This product of movement, which it is so easy for us to describe, would be difficult for him to conceive. But this difficulty is connected rather with its totality than with any particular part of it.
Any line, or plane of this, to him higher, solid we could show to him, and put in his sensible world.
We have already seen how the pink square could be put in his world by a turning of the cube about the red line. And any section which we can conceive made of the cube could be exhibited to him. You have simply to turn the cube and push it through, so that the plane of his existence is the plane which cuts out the given section of the cube, then the section would appear to him as a solid. In his world he would see the contour, get to any part of it by digging down into it.
The Process by which a Plane Being would gain a Notion of a Solid.
If we suppose the plane being to have a general idea of the existence of a higher solid—our solid—we must next trace out in detail the method, the discipline, by which he would acquire a working familiarity with our space existence. The process begins with an adequate realisation of a simple solid figure. For this purpose we will suppose eight cubes forming a larger cube, and first we will suppose each cube to be coloured throughout uniformly. Let the cubes in [fig. 91] be the eight making a larger cube.
Fig. 91.
Now, although each cube is supposed to be coloured entirely through with the colour, the name of which is written on it, still we can speak of the faces, edges, and corners of each cube as if the colour scheme we have investigated held for it. Thus, on the null cube we can speak of a null point, a red line, a white line, a pink face, and so on. These colour designations are shown on No. 1 of the views of the tesseract in the plate. Here these colour names are used simply in their geometrical significance. They denote what the particular line, etc., referred to would have as its colour, if in reference to the particular cube the colour scheme described previously were carried out.
If such a block of cubes were put against the plane and then passed through it from right to left, at the rate of an inch a minute, each cube being an inch each way, the plane being would have the following appearances:—
First of all, four squares null, yellow, red, orange, lasting each a minute; and secondly, taking the exact places of these four squares, four others, coloured white, light yellow, pink, ochre. Thus, to make a catalogue of the solid body, he would have to put side by side in his world two sets of four squares each, as in [fig. 92]. The first are supposed to last a minute, and then the others to come in in place of them, and also last a minute.