362 
that the number-system of arithmetic 
closes with the complex number, and that 
the operations of algebra give no indication 
of any higher imaginary form. For in- 
stance, in an article on ‘Monism in Arith- 
metic,’ Professor Schubert says: ‘ In the 
numerical combination a+ ib, which we 
also call number, we have found the 
most general numerical form to which 
the laws of arithmetic can lead, even 
though we wished to extend the limits of 
arithmetic still further. * * * With re- 
spect to quaternions which many might be 
disposed to regard as new numbers it will 
be evident that though quaternions are 
valuable means of investigation in geom- 
try and mechanics they are not numbers of 
arithmetic, because the rules of arithmetic 
are not unconditionally applicable to 
them.’’ When the plane of the complex 
quantity is that of the axes of x and y it 
is true that no higher form appears, because 
in multiplication we get only k and k’, which 
is—!. But when Hamilton took for the 
common plane a general plane passing 
through the axis of « he immediately en- 
countered a higher form jk, and the prob- 
lem resolved itself into finding the meaning 
of that new imaginary combination. He had 
a great difficulty in emerging out of ‘ Flat- 
land,’ but he succeeded in doing it. The 
reason given for excluding the quaternion 
cannot apply, for it would exclude infinite 
series, as the rules of arithmetic are not 
unconditionally applicable to them. 
Last year there appeared the first volume 
of a ‘Treatise on Universal Algebra,’ by 
Mr. Whitehead, of Trinity College, Cam- 
bridge. By universal algebra the author 
means the various systems of symbolic 
reasoning allied to ordinary algebra, the 
chief examples being Hamilton’s Quater- 
nions, Grassmann’s Calculus of Extension 
and Boole’s Symbolic Logic. The author 
does not include ordinary algebra in his 
treatment, and the main idea of the work 
SCIENCE. 
ria 
[N.S. Von. X. No. 246. 
is not unification of the methods, nor 
generalization of algebra so as to include 
them, but a detailed study of each structure, 
to be followed by a comparative anatomy. 
In this idea of comparative anatomy there 
is involved the assumption that these 
methods are essentially distinct and inde- 
pendent. But that they overlap to a large 
extent is very evident. 
The author preaches the view of the ex- 
treme formalist ; nevertheless, at various 
places he makes admissions which are very 
damaging to it. As regards the fundamental 
rules he says: ‘“ The justification of the 
rules of inference in any branch of mathe- 
matics is not properly part of mathematics; 
it is the business of experience or philoso- 
phy. The business of mathematics is simply 
to follow the rules. In this sense all mathe- 
matical reasoning is necessary; namely, it 
has followed the rule.’”’? Must the mathe- 
matician wait for the experimenter or the 
philosopher to justify the rules of algebra? 
Was it no part of Hamilton’s business to 
test whether the associative law is true ofa 
product of spherical quaternions? To ad- 
vance the principles of analysis is surely the 
special work of the mathematician ; to fol- 
low the rules discovered is work of a lower 
order. 
Mr. Whitehead thus describes a calculus : 
‘In order that reasoning may be conducted 
by means of substitutive signs it is necessary 
that rules be given for the manipulation of 
the signs. The rules should be such that 
the final state of the signs after a series of 
operations according to rule denotes, when 
the signs are interpreted in terms of the 
things for which they are substituted, a 
proposition true for the things represented 
by the signs. The art of manipulation of 
substitutive signs according to fixed rules, 
and of the deduction therefrom of true 
propositions, is a calculus.”’ By substitutive 
sign is meant one such that in thought it 
takes the place of that for which it is sub- 
