SEPTEMBER 15, 1899. ] 
tion of ratios includes numerical multipli- 
cation as a particular case, and combines 
in the same way with addition and subtrac- 
tion. So close and far-reaching is this anal- 
ogy that the processes and results of the two 
sciences are expressed in the same language, 
verbal and symbolical, while no confusion 
is produced by this ambiguity of meaning, 
except in the minds of those who try to 
make familiarity with language do duty for 
knowledge of things.” 
What is the analogy here spoken of? It 
cannot be a mere rhetorical analogy; it isa 
true logical analogy. But what is a logical 
analogy, except that the subjects have some- 
thing in common, which is the basis of the 
common properties. The logical relation 
of number to quantity is that of subordina- 
tion ; we cannot pass deductively from the 
former to the latter, but we can pass de- 
ductively from the latter to the former. It 
is easy to pass downwards from quantity to 
number; the difficulty is in passing up- 
wards from number to quantity. 
The most elaborate treatise on algebra 
written in the English language within re- 
cent times is Chrystal’s ‘Text-book of Al- 
gebra,’ published in two volumes. The 
task which the author sets before himself is 
the same as that which Peacock undertook 
—namely, to place the teaching of the ele- 
ments of algebra on a scientific basis, and 
abreast of what may be called the technical 
knowledge of the day. In the first volume 
he starts out with the idea of building up 
the science on the three laws of association, 
commutation and distribution, the two 
former being applicable to addition and 
subtraction, multiplication and division, 
and the third to multiplication. The view 
which he takes of these laws is expressed 
by the phrase ‘ canons of the science,’ as is 
evidenced by the following passage: ‘‘ As 
we have now completed the establishment 
of the fundamental laws of ordinary al- 
gebra, it may be well to insist once more 
SCIENCE. 
361 
upon the exact position which they hold in 
the science. To speak, as is sometimes 
done, of the proof of these laws in all their 
generality is an abuse of terms. They are 
simply laid down as the canons of the sci- 
ence. The best evidence that this is their 
real position is the fact that algebras are in 
use whose fundamental laws differ from 
those of algebra. In the algebra of qua- 
ternions, for example, the law of commuta- 
tion for multiplication and division does 
not hold generally.” 
If it is an abuse of terms to speak of the 
proof of these laws why does Hamilton de- 
vote page upon page to the proof of the as- 
sociative law for a product of quaternions ? 
He is not content with laying it down as a 
canon ; he investigates whether it corre- 
sponds to nature. No doubt, the function 
of the expositor is different from that of the 
investigator ; the latter must establish prin- 
ciples in the best way he can ; the former 
may proceed deductively from these prin- 
ciples as the axioms of the science. But 
the idea of ‘canon’ involves something ar- 
bitrary and formal which is not involved in 
the idea of an ‘ axiom.’ 
But if we turn to the second volume we 
find evidence against the canonical nature 
of these laws, for the author admits that 
they must be modified within the bounds of 
algebra itself. The law of association can- 
not be applied to the terms of an infinite 
series, unless it is convergent; the law of 
commutation cannot be applied to the terms 
of an infinite series, unless it is absolutely 
convergent ; and the law of distribution re- 
quires modification when applied to the 
product of two infinite series. If, in any 
case, the so-called canons are modified 
there must be some higher authority to 
which appeal is made. The only conclusion 
left is that the rules in question are not 
canons at all, excepting in so far as they 
represent properties of the subject analyzed. 
I may here refer to the prevalent doctrine 
