and on Algebra as the Science of Pure Time. 361 
of which the latter has been shown to be impossible. Hence finally, if a and B be 
any two distinct moments, and if we choose a third moment c such that 
the moment c will not be uniserial with a and B, that is, no one equi-distant series of 
moments can be imagined, comprising all the three. And all that has here been 
sliown respecting the square-root of two, extends to the square-root of three, and 
may be illustrated and applied in an infinite variety of other examples. We must 
then admit the existence of pairs of steps which have no common measurer ; and 
may call the ratio between any two such steps an incommensurable ratio, or incom- 
mensurable number. 
More formal proof of the general existence of a determined positive square-root, 
commensurable or incommensurable, for every determined positive ratio: conti- 
nuity of progression of the square, and principles connected with this continuity. 
26. The existence of these incommensurables, (or the necessity of conceiving 
them to exist,) is so curious and remarkable a result, that it may be usefully con- 
firmed by an additional proof of the general existence of square-roots of positive 
ratios, which will also offer an opportunity of considering some other important prin- 
ciples. 
The existence of a positive square-root a= /}, of any proposed ratio of largeness 
b> 1, was proved in the foregoing article, by the comparison of the two opposite 
progressions of the two ratios w and u ax, from the states r=1, u ax 6=5, for 
which u «xb >a, to the states r=b, uxxb=1, for which uxxb<w; for this 
comparison obliged us to conceive the existence of an intermediate state or ratio « 
between the limits 1 and 4, as a common state or state of meeting of these two oppo- 
site progressions, corresponding to the conception of a moment at which the de- 
creasing ratio u # x 6 becomes exactly equal to the increasing ratio w, having been 
previously a greater ratio (or a ratio of greater relative largeness between steps), and 
becoming afterwards a lesser ratio (or a ratio of less relative largeness). And it 
was remarked that an exactly similar comparison of two other inverse progressions 
would prove the existence of a positive square-root / of any proposed positive 
