and on Algebra as the Science of Pure Time. 341 
and shall enunciate these two cases respectively, by saying that in the first case the 
. y . rd . . . . 
fractional number — is on the positive side, and that in the second case it is on the 
ord . . v . v . 
contra-positive side, of the other fractional number —; or that in the first case —, _fol- 
. . v . . 
lows and that in the second it precedes —, in the general progression of numbers, 
ia 
from contra-positive to positive: definitions which may easily be shown to be con- 
sistent with each other, and which extend to whole numbers and their reciprocals, as 
included in fractional numbers, and to the number zero itself as compared with any of 
these. Thus, every positive number is on the positive side of zero and of every 
contra-positive number; while zero is on the positive side of all contra-positive 
numbers, but on the contra-positive side of all positive numbers: for example, 
2>0,2>03,03<0,08<2,0>03,0<2 (143.) 
Of two unequal positive whole numbers, the one which has the greater quotity is on 
the positive side, but among contra-positive numbers the reverse is the case; for 
example, 
3>2,08<02: (144) 
and in general a relation of subsequence or precedence between any two whole or 
fractional numbers is changed to the opposite relation of precedence or subsequence, 
by altering those numbers to their opposites, though a relation of equality or coinci- 
dence remains unaltered after such a change. Among reciprocals of positive whole 
numbers, the reciprocal of that which has the lesser quotity is on the positive side of 
the other, while reciprocals of contra-positive numbers are related by the opposite 
rule; thus 
! 1 1 : 
57 3? 62 ~*63’ that is, u2>u3, uO2<193z (145.) 
In general, to determine the ordinal relation of any one fractional number ~ to 
v . . - . 
another ie as subsequent, or coincident, or precedent, in the general progression 
of numbers, it is sufficient to prepare them by the principle (185.) so that their deno- 
minators may be equal and positive, and then to compare their numerators; for 
which reason it is always sufficient to compare the two whole numbers p xp xy xv’ 
and »’ x xu xv, and we have 
t 
> 
, according asuxpxpe xv =n xp xpxv: (146.) 
< 
All Vv 
Rs 
ee 
> . 
the abridged notation = implying the same thing as if we had written more fully 
< 
