and on Algebra as the Science of Pure Time. 391 
b over the number a, or the remainder obtained by the algebraical subtraction of the 
latter number from the former ; and we should have had the equation, 
b6-—a=Oa+4, (357.) 
which is, with respect to nwmbers, or ratios, what the equation (355.) is, with respect 
to steps. And when such symbols of remainders, ») — a or 5—a, are to be combined 
with other symbols in the way of algebraical addition, it results, from principles 
already explained, that they need not be enclosed in parentheses; for example, we 
may write simply ¢ + b —a instead of ¢ +(» —a), because the sum denoted by this 
last notation is equivalent to the remainder (ce +b) —a. But the parentheses (or 
some other combining mark) must be used, when a remainder is to be subtracted ; 
thus the symbol ¢ — » — a is to be interpreted as (ec — b) — a, and not as ec — (b — a), 
which latter symbol is equivalent to (¢—b) +a, or e—b+a. 
35. With this way of denoting the algebraical subtraction of steps, and that of 
numbers, we have the formula, 
O—a=O9a, O-—a=0 a, (358.) 
O denoting in the one a null step, and in the other a null number. And if we farther 
agree to suppress (for abridgement) this symbol O when it occurs in such combina- 
nations as the following, 0 + a, O—a, 0+a, O—a, writing, in the case of steps, 
O+a=>+a, O—a=—a, (359.) 
and similarly, in the case of numbers, 
0+a= +a, O—a=—a, (360.) 
and, in like manner, 
oe (361.) 
O+a+b=+a+tb, O-at+b=-atb, 
we shall then have the formula 
+asa, —a=QOa, (362,) 
and 
+a=a, —a=04, (363.) 
of which the one refers to steps and the other to numbers. With these conventions, 
