34 . REPORT—1883. 
number p is by definition not the product of two actual numbers, but in the 
example just referred to the number p is the product of two ideal numbers 
having for their cubes the two actual numbers A, B, respectively, and we 
thus’ ares p?=AB. It is, I think, in this way that we most easily get some 
notion of the meaning of an ideal number, but the mode of treatment (in 
Kummer’s great memoir, ‘ Ueber die Zerlegung der aus Wurzeln der 
Hinheit gebildeten Complexen-Zahlen in ihre Primfactoren, Crelle, t. xxxv. 
1847) is a much more refined one; an ideal number, without ever being 
isolated, is made to manifest itself in the properties of the prime number 
of which it is a factor, and without reference to the 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 Num- ~ 
bers by Dedekind, the units, or incommensurables, are the roots of any M 
irreducible equation having for its coefficients ordinary integer numbers, 7 
and with the coefficient unity for the highest power of w. The question ~ 
arises, What is the analogue of a whole number ? thus for the very simple 
case of the equation 2?+3=0,-we have as a whole number the apparently 
fractional form }(1+ 7/3) which is the imaginary cube root of unity, 
the p of Hisenstein’s theory. We have, moreover, the (as far as appears) 
wholly distinct complex theory ofthe numbers composed with the con- 
gruence-imaginaries of Galois: viz: these are imaginary numbers assumed. 
to satisfy a congruence mae is not satisfied by any real number; for 
instance the congruence v7 —2=0 (mod 5) has no real root, but we assume 
an imaginary root 7, the thet root is then = —i, and we then consider — 
the system of eoaglies numbers a+bi (mod 5), viz. we have thus the 5? 
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 , to any prime modulus p, 
gives rise to an imaginary congruence root i, and to complex numbers © 
of the form a+bi+tci?-. +k", where a, b,...k are ordinary integers — 
each = 0, 1, 2,-- p—l. é 
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 ibe theories’ of the repre- — 
sentation of a numbel as a sum of four, five or moze squares), and also 
binary. cubic and quartic forms, and ternary cubic forms, in regard to all — 
which something has been done; the binary quadratic forms haye been — 
studied in the theory of the complex numbers w + bi. 4 
A seemingly isolated question in the Theory of Numbers, the demon-_ 
stration of Fermat’stheorem of the impossibility for any exponent A greater 
than 3, of the equation z+ y\=~, has given rise to investigations ole 
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 
