360 Professor Hamiiton on Conjugate Functions, 
this more full examination what we before assumed to be true, that every positive 
number or ratio } has a positive (and therefore also a contra-positive) square-root. 
And hence we can easily prove another important property of ratios, which has 
been already mentioned without proof; namely that several ratios can and must be 
conceived to exist, which are incapable of being expressed under the form of whole 
or fractional numbers ; or, in other words, that every effective step a has other steps 
incommensurable with it ; and therefore that when any two distinct moments a and 
B are given, it is possible to assign (in various ways) a third moment c which shall 
not be wniserial with these two, in the sense of the Sth article, that is, shall not 
belong in common with them to any one equi-distant series of moments, comprising 
all the three. For example, the positive square-root of 2, which is evidently inter- 
mediate between 1 and 2 in the general progression of numbers, and which therefore 
is not a whole number, cannot be expressed as a fractional number either ; since if it 
could be put under the fractional form = , so that 
Ope deen (220.) 
we should then have 
@=~x7=7** (221.) 
m ™m mxm 
that is, 
MXn=2Qxmxm; (222.) 
but the arithmetical properties of quotities are sufficient to prove that this last equa- 
tion is impossible, whatever positive whole numbers may be denoted by m and n. 
And hence, if we imagine that 
b= /2xX a, a >O, (223.) 
the step b which is a mean proportional between the two effective and co-directional 
steps a and 2a (of which the latter is double the former) will be zncommensurable 
with the step a (and therefore also with the double step 2); that is, we cannot 
find nor conceive any other step c which shall be a common measurer of the steps 
a andb, so as to satisfy the conditions 
a=Mc, b=Mc, (224) 
whatever positive or contra-positive whole numbers we may denote m and n ; be- 
cause, if we could do this, we should then have the relations, 
2 n 
bimini, « AA Sms (225.) 
