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theorem afterwards arrived at, that there is always some power of the ideal number 
which is an actual number. In the still later developments of the Theory of Numbers 
by Dedekind, the units, or incommensurables, are the roots of any irreducible equation 
having for its coefficients ordinary integer numbers, and with the coefficient unity for 
the highest power of x. The question arises, What is the analogue of a whole 
number ? thus, for the very simple case of the equation x 2 + 3 = 0, we have as a whole 
number the apparently fractional form ^ (1 + i V3) which is the imaginary cube root 
of unity, the p of Eisenstein’s theory. We have, moreover, the (as far as appears) 
wholly distinct complex theory of the numbers composed with the congruence-imaginaries 
of Galois : viz. these are imaginary numbers assumed to satisfy a congruence which is 
not satisfied by any real number ; for instance, the congruence x 2 — 2 = 0 (mod 5) has 
no real root, but we assume an imaginary root i, the other root is then = — i, and 
we then consider the system of complex numbers a + hi (mod 5), viz. we have thus 
the 5 2 numbers obtained by giving to each of the numbers a, b, the values 0, 1, 2, 3, 4, 
successively. And so in general, the consideration of an irreducible congruence F (x) = 0 
(mod p) of the order n, to any prime modulus p, gives rise to an imaginary con 
gruence root i, and to complex numbers of the form a + bi + ci 2 + ... + ki n-1 , where 
a, b, k,... &c., are ordinary integers each =0, 1, 2,..., p — 1. 
As regards the theory of forms, we have in the ordinary theory, in addition to 
the binary and ternary quadratic forms, which have been very thoroughly studied, the 
quaternary and higher quadratic forms (to these last belong, as very particular cases, 
the theories of the representation of a number as a sum of four, five or more squares), 
and also binary cubic and quartic forms, and ternary cubic forms, in regard to all 
of which something has been done ; the binary quadratic forms have been studied in the 
theory of the complex numbers a + bi. 
A seemingly isolated question in the Theory of Numbers, the demonstration of 
Fermat’s theorem of the impossibility for any exponent A. greater than 3, of the equation 
æ K + y K = z k , has given rise to investigations of very great interest and difficulty. 
Outside of ordinary mathematics, we have some theories which must be referred 
to: algebraical, geometrical, logical. It is, as in many other cases, difficult to draw 
the line; we do in ordinary mathematics use symbols not denoting quantities, which 
we nevertheless combine in the way of addition and multiplication, a+b, and ab, and 
which may be such as not to obey the commutative law ab — ba: in particular, this is 
or may be so in regard to symbols of operation; and it could hardly be said that 
any development whatever of the theory of such symbols of operation did not belong 
to ordinary algebra. But I do separate from ordinary mathematics the system of 
multiple algebra or linear associative algebra, developed in the valuable memoir by the 
late Benjamin Peirce, Linear Associative Algebra (1870, reprinted 1881 in the American 
Journal of Mathematics, vol. iv., with notes and addenda by his son, C. S. Peirce); we 
here consider symbols A, B, &c. which are linear functions of a determinate number 
of letters or units i, j, k, l, &c., with coefficients which are ordinary analytical magni 
tudes, real or imaginary, viz. the coefficients are in general of the form x + iy, where 
C. XI. 58