fc)> 
}hat of 
■ space 
ed and 
it ’ and 
ged in 
? words 
j them 
th the 
, &c.— 
d, &c., 
.he set, 
without 
, P, Q, 
is that 
.rdinal) 
e rank 
of the 
third, 
)earmg 
hi = 4, 
l + &c. 
(n terms) = n, but which is better expressed in the form l x + 1 2 + ••• +l n = w, where the 
subscript numbers merely distinguish between the different unities which are added 
altogether. 
I use the term Algebra in a wide sense as including, or indeed I might say 
identical with, Finite Analysis, and excluding Infinite Analysis; but in speaking of it 
as identical with Finite Analysis I include in that term part of what might be con 
sidered Infinite Analysis; viz. many of the theorems relating to infinite series or other 
successions of operations, e.g. 
(1 — x) (1 + x + x l + ... ad inf.) — 1, 
really belong to Finite Analysis; for what is asserted is that the coefficient of the 
term of indefinite rank, say x 11 , is a finite series equal in value to zero (this coefficient 
in fact is 1 — 1 which is = 0). On the other hand the theorem 
1 - 
+ 
1.2 1.2.3.4 
— &c. 
the truth whereof depends on the equations 
T = 'fa+ ^ + ^3 + • • • ™ l f-> & c -> 
which are not arithmetically verifiable, belongs strictly to Infinite Analysis. 
Algebra is an Art and a Science; qua Art, it defines and prescribes operations 
which are either tactical or else logistical; viz. a tactical operation is one relating to 
the arrangement in any manner of a set of things; a logistical operation (I prefer to 
use the new expression instead of arithmetical) is the actual performance, so as to obtain 
for the result a number, of any arithmetical operations (of course, given operations) 
finite in number, since these alone can be actually performed, upon given numbers. 
And qua Science Algebra affirms cl priori, or predicts, the result of any such tactical 
or logistical (or tactical and logistical) operations. An equation such as 1 + 4 + 10 = 15 
is not an algebraical theorem; it is merely the assertion that the sum of the numbers 
1, 4, 10 is that number, viz. 15, which is the sum of the numbers in question. And, 
similarly, the equation 1 + 1 + 1 = 3 is not an algebraical theorem. But on the other 
hand, the equation 1 + 1 + 1 + ... (w terms) = n, is an algebraical theorem; in the 
equivalent form l x +1 2 + ••• +1» = n, (where 1^ = 1) it is not different in kind from the 
equation 1 + 2 + 3 ... +n = (n + 1), or say l x + 1 2 + ... + l n = \n (n + 1), (where 1* = k) 
which is certainly an algebraical theorem. And this leads to the remark, that every 
algebraical theorem rests ultimately on a tactical foundation. In fact, whether we prove 
the last-mentioned theorem in the easiest way by writing 
1+2 +3 .,. + n = S, 
n + (rc-l) + (w-2)... + l =S,