40 
SECTION A. 
other function involving y / —1 can be reduced to the form P -f- Q -\/ —1- 
But it cannot be proved that this reduction is always possible, unless on 
the assumption that all the imaginarles refer to one plane. l’or example, 
De Morgan, in his Doxible Algebra, does not interpret directly e ay ~ 1 or the 
more general expression (a-{-b\/ — l) i> + 3 ’ y '~ 1 , but the expression is reduced 
to significance by being reduced to the form P -f- Q y—1. And this is the 
current mode in modern analysis of explaining functions of the imaginary. 
In a subsequent paper Argand adopted the notation of Français for a 
line in a plane ; but used instead of ~ to denote the quadrant, which, as 
Français pointed out, is not an improvement. So imbued was he with the 
direction theory of y —1 that he sought to express any direction in space 
by means of an imaginary function. He arrived at the view that the third 
mutual perpendicular KP (fig. 6) is expressed by y—ll 7-1 , the opposite 
line KQ by — 1 > an d any line KM in the perpendicular plane by 
denotes the angle between KB and KM. 
He remarks that if the above be the cor 
rect meaning of \/—then it is not 
true that every function can be reduced to 
the form p -f- q |/—1 and he doubts the 
validity of the current demonstration 
which aims at proving that the function 
(a + b i/=I) B * + w y- 1 can always be re 
duced to the form^ q ]/—1. Accord 
ing to that reduction, as was shown by 
Euler, \/—1 , —1 = e — 2", and this mean 
ing of the expression was maintained by 
Français and Servois. The latter, fol 
lowing the analogy of a -f- & ]/—1 for a line in one plane, suggested that 
the expression for a line in space had the form 
p cos a -f- q cos /5 + r cos y, 
where p, q, r are imaginarles of some sort, but he questioned whether they 
are each reducible to the form A -f- B \/—1. In reply to the criticisms 
of Français and Servois, Argand maintained that Euler had not demon 
strated that 
e x /~l = cos x -\~i/—1 sin x 
but had defined the meaning of e x ^~ 1 by extending the theorem 
= etc. 
It will be shown afterwards that in the equation of Euler, namely 
|/ i c °s m + V— 1 sin M where p 
Fig. 6.