ADDRESS. 7 
we can form any judgment, would seem to be inconsistent with the 
physical constitution of our planet at least, if not of the universal [sic]. 
To get rid of this difficulty and at the same time to save the credit of the 
supposed system of necessary truth, it is customary to say that the points, 
lines, circles and squares which are the subjects of geometry exist in our 
conceptions merely and are parts of our minds ; which minds by working 
on their own materials construct an a priori science, the evidence of which 
is purely mental and has nothing to do with outward experience. By 
howsoever high authority this doctrine has been sanctioned, it appears to 
me psychologically incorrect. The points, lines and squares which any- 
one has in his mind are (as I apprehend) simply copies of the points, lines 
and squares which he has known in his experience. Our idea of a point 
LIapprehend to be simply our idea of the minimwm visibile, the small 
portion of surfacé which we can see. We can reason about a line as if it 
had no breadth, because we have a power which we can exercise over the 
operations of our minds: the power, when a perception is present to our 
senses or a conception to our intellects, of attending toa part only of that 
perception or conception instead of the whole. But we cannot conceive a 
line without breadth: we can form no mental picture of such a line; ail 
the lines which we have in our mind are lines possessing breadth. If any- 
one doubt this, we may refer him to his ownexperience. I much question 
if anyone who fancies that he can conceive of a mathematical line thinks 
so from the evidence of his own consciousness. I suspect it is rather 
because he supposes that unless such a perception be possible, mathe- 
matics could not exist as a science: a supposition which there will be no 
difficulty in showing to be groundless.’ 
I think it may be at once conceded that the truths of geometry are 
truths precisely because they relate to and express the properties of what 
Mill calls ‘ purely imaginary objects; ’ that these objects do not exist in 
_ Mill’s sense, that they do not exist in nature, may also be granted; that 
they are ‘not even possible,’ if this means not possible in an existing 
nature, may also be granted. That we cannot ‘conceive’ them depends 
on the meaning which we attach to the word conceive. I would myself 
say that the purely imaginary objects are the only realities, the dyrwe 
éyra, in regard to which the corresponding physical objects are as the 
shadows in the cave; and it is only by means of them that we are able 
to deny the existence of a corresponding physical object; if there is no 
conception of straightness, then it is meaningless to deny the existence of 
a perfectly straight line. 
But at any rate the objects of geometrical truth are the so-called 
imaginary objects of Mill, and the truths of geometry are only true, and 
@ fortiori are only necessarily true, in regard to these so-called imaginary 
objects ; and these objects, points, lines, circles, &c., in the mathematical 
sense of the terms, have a likeness to and are represented more or less im- 
perfectly, and from a geometer’s point of view no matter how imperfectly, 
by corresponding physical points, lines, circles, &c. Ishall have to return 
