104 Analyses of Boohs. [February, 
length, breadth, and depth, — and we have a clear conception of 
a simple body, e.g ., a quadrangular pillar, when we state its 
dimensions in length, breadth, and height. Concerning a fourth 
dimension we know nothing, and are absolutely unable even to 
conceive it. But this may possibly be due to the limitation of 
our senses. Let us imagine a being which, in virtue of its 
peculiar organisation, lived in a plane, and was incapable of 
recognising any objedt outside of such plane. This being will 
possess a conception of the extent of its plane in length and 
breadth, but, to it, the conception of depth or height — and con- 
sequently of space — would be totally wanting. All objedts would 
become visible to this being only when they entered its plane of 
perception, and on leaving it they would vanish. A human 
finger touching this plane with its point would, e.g., become 
visible to such a being as an oval spot at the point of contadt, 
whilst the part of the finger standing above the plane would 
remain, to it, invisible. If the finger touches the plane, not with 
its tip, but with its side, our supposed being will obtain a quite 
different view of the finger, which will now appear as a long bent 
figure. Such a being will scarcely believe that the oval spot and 
the long figure proceed from one and the same objedt. If we 
remove the finger it will vanish from the sight of the supposed 
being, who will be unable to conceive what has become of it. 
Let us suppose a circle drawn in the plane ; it will appear to our 
imagined bi-dimensional being as a perfedtly closed figure. 
That the circle is open above and below it will not perceive. If 
it wishes to go to the centre of the circle it must break through 
the boundary-line, and if we, from above or from below, — i.e. f 
from the third dimension , — put a finger in the middle of the 
circle without breaking through the circumference, it will, to the 
being in question, appear a miracle. 
“ It would be quite the same with us if there existed, in addi- 
tion to the three dimensions of space which we perceive, length, 
breadth, and thickness, yet a fourth, which we do not perceive. 
If an objedt passes into the fourth dimension, it vanishes for us, 
and suddenly re-appears when it enters into our region of per- 
ception. It is even conceivable that a hollow ball, which to us 
appears as a body closed on all sides, is open in the fourth 
dimension, just as the circle which to a bi-dimensional being 
appears completely closed, is open above and below. 
“ To a bi-dimensional being the tying an ordinary knot on a 
piece of string free at both ends, would appear inconceivable, as 
for this purpose one end of the string must be lifted up out of the 
plane in order to be passed over and then under the other part of 
the string, whilst a man, with his conceptions of height, breadth, 
and length, at once understands this operation. 
“ On the other hand, we are not able to tie a knot in a string 
whose two ends are connected so as to form a ring without first 
opening the string. But if we suppose a fourth dimension, and 
