and on Algebra as the Science of Pure Time. 34.9 
whole numbers; since the fractional ratio denoted by the symbol ~ is the ratio of 
le 
the multiple step » x a to the multiple step » x a. In like manner it follows, from 
the same conceptions and definitions, that 
a iD) aad reciprocally v = : x bif x = =; (166.) 
and more generally, that 
and reciprocally, 
sae e ft Fs (168.) 
a 
whatever effective steps may be denoted by a, b, c,d, and whatever fraction by ~. 
We may also conceive combinations of ratios with each other, by operations which 
we may call Addition, Subtraction, Multiplication, and Division of Ratios, or of 
general algebraic numbers, from the analogy of these operations to those which we 
have already called by the same names, in the theories of whole numbers and of 
fractions. And as we wrote, in treating of whole numbers, 
w=v+m when w x a= (vx a)t+(u Xa), (107.) 
and 
w=vX mu when w x a=v X (u Xa); (111.) 
and, in the theory of fractions, 
wr v Us J 
a= pe when “xp=(=x b) +(% x b), (149.) 
eye lu le B 
and 
n v’ un v’ 
~=+ x? when vxva=%x(% x), (152.) 
jee ot a ft NE 
with other similar expressions ; so we shall now write, in the more general theory of 
ratios, 
a =e when axe=(3xe)t+(2x e), (169.) 
and 
bei ob b’ vb! (bd 
ae eae when ax esa x(>x e): (170.) 
