292 
[347 
347. 
ON THE NOTION AND BOUNDARIES OF ALGEBRA. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vi. (18G4), 
pp. 382—384.] 
I DO not admit the assertion, that the idea of number is derived from that of 
time, it appears to me that it is derived from that of succession in time or space 
indifferently. But I would rather say that the idea of cardinal number is derived and 
abstracted from that of ordinal number, viz. (distinguishing the expressions ‘ set ’ and 
‘ series,’ the latter being used to designate a set of things considered as arranged in 
a definite order), if we have a series of things a, h, c, d, &c., or say a series of words 
first, second, third, fourth, &c.; then any set of things X, N, Y, P, Q, &c., taking them 
up one after the other, no matter in what order, and coordinating them with the 
terms of the series a, h, c, d, &c. or with the words, first, second, third, fourth, &c.— 
the last of them will be coordinated with a definite term of the series a, h, c, d, &c., 
or with a definite term of the series first, second, third, fourth, &c.; that is, the set, 
whatever be the assumed order of the terms, or (what is the same thing) without 
assuming any order therein, will have a certain property; viz. in the set X, N, Y, P, Q, 
where the last term is coordinated with e or with the word fifth, the property is that 
the set consists of five things : and so in general the set consists of a certain (cardinal) 
number of things, such cardinal number being the number corresponding to the rank 
in the series a, h, c, d, &c., of the term wherewith is coordinated the last term of the 
set, or corresponding with the like ordinal number in the series first, second, third, 
fourth, &c. 
The foregoing remarks are made to some extent incidentally, but they have a bearing 
on the distinction in kind which exists e.g. between the proposition 1 + 1 + 1 + 1=4, 
and the proposition which for ordinary purposes would be expressed as 1 +1 + &c.