for examining the ordinal relation of any two unequal fractions, in the 
general progression of numbers. This ordinal relation between any two fractional 
