The Fourth Dimension

by Charles Howard Hinton

Looking now at the sides of this figure we see that there is a unit square on each of them—the two squares contain no points, but have four corner points each, which gives the point value of each as one point.

Hence we see that the square on the diagonal is equal to the squares on the two sides; or as it is generally expressed, the square on the hypothenuse is equal to the sum of the squares on the sides.

Fig. 18.

Noticing this fact we can proceed to ask if it is always true. Drawing the square shown in [fig. 18], we can count the number of its points. There are five altogether. There are four points inside the square on the diagonal, and hence, with the four points at its corners the point value is 5—that is, the area is 5. Now the squares on the sides are respectively of the area 4 and 1. Hence in this case also the square on the diagonal is equal to the sum of the square on the sides. This property of matter is one of the first great discoveries of applied mathematics. We shall prove afterwards that it is not a property of space. For the present it is enough to remark that the positions in which the points are arranged is entirely experimental. It is by means of equal pieces of some material, or the same piece of material moved from one place to another, that the points are arranged.

Pythagoras next enquired what the relation must be so that a square drawn slanting-wise should be equal to one straight-wise. He found that a square whose side is five can be placed either rectangularly along the lines of points, or in a slanting position. And this square is equivalent to two squares of sides 4 and 3.

Here he came upon a numerical relation embodied in a property of matter. Numbers immanent in the objects produced the equality so satisfactory for intellectual apprehension. And he found that numbers when immanent in sound—when the strings of a musical instrument were given certain definite proportions of length—were no less captivating to the ear than the equality of squares was to the reason. What wonder then that he ascribed an active power to number!

We must remember that, sharing like ourselves the search for the permanent in changing phenomena, the Greeks had not that conception of the permanent in matter that we have. To them material things were not permanent. In fire solid things would vanish; absolutely disappear. Rock and earth had a more stable existence, but they too grew and decayed. The permanence of matter, the conservation of energy, were unknown to them. And that distinction which we draw so readily between the fleeting and permanent causes of sensation, between a sound and a material object, for instance, had not the same meaning to them which it has for us. Let us but imagine for a moment that material things are fleeting, disappearing, and we shall enter with a far better appreciation into that search for the permanent which, with the Greeks, as with us, is the primary intellectual demand.

What is that which amid a thousand forms is ever the same, which we can recognise under all its vicissitudes, of which the diverse phenomena are the appearances?

To think that this is number is not so very wide of the mark. With an intellectual apprehension which far outran the evidences for its application, the atomists asserted that there were everlasting material particles, which, by their union, produced all the varying forms and states of bodies. But in view of the observed facts of nature as then known, Aristotle, with perfect reason, refused to accept this hypothesis.