14 REPORT— 1883. 
meaningless to assert that the line passes through their points of inter- 
section. The difficulty is not got over by the analytical method before 
referred to, for this introduces difficulties of its own: is there in a plane 
a point the coordinates of which have given imaginary values? As a 
matter of fact, we do consider in plane geometry imaginary points intro- 
duced into the theory analytically or geometrically as above. 
The like considerations apply to solid geometry, and we thus arrive 
at the notion of imaginary space as a locus in quo of imaginary points 
and figures. 
I have used the word imaginary rather than complex, and I repeat 
that the word has been used as including real. But, this once under- 
stood, the word becomes in many cases superfluous, and the use of it 
would even be misleading. Thus,‘a problem has so many solutions: ’ 
this means, so many imaginary (including real) solutions. But if it 
were said that the problem had ‘so many imaginary solutions,’ the word 
‘imaginary’ would here be understood to be used in opposition to real. 
I give this explanation the better to point out how wide the application 
of the notion of the imaginary is—viz. (unless expressly or by implication 
excluded), it is a notion implied and presupposed in all the conclusions 
of modern analysis and geometry. It is, as I have said, the fundamental 
notion underlying and pervading the whole of these branches of mathe- 
matical science. 
I shall speak iater on of the great extension which is thereby given 
to geometry, but | wish now to consider the effect as regards the theory 
of afanction. In the original point of view, and for the original purposes, 
a function, algebraic or transcendental, such as /z, sin a, or log a, was 
considered as known, when the value was known for every real value 
(positive or negative) of the argument; or if for any such values the 
value of the function became imaginary, then it was enough to know that 
for such values of the argument there was no real value of the function. 
But now this is not enough, and to know the function means to know its 
value—of course, in general, an imaginary value X + 7Y,—for every 
imaginary value « + iy whatever of the argument. 
And this leads naturally to the question of the geometrical repre- 
sentation of an imaginary variable. We represent the imaginary variable 
z+ 7iy by means of a point in a plane, the coordinates of which are 
(z, y). This idea, due to Gauss, dates from about the year 1831. We 
thus picture to ourselves the succession of values of the imaginary variable 
z + iy by means of the motion of the representative point : for instance, 
the succession of values corresponding to the motion of the point along 
a closed curve to its original position. The value X + 7Y of the function 
can of course be represented by means of a point (taken for greater con- 
venience in a different plane), the coordinates of which are X,Y. 
We may consider in general two points, moving each in its own 
plane, so that the position of one of them determines the position of the 
