784] 
BRITISH ASSOCIATION, SEPTEMBER 1883. 
439 
Of course it may be said in reply that the conclusion is a very natural one, 
provided we assume the existence of imaginary points; and that, this assumption not 
being made, then, if the circles do not intersect, it is meaningless to assert that the 
line passes through their points of intersection. 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 introduced 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 understood, 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 mathematical science. 
I shall speak later on of the great extension which is thereby given to geometry, 
but I wish now to consider the effect as regards the theory of a function. In the 
original point of view, and for the original purposes, a function, algebraic or transcen 
dental, such as V«, sin«, or log«, 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 + iY,—for every imaginary value « + iy whatever of the 
argument. 
And this leads naturally to the question of the geometrical representation of an 
imaginary variable. We represent the imaginary variable x + iy by means of a point 
in a plane, the coordinates of which are («, 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 x + 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 + iY of the function can of course 
be represented by means of a point (taken for greater convenience 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 other, and consequently