MATHEMATICS AND ASTRONOMY. 35 
The above principle of generalization may be tested in another way. 
If r denote the ordinary algebraic quantity which may he positive or neg 
ative, r • 0 may represent that quantity when generalized so as to have any 
angle 6 with an initial line in a given plane. For this generalized magni 
tude 
r • 6 X r' • 0' — rr' ’ e + 0'; 
in words, the length of the product is the product of the lengths, and 
the angle of the product is the sum of the angles. Now the principle of 
the permanence of equivalent forms does not help us to generalize this 
proposition for space. A plausible hypothesis likely to present itself at 
first is : Let w denote the angle between the given plane and a fixed plane, is 
(r • 6 ' (f) X (r' ■ 0' * <p ,s ) = rr 1 • 0 -(- 0' ’ <p -f- (p 1 ? 
This is a question not of symbolism, but of truth. 
At the time of De Morgan there was no adequate theory of ]/—1, as is 
evident from the quotation prefixed; nor is there at the present time. 
The view at present held about i =\/—1 by analysts is thus stated by 
Cayley in a paper “On Multiple Algebra,” printed in the Quarterly Journal 
of Mathematics, vol. xxn. 
“We have come to regard a + hi as an ordinary analytical magnitude, 
viz.: in every case an ordinary symbol represents or may represent such 
a magnitude, and the magnitude (and as a particular case thereof the 
symbol i) is commutable with the extfaordinaries of any system of mul 
tiple algebra; and similarly in analytical geometry without seeking for 
any real representation we deal with imaginary points, lines, etc., that is, 
with points, lines, etc., depending on parameters of the form a -j- hi.” 
I propose to review critically the different explanations or elements of 
explanation which have been contributed, with the hope of finding a theory 
which will tend to unify them, and to diminish still further that region of 
analysis where we have mere symbolism without real definition. 
The investigation of this subject arose with the celebrated controversy 
about the nature of the logarithms of negative numbers; whether they are 
real or impossible. Leibnitz maintained that the logarithm of a negative 
number is impossible, because if log (—2) is real, so is h log (—2), that 
is log ■■/ o, which would lead to the supposed absurdity of the logarithm 
of an impossible quantity being real. John Bernoulli held that the log 
arithm of a negative number is as real as the logarithm of a positive 
number; for the ratio — m : — n does not differ from that of -f- w •’ 
+ n. The former view was afterwards maintained by Euler, the latter 
by D’Alembert. Euler claimed to demonstrate that every positive number 
has an infinite number of logarithms, of which only one is possible; fur 
ther, that every negative as well as every impossible number has an infi 
nite number of logarithms, which are all impossible. He reasoned from 
the values of the n th root of -f- 1 and of — 1, viewing + as denoting an 
even number, and — as denoting an odd number, of half revolutions. 
D’Alembert pointed out that the logarithm of a negative number may be