and on Algebra as the Science of Pure Time. 403 
On the Addition, Subtraction, Multiplication, and Division, of Number- Couples, 
as combined with each other. 
6. Proceeding to operations upon number-couples, considered in combination with 
each other, it is easy now to see the reasonableness of the following definitions, and 
even their necessity, if we would preserve in the simplest way, the analogy of the the- 
ory of couples to the theory of singles : 
(1, b)+ (Gy, G)=(i+a, b+); (52.) 
(A, by —-(4,, a,jy=(h —, b,—a.) B (53.) 
(, ’ b,) (a, ’ ar=(b, > by) x (Q ) a)=(b, (oh Si), Gs; b, aq,t+ b, as) : 6 1.) 
(ho) (ba +h, boa —b i; 
“ (Q 5 a2) ( Gy Fas OPN a?-Fa? ) (55.) 
« 
Were these definitions even altogether arbitrary, they would at least not contradict 
each other, nor the earlier principles of Algebra, and it would be possible to draw 
legitimate conclusions, by rigorous mathematical reasoning, from premises thus arbi- 
trarily assumed : but the persons who have read with attention the foregoing remarks 
of this theory, and have compared them with the Preliminary Essay, will see that 
these definitions are really not arbitrarily chosen, and that though others might have 
been assumed, no others would be equally proper. 
With these definitions, addition and subtraction of number-couples are mutually 
inyerse operations, and so are multiplication and division ; and we have the relations, 
(1, 5) + (2), a=(a, , %) + (21, Fy), (56.) 
(2,5 2) x (4, %)=(Q, 4) x (4, be)s (57.) 
(A> i) f(a a’) + (4, a,)} = (4, b,) (a, @) + (1, b,) (@,, @): (58.) 
we may, therefore, extend to number-couples all those results respecting numbers, 
which have been deduced from principles corresponding to these last relations. For 
example, 
{(4,, by) +(a, a2)} x {(h, by) + (a, &)}= 
(h, b2) (i, b,) +2(6, b,) (M4, a2) + (a, a2) (M, 1), (59.) 
VOL. XVII. 4 c¢ 
