and on Algebra as the Science of Pure Time. 351 
And as, by our conceptions and notations respecting the ordinal relation of one 
fractional number to another, (as subsequent, or coincident, or precedent, in the 
general progression of such numbers from contra-positive to positive,) we had the 
relations, 
> 
, 
v Vv v 
=, when = x= — x a,,8 > 05 
woe lH < He 
so we may now establish, by analogous conceptions and notations respecting ratios, 
the relations, 
fe ity ” > 
by ee aee b 2 iy di 
ae a) NOS nt aie iP app O:: (177-) 
that is, more fully, 
BS pa 
= > ywhen (2 xa) +a>(Yxa) +a, (178.) 
HW” , “a 4 hs 
a = a when (3, x a) +a=(Fxe)+a, (179.) 
and 
b’ b’ b” b’ 
mw <g> When (5 xe) +a<(Zxa) +a; (180.) 
the symbol a denoting any moment of time, and a any late-making step. ‘The rela- 
tion (179.) is indeed an immediate consequence of the first conceptions of steps and 
ratios; but it is inserted here along with the relations (178.) and (180.), to show 
more distinctly in what manner the comparison and arrangement of the moments 
, 
(x s)+a (Gx e)+a, &. (181.) 
which are suggested and determined by the ratios or numbers = ; = , &c., (in combi 
nation with a standard moment a and with a late-making step a,) enable us to com- 
pare and arrange those ratios or numbers themselves, and to conceive an indefinite 
progression of ratio from contra-positive to positive, including the indefinite pro- 
gression of whole numbers (103.), and the more comprehensive progression of frac- 
tional numbers considered in the 17th article: for it will soon be shown, that though 
every fractional number is a ratio, yet there are many ratios which cannot be ex- 
pressed under the form of fractional numbers. Meanwhile we may observe, that the 
theorems (151.) (157.) (158.) respecting the ordinal relations of fractions in the 
general progression of number, are true, even when the symbols a B y denote ratios 
which are not reducible to the fractional form; and that this indefinite progression 
