and on Algebra as the Science of Pure Time. 347 
But it is not so immediately clear what ought to be regarded as the meaning of a 
fractional sign, in the case when the denominator is null, and when therefore the act 
of fractioning prescribed by the notation involves a sub-multipling by zero. ‘To 
discuss this case, we must remember that to sub-multiple a step b by a whole number 
u, is, by its definition, to find another step a, which, when multipled by that whole 
number p, shall produce the proposed step b; but, whatever step a may be, the theory 
of multiple steps (explained im the 13th article) shows that it necessarily produces 
the null step 0, when it is multipled by the null number zero ; that is, the equation 
Ova 0) (1€0.) 
is true independently of a, and consequently we have always 
Oxa+bd, ifo+o. (161.) 
It is, therefore, impossible to find any step a, in the whole progression of time, which 
shall satisfy the equation 
Mab = AOL. Ora. — b's (162.) 
= 
if the given step b be effective ; or, in other words, it is impossible to sub-multiple an 
effective step by zero. The fractional sign 6 denotes therefore an ¢mpossible act, if it 
be applied to an effective step: and the zero-submultiple of an effective step is a phrase 
which involves a contradiction. On the other hand, if the given step » be null, it is 
not only possible to choose some one step a which shall satisfy the equations (162.), 
but every conceivable step possesses the same proposed property ; in this case, there- 
fore, the proposed conditions lay no restriction on the result, but at the same time, 
and for the same reason, they fail to give any information respecting it: and the act 
of sub-multipling a null step by zero, is indeed a possible, but it is also an indeter- 
minate act, or an act with an indeterminate result ; so that the zero-submultiple of a 
: 1 : ; : 
null step, and the written symbol 9 x O% are spoken or written signs which do not 
specify any thing, although they do not involve a contradiction. We see then that 
while a fractional number is in general the sign of a possible and determinate act of 
fractioning, it loses one or other of those two essential characters whenever its deno- 
minator is zero ; for which reason it becomes comparatively unfit, or at least inconve- 
nient, in this case, for the purposes of mathematical reasoning. And to prevent the 
confusion which might arise from the mixture of such cases with others, it is conve- 
nient to lay down this general rule, to which we shall henceforth adhere: that all 
VOL. XVII. 3T 
