Fig. 2.
If now I take account of another quality, say length, they can be arranged in a plane. Starting from A, B, C, I can find points to represent different degrees of length along such lines as AF, BD, CE, drawn from A and B and C. Points on these lines represent different degrees of length with the same degree of brightness. Thus the whole plane is occupied by points representing all conceivable varieties of brightness and length.
Fig. 3.
Bringing in a third quality, say sharpness, I can draw, as in [fig. 3], any number of upright lines. Let distances along these upright lines represent degrees of sharpness, thus the points F and G will represent swords of certain definite degrees of the three qualities mentioned, and the whole of space will serve to represent all conceivable degrees of these three qualities.
If now I bring in a fourth quality, such as weight, and try to find a means of representing it as I did the other three qualities, I find a difficulty. Every point in space is taken up by some conceivable combination of the three qualities already taken.
To represent four qualities in the same way as that in which I have represented three, I should need another dimension of space.
Thus we may indicate the nature of four-dimensional space by saying that it is a kind of space which would give positions representative of four qualities, as three-dimensional space gives positions representative of three qualities.
CHAPTER II
THE ANALOGY OF A PLANE WORLD
At the risk of some prolixity I will go fully into the experience of a hypothetical creature confined to motion on a plane surface. By so doing I shall obtain an analogy which will serve in our subsequent enquiries, because the change in our conception, which we make in passing from the shapes and motions in two dimensions to those in three, affords a pattern by which we can pass on still further to the conception of an existence in four-dimensional space.