and on Algebra as the Science of Pure Time. 825 
number contra-positive three, as a possible answer to the foregoing general question ; 
and it implies, when prefixed to the sign of the base a, in the complex written sign 
3 © a of the corresponding multiple step, that this multiple step has been formed, 
(as already shown in the equations (102.), ) by making three steps equal to the base 
a in length, but in the direction opposite thereto. Again, the mark 1 may be re- 
garded as a written sign of the cardinal number positive one, and 1 a denotes (in 
this view) the step formed by making one such step as a, and in the same direction, 
that is, (as before,) the original step a itself ; and O denotes the cardinal number 
none, so that O a is (as before) a symbol for the null step from a to a, which step we 
haye also marked before by the simple symbol 0, and which is here considered as 
formed by making no effective step like a. In general, this view of the numeral 
signs (103.), as denoting cardinal numbers, conducts to the same ultimate interpre- 
tations of the symbols (99.), for the steps of the series (98.), as the former view, 
which regarded those signs (103.) as denoting ordinal numbers. ; 
If we adopt the latter view of those numeral signs (103.), which we shall call by the 
common name of whole (or integer) numbers, (as distinguished from certain broken 
or fractional numbers to be considered afterwards,) we may conveniently continue to 
use the word mudtiple (occasionally) as a verb active, and may speak of the several 
multiple steps of the series (98.), or (99.), as formed from the base a, by mudtipling 
that base by the several whole (cardinal) numbers: because every multiple step may be 
conceived as generated (in thought) from the base, by a certain mental act, of which 
the cardinal number is the mark. Thus we may describe the multiple step 3 0 a, 
(which is, in the ordinal view, the third contra-positive multiple of a,) as formed from 
the base a by multipling it by contra-positive three. Some particular acts of multi- 
pling have familiar and special names, and we may speak (for instance) of doubling 
or tripling a step, instead of describing that step as being multipled by positive two, 
or by positive three. In general, to distinguish more clearly, in the written symbol 
of a multiple step, between the base and the determining number (ordinal or cardi- 
nal), and to indicate more fully the performance of that mental act (directed by the 
number) which generates the multiple from the base, the mark x may be inserted 
between the sign of the base, and the sign of the number; and thus we may 
denote the series of multiple steps (99.) by the following fuller symbols, 
SLOmas we OLE ia, UO Xsa) OLX a, ell X. a, eX B08 Xmen OCC ee LOd, ) 
and which 1 x a (for example) denotes the original step a itself, and 2 x a represents 
the double of that original step. 
