and on Algebra as the Science of Pure Time. 401 
this choice, we ought to take care to satisfy, if possible, the essential condition that 
there shall be always one determined number-couple to express the ratio of any one 
determined step-couple to any other, at least when the latter is not null : since this was 
the very object, to accomplish which we were led to introduce the conception of these 
number-couples. It is easy to show that no choice simpler than the following, 
yi——I1, y= 0, (36.) 
would satisfy this essential condition : and for that reason we shall now select these 
two numbers, contra-positive one and zero, for the two constants of multiplication, 
and shall establish finally this formula for the multiplication of any step-couple (4, 42) 
by any number-couple (a,, 0), 
(4; %) (1, By) = (4 a, —@, a5, @, a, + @ ay), (37.) 
5. In fact, whatever constants of multiplication y, y, we may select, if we denote 
by (»,,>,) the product of the step-couple (4,, 4,) by the number-couple (4, @,), so 
that 
(>, b) = (4, ay) x (4, ay), (38.) 
we have by (35.) the following expressions for the two coupled steps b,, »., of the 
product, 
Wat CRRA ea & Ay Any 
a SV | 9 eae Was \ (39.) 
= @% a, + Gy a, + Ya % Foy 
and therefore 
Bi =a, a, +Y1 U2 ay 5 t (40.) 
Py = A, a2 + Az a, +2 Ay a2 y 
if a, a, (3; 2. denote respectively the ratios of the four steps 4, 4, b, b, to one effective 
step c, so that 
8a) %, 4 =a °¢, {41.) 
and 
B= Gite ans (42.) 
from which it follows that 2 
a, fa, (a; + y2 a) —y, a2} =P: (a +2 a2) — Pa ys as t (43.) 
Ay fa; (a! + y2 a2) — yi a2 $ =P: at— Py a2; 
in order therefore that the numbers a, a, should always be determined by the equa- 
