4 
SEPTEMBER 15, 1899. ] 
cepting that ( ) is omitted. The law of 
signs for + and — is the same, but none is 
given for x and ~ on account of the am- 
biguity of the reciprocal ; the commutative 
law applies to both sum and product; the 
distributive law applies to the product of 
sums ; there are no index laws, excepting 
the peculiar one a =a. The law of reduc- 
tion a—a=0O remains, but the comple- 
a ‘ i 
mentary law — = 1 is not true in general. 
a 
How is the truth or suitability of these 
laws established ? He says that it would be 
mere hypothesis to borrow the notation of 
the analysis of quantity, and to assume that 
in its new application the laws by which its 
use is governed would remain unchanged ; to 
establish them he investigates the opera- 
tions of the mind in reasoning as expressed 
by language, and applies Kant’s theory of 
seeing the general truth in a particular in- 
stance. As regards the commutative law it 
may be remarked that Boole overlooks the 
fact that two notions may in their definition 
be coordinate with one another, or subordi- 
nate the one to the other, just as in the theory 
of probability there is a difference between 
two events which are independent of one 
another, and two events which are dependent 
the one on the other ; and in the latter case 
it is not true that the order of the notions 
is indifferent. This is not the place to enter 
into a discussion of these so-called laws of 
thought ; I wish merely to point out that 
Boole’s view is essentially that of the real- 
ist ; the fundamental rules of an analysis 
are not to be assumed arbitrarily, but must 
be found out by investigation of the subject 
analyzed. 
Contemporaneously with Boole, and liv- 
ing on the same Emerald Isle, another 
mathematician spent many days reflecting 
on the fundamental principles of algebra— 
Sir W. R. Hamilton. His investigations 
started from the reading of some passages 
in Kant’s ‘Critique of the Pure Reason’ 
SCIENCE. 3038 
which appeared to justify the expectation 
that it should be possible to construct 
a priori a science of time as well as a science 
of space. The principal passage is as fol- 
lows: ‘‘ Time and space are two sources of 
knowledge from which various a priort syn- 
thetical cognitions can be derived. © Of this 
pure mathematics gives a splendid example 
in the case of our cognitions of space and 
its various relations. As they are both pure 
forms of sensuous intuition, they render 
synthetical propositions a priori possible.’’ 
Thus, according to Kant, space and time 
are forms of the intellect; and Hamilton - 
reasoned that, as geometry is the science of 
the former, so algebra must be the science 
of the latter. He amplifies that view as 
follows: ‘‘ It early appeared to me that 
these ends might de attained by consenting 
to regard algebra as being no mere art, nor 
language, nor primarily a science of quan- 
tity, but rather as the science of order in 
progression. It was, however, a part of 
this conception that the progression here 
spoken of was understood to be continuous 
and unidimensional, extending indefinitely 
forward and backward, but not in any lat- 
eral direction. And although the success- 
ive states of such a progression might, no 
doubt, be represented by points upon a line, 
yet I thought that their simple successive- 
ness was better conceived by comparing 
them with moments of time, divested, how- 
ever, of all reference to cause and effect ; 
so that the ‘time’ here considered might be 
said to be abstract, ideal or pure, lke 
that ‘space’ which is the object of geom- 
etry. In this manner I was led to regard 
algebra as the science of pure time, and an 
essay containing my views respecting it as 
such was published in 1835.” (Preface to 
‘Lectures on Quaternions,’ p. 2.) If al- 
gebra is based on any unidimensional sub- 
ject a difficulty arises in explaining the 
roots of a quadratic equation when they are 
imaginary. To get over the difficulty 
