1885.] 
Space and its Dimensions . 
79 
depth. How then do we escape from an experience so uni- 
versal that it affefts even our powers of conception ? How 
are we led even to raise the question, “ Why should there 
be three and only three dimensions ?” 
Certain very elementary mathematical considerations fully 
warrant us in making this inquiry, and even in seeing that 
space does not necessarily involve three dimensions, and 
neither more nor less. 
We take for instance a line 2 inches or 2 centimetres long. 
If we multiply 2 by itself we have 2 2 , or, as it is expressed in 
words, two in the square. This number, thus obtained by 
multiplying 2 by itself, we can symbolise as a square, each of 
whose sides is 2 inches long. We then again multiply 2x2 
by itself and get 2 3 , called 2 in the cube, or the cube of two. 
This result we again symbolise by a cube, a solid figure 2 
inches long, 2 broad, and 2 thick or deep. Here, then, we 
have passed from the finite right line 2, to the plane figure 
2 3 , and again to the solid cube 2 3 . But what if we go on 
and raise our original 2 to the fourth power or biquadrate, 
2 x 2 x 2 x 2 or 2 4 ? This we can no longer symbolise by 
any figure we can see or touch, or even definitely conceive. 
We know that a finite line is bounded by 2 points, a plane 
square figure by 2 x 2 right lines, and a cube by 2 x 2 x 2 
square planes. Therefore, passing on in the same serial 
order, a figure representing 2 4 must be bounded by 
2 x 2 x 2 x 2 solids. Now, we cannot realise to ourselves 
such figure. It must not be merely a larger cube; it must 
differ from the cube just as essentially as the cube differs 
from the square, or as the square differs from the right line. 
We cannot obtain the cube by superimposing any number 
of plane squares upon each other, because the plane figure 
has merely two dimensions. If we wish to generate a 
square plane figure from a straight line we must suppose the 
line to be moved in a direction not contained within itself , since 
the only possibility of motion in a straight line is from one of 
its extremities towards the other. In like manner, if we 
wish to generate a cube from a plane square we must move 
the plane in a direction not contained within itself, that is not 
in length or breadth, from the left to the right, or the front to 
the back, but upwards or downwards. All this is perfectly 
clear and simple. But suppose we wish to generate the un- 
known biquadrate. We must then in like manner move 
the cube in some direction not contained within itself. And 
here lies the difficulty, for the direction wanted lies outside 
the pale of our experience. But does our inability to point 
out this direction inhere in the very nature of space, or is it 
merely due to the limitation of human faculties ? 
