From "The Education of the Imagination" (1898)
by C.H. Hinton, M.A.
Quod cubus primum corporum et inter altissimos planetas. There follow no less than nine reasons why the cube should be considered as the first amongst the solids.
The second of these reasons is: "The cube is the only solid that can be divided into homogenous cubes without prisms being left over."
The seventh is: "It is the most simple of all the rectilinear solids; even if there be a doubt in the case of the pyramid, the tetrahedron, the difficulty is easily solved by the consideration that the cube is the measure of the pyramid, and it must be that the measure is prior to the thing measured. The cube is a measure by ordinance of men, for, when they measure any of the solids, they conceive its quantity divided up into small cubes. But it is also a measure in the nature of things. For one right angle is equal to another in whatever plane it is laid out; therefore, it is continually equal to itself, and so stands singly, for of others larger and smaller than itself there are an infinite number. Now, a measure must be one and the same, and also finite." And the last argument the ninth, is: "But it must not be omitted, that experienced (artful) nature has given to the most perfect animal the same six limits (as a cube has) most perfectly marked, and this is no mean argument how this body approaches herself in worth. For man himself is, as it were, a cube; for there are as it were, six boundaries to him - Upwards, Downwards, Forwards, Backwards, To the right hand, To the left hand."
Without perhaps, altogether sharing to its full extent this enthusiasm for the cube, it will be well worth our while if we follow up the suggestion contained in the seventh argument, namely, that it is a natural measure of all bodies. It is used as a measure of quantity familiarly enough; but if we recall the first passage of all, we are reminded that the essence of quantity is the comparison of straight with curved, or, more generally *the measure of form.*
Hence it is a simple carrying out of two different principles enunciated by Kepler, but not actually brought together by him, to apply cubes as a measure of form. And in order to do this, let us begin by using them to register *position*, which is the first and most natural approach to the study of form.
The idea is exemplified in the case of a large house. Suppose four people are spoken of as being in four different rooms of a well-known house; their positions in space with regard to one another are thereby defined. Three of them, for instance, may be in three rooms, so as to form a triangle on one story, the fourth, at some distance above one of them, on a higher story.
If there were six rooms in the front of the house on the ground floor, and all the four people were in them, there would be two ways of naming their positions. They might be said to be in room number one, two, five, and six respectively, or else to be in the green room, the white room, the dining-room, and the library, if those happened to be the names of the rooms. Thus, for the rooms in the front of the house, there are two interchangeable sets of names -the numbers, and the names they ordinarily bear. But for rooms in the interior of the house there are no names, except for such as have obtained them by use, e.g., master's dressing -room.
Now, a person going from one house to another similarly built, would naturally, if he had to give directions, use the names he was familiar within the old house for the corresponding rooms in the new house, even though the green room, for instance, might not be furnished with green.
This suggests the plan of taking a typical house, and using the names of its rooms so as to designate corresponding positions in any other house.
Instead of doing this, let us arrange a heap of small cubes, so as to form a larger one, and give to each of them a name. In this way we shall get a more regular and accurate scale of comparison; and the names may be employed to denote the position of any objects with regard to one another in space, exactly as numbers denote the position with regard to one another of objects in a line.
For the sake of simplicity, let us take at first 27 cubes and arrange them to form a larger cube. The numbers will serve temporarily, as it is often convenient to take a larger set of cubes, say 64 or 125. The first step, then, in the cultivation of the imagination, is to give a child 27 cubes, and make him name each of them according to its place, as he puts them up.
The only difference of the cubes from one another is their position in the heap; but it is not a bad plan to mark each, or write its name on it, and each time the heap is re-made to put the same cube in the same place. It should be a rule, that a cube is never to be used without its name being said.
When even such a small system as this is learnt, a child becomes possessed of a new power. He can be made to build up brick houses of any form, by simply being told the names of the cubes in the order in which he has to put them. With even this limited number of cubes or blocks, it is possible to make arrangements of any complexity.
The key to this is given by Kepler's second argument, namely, that a cube can be exactly divided into smaller cubes, for taking as the cube (1), for instance, not one of the blocks but twenty-seven of them arranged in a cube, each of these can easily be given a name. The first of all will be (1) in (1), the second (2) in (1), and so on.
Next to the big cube (1) comes the big cube (2), containing likewise 27 blocks. The first of these is (1) in (2), and so on in succession. The fourth cube above (1) in (1) is for instance (1) in (10). The smallest children do not find the least difficulty in understanding this principle, if there are enough blocks supplied them to carry it out practically to some extent. But it is by far the best plan not to show the child this way until he has learnt a cube containing five blocks in a side, for thus all his interest is concentrated in a desire to learn the names or places of more cubes, in order to make larger buildings; and it is only those thus learnt by heart that are really known at all. The right use of the intellect is to determine what knowledge shall be made intuitive.
When a child has learnt a set of cubes perfectly, it will be found that his power of imagination, as defined above, has been greatly increased. The imagination is, as it were, a power of inward drawing or modeling; and what corresponds to the actual delineation of a form on paper or modeling in clay, is in the mind the affixing a name. When a shape of cubes is thought of, and each of them is named, the mind can recur to each part of it and note its relations exactly in the same way as, when a form is put on paper, each portion of it can be looked at and gone over again.