Thus in a rough and ready manner there is something which we can take. If we do not inquire about one of the cubes itself, we are all right; that being granted we can know the block.
But if we look into what each of these unit cubes, or what each of these unit positions is, we find quite an infinity opening before us. There is nothing definitely of which we can say that the whole unit cube is built up, and each of the positions has a perfectly endless number of positions in it, if we come to examine it closely. All that we can say is that our ignorance about each of the unit positions is of the same kind as our ignorance about every other, and, taking one as granted, we may as well take the 27 as granted; and so out of a lot of similar ignorances we get a kind of knowledge of the whole. And this knowledge is not a mere indefinite thing, but it can be worked at, improved, and made perfect after its kind. For suppose we limit ourselves to the 27 positions numbered in Diagram I. Two of these positions form one shape, three of them will form another shape, and so on. And in going over each of these arrangements we gradually get to know the whole set of them which form the block.
Having given up for the time any question as to the possible subdivisions of the cube, and looking on each cube as a unit position, we have 27 positions. These positions can be taken in different selections, and each selection is a shape. To know the block or set of positions means to form a clear idea of every shape, consisting of selections of positions, which can be formed out of the 27.
But each of the cubes, 27 of which form the whole block, can be divided up. Each of these cubes contains a great many positions. There must, for instance, be positions in each cube for every one of its molecules.
Thus it is evident that the cube supplies an inexhaustible number of positions to be learnt. I call the cube unknown in the sense that there are a great number of positions in it which are not clearly realized by the mind.
By a very simple device it is possible to penetrate a little into the unknown part. The whole set of cubes forms a cube. Let us consider the small cube to be a model of the whole cube. Let us consider it as consisting of 27 parts, each related to the other as the 27 first cubes were related amongst themselves. Thus the unknown part, the material cube, which is used to build up the whole, becomes reduced in size. Diagram II. represents such a cube.
This is the theory. The practical work consisted in learning the names denoting these smaller cubes in connection with their positions, so that, the names being said, the small cubes meant were present to the mind, and a set of names being said, the shape, consisting of a set of cubes in definite relations to each other, came vividly before one. A complete knowledge of the block of cubes would be a complete appreciation of all the possible shapes which selections of the cubes would form, and this I strove to attain. Here at length I found real knowledge, and after a time I was able to reduce the size of the unknown still further, and to obtain a solid mass of knowledge fairly well worked all through.
And now it all seemed satisfactory enough. There was real knowledge in knowledge of the arrangement; and the material cube, which must be assumed, could be made smaller and smaller, it could be turned into knowledge, thus affording a prospect of obtaining endless knowledge. Thus I found the real home of my mind, the only knowledge I had ever had, and I hoped always to continue to add to it, and always to reduce the unknown in size.
Presently, too, the forms of the outward world began to fall in with this knowledge; and as the mass of known cubes became larger in number, a group of them would fairly well represent a wall, a door, a house, a simple natural object such as a stone or a fruit.
Yet amidst all this delight I became conscious, dimly enough, of a self-element in the knowledge of blocks.