and on Algebra as the Science of Pure Time. 327 
w—=v+p, when w x a=(v X a)+(m x a), (107.) 
and 
wr=E+vd4-p, when w x a= (e x a)+ x a) +(u x a), (108.) 
together with other similar expressions for the case of more component steps than 
three. In this notation, ; 
(Exa)t@ xa) tx a)aE+vtu) xe, (109.) 
&e. 
(v x a)+(m x a)=(vtn) x a, \ 
whatever the whole numbers » v & may be; equations which are to be regarded here 
as true by definition, and as only serving to explain the meaning attributed to such 
complex signs as v+p, or €+v+p, when wv & are any symbols of whole nun- 
bers: although when we farther assert that the equations (109.) are true inde- 
pendently of the base or unit-step a, so that symbols of the form v + » or §+v +p, 
denote whole numbers independent of that base, we express in a new way a theorem 
which we had before assumed to be evidently true, as an axiom and not a definition, 
respecting the composition of multiple steps. 
In the particular case when the whole numbers denoted by , v & are positive, the 
law of composition of those numbers expressed by the notation v + or E+v+ 4, 
as explained by the equations (109.), is easily seen to be the law called addition of 
numbers (that is of quotities) in elementary arithmetic ; and the quotity of the com- 
pound or resulting whole number is the arithmetical swm of the quotities of the com- 
ponent numbers, this arithmetical swm being the answer to the question, How many 
things or thoughts does a total group contain, if it be composed of partial groups 
of which the quotities are given, namely the numbers to be arithmetically added. 
For example, since (3 x a) +(2 x a) is the symbol for the total or compound mul- 
tiple step composed of the double and the triple of the base a, it must denote the 
quintuple or fifth positive multiple of that base, namely 5 x a; and since we have 
agreed to write 
(8 x a)+(2xa)=(3 + 2) xa, 
we must interpret the complex symbol 3 + 2 as equivalent to the simple symbol 5; 
in seeking for which latter number five, we added, in the arithmetical sense, the given 
component numbers fwo and three together, that is, we formed their arithmetical 
sum, by considering how many steps are contained in a total group of steps, if the 
component or partial groups contain two steps and three steps respectively. In like 
