and on Algebra as the Science of Pure Time. 329 
ing the composition of steps. In the first place, the numbers to be added may be 
added in any arbitrary order ; that is, 
vtpoputy, 
E+vtyr=ut+éEt+v= &e., (110.) 
&e. ; 
we may therefore collect the positive numbers into one algebraical sum, and the con- 
tra-positive into another, and then add these two partial sums to find the total sum, 
omitting (if it anywhere occur) the number None or Zero, as not capable of altering 
the result. In the next place, positive numbers are algebraically added to each other, 
by arithmetically adding the corresponding arithmetical numbers or quotities, and 
considering the result as a positive number ; thus positive two and positive three, 
when added, give positive five: and contra-positive numbers, in like manner, are al- 
gebraically added to each other, by arithmetically adding their quotities, and consi- 
dering the result as a contra-positive number ; thus,contra-positive two and contra-po- 
sitive three have contra-positive five for their algebraic sum. In the third place, a 
positive number and a contra-positive, when the quotity of the positive exceeds that 
of the contra-positive, give a positive algebraic sum, in which the quotity is equal to 
that excess ; thus positive five added to contra-positive three, gives positive two for 
the algebraic sum: and similarly, a positive number and a contra-positive number, if 
the quotity of the contra-positive exceed that of the positive, give a contra-positive 
algebraic sum, with a quotity equal to the excess; for example, if we add positive 
three to contra-positive five, we get contra-positive two for the result. Tinally, a posi- 
tive number and a contra-positive, with equal quotities, (such as positive three and 
contra-positive three,) destroy each other by addition ; that is, they generate as their 
algebraic sum the number None or Zero. 
It is unnecessary to dwell on the algebraical operation of decomposition of multiple 
steps, and consequently of whole or multipling numbers, which corresponds to and 
includes the operation of arithmetical subtraction ; since it follows manifestly from 
the foregoing articles of this Essay, that the decomposition of numbers (like that of 
steps) can always be performed by compounding with the given compound number 
‘(that is, by algebraically adding thereto) the opposite or opposites of the given com- 
ponent or components: the number or numbers proposed to be subtracted are there- 
fore either to be neglected if they be null, since in that case they have no effect, or 
else to be changed from positive to contra-positive, or from contra-positive to positive, 
(their quotities being preserved,) and then added (algebraically) in this altered state. 
Thus, positive five is subtracted algebraically from positive two by adding contra-posi- 
