112 IMAGINARY QUANTITIES—THEIR PHILOSOPHY. 
Dividing by the co-efficients of y and Z, we have 
Mie & 
Yyt2z2= na 
Oe Pe aR:. q 
Ce an’—a 
the left hand members of which are identical. Since a 
is any difference, and na and n’a any multiples of a, all 
equations will come out the same way.—Q. E. D. 
It can be shown inthe same way that four, or any num- 
ber of equations, whose coefficients are in arithmetical 
progression, are indeterminate. 
IMAGINARY QUANTITIES—THEIR PHILOSOPHY . 
BY C. B. WARRING, Ph.D. 
An imaginary quantity is defined to be the even root 
of a negative quantity. for example, ”-a@; and aroot of 
a number isa number which, multiplied by itself, will 
produce thatnumber. It follows from the idea of, oppo- 
sition, which is the fundamental difference between ++ 
and —, that while + x-++ produces +, so also —X— pro- 
duces -++. oe 
I may write on the board the expression ,/—1. But 
since +1x+1=+1 and —1x -1=-+1, there can be no 
such number as the square root of —1, 7. e., no number 
which, multiplied by itself, will produce —1. The mind 
is baffled when it endeavors to form a conception of such 
anumber. For this reason, such expressions have very 
justly been called impossible quantities, and, very incor- 
rectly, imaginary quantities, for the imagination refuses 
to recognize them. A better name would be pseuds 
(from #&vdos). Their claim to this title will appear 
later on. 
Yet certain very real results are obtained by their use. 
If, e. g., I multiply (6—5,/—1) by (6+5,/—1), I get 61, 
a result from which both imaginaries have vanished, 
62 
