504 The British Association. [October, 
and thirdly, that of Geometry not according to Euclid. He 
explained Imaginary Quantities as follows ; — 
To fix our ideas, consider the measurement of a line, or 
the reckoning of time, or the performance of any mathe- 
matical operation. A line maybe measured in one direction 
or in the opposite ; time may be reckoned forward or back- 
ward ; an operation may be performed or be reversed, it 
may be done or it may be undone ; and if having once re- 
versed any of these processes we reverse it a second time, 
we shall find that we have come back to the original direc- 
tion of measurement or of reckoning, or to the original kind 
of operation. Suppose, however, that at some stage of a 
calculation our formulae indicate an alteration in the mode 
of measurement such that, if the alteration be repeated, a 
condition of things not the same as, but the reverse of, the 
original will be produced. Or suppose that, at a certain 
stage, our transformations indicate that time is to be 
reckoned in some manner different from future or past, but 
still in a way having definite algebraical connection with 
time which is gone and time which is to come. It is clear 
that in adtual experience there is no process to which such 
measurements correspond. Time has no meaning except as 
future or past ; and the present is but the meeting point of 
the two. Or, once more, suppose that we are gravely told 
that all circles pass through the same two imaginary points 
at an infinite distance, and that every line drawn through 
one of these points is perpendicular to itself. On hearing 
this statement we shall probably whisper, with a smile or a 
sigh, that we hope it is not true ; but that in any case it is 
a long way off, and perhaps, after all, it does not very much 
signify. If, however, as mathematicians we are not satis- 
fied to dismiss the question on these terms, we ourselves 
must admit that we have here reached a definite point of 
issue. Our science must either give a rational account of 
the dilemma or yield the position as no longer tenable. 
Special modes of explaining this anomalous state of things 
have occurred to mathematicians. But, omitting details as 
unsuited to the present occasion, it will, I think, be sufficient 
to point out in general terms that a solution of the difficulty 
is to be found in the fadt that the formulae which give rise 
to these results are more comprehensive than the significa- 
tion assigned to them ; and when we pass out of the 
condition of things first contemplated they cannot (as it is 
obvious they ought not) give us any results intelligible on 
that basis. But it does not therefore by any means follow 
that upon a more enlarged basis the formulae are incapable 
