Page:Harper's New Monthly Magazine - v109.djvu/252

matter in this unknown direction the "paron" (from "para," alongside, and "on," being). The paron corresponds to the sheet of which we should have to tell the plane being.

As we have to tell the plane being of an unknown extension of his matter to the right away from his sheet, so we have to admit that our matter has an extension away from the paron in the unknown direction. Let us call the direction in which our matter extends away from the paron the "apo" direction (apo meaning away), and call the opposite direction from our matter towards the paron the "eiso" direction (eiso meaning within). Then "apo" and "eiso" correspond to the words "right" and "left" which we should have to teach the plane being to use. As in his case, so in ours, there is nothing in our conscious experience which corresponds to these words. They have reference to an unknown direction, and by attending to the possibilities which such a new direction gives we can gain the means of putting the question rationally as to whether it exists or not.

Triangles which, if kept against the Sheet, cannot be moved so that one will occupy the Space of the other

I will now briefly describe three cases in which an attempt has been made to find evidence for the reality of a fourth dimension. Cases 1 and 2 are such as would obviously suggest themselves to any inquirer. Case 3 I shall also merely touch upon, as its general argument has been published (see Bulletin of the Philosophical Society of Washington, April, 1902), while the mathematical method used was exemplified in a paper printed in the Proceedings of the Royal Irish Academy, November, 1902.

Case 1 depends on the properties of configuration. In a plane three points can be found equally distant from one another, such as the vertices of an equilateral triangle. In our space four such points can be found, such as the vertices of a tetrahedron. In four-dimensional space five points equally distant from one another can be found.

Figure symmetrical about a
Line

Now to account for the properties of organic compounds it has been necessary to assume that the carbon atoms in the molecules of certain substances are related as the four vertices of a tetrahedron. If it became necessary to assume the existence of five atoms at equal distances from one another in a molecule, there would be evidence of a fourth dimension.

Case 2 depends on the properties of rotation in four-dimensional space. We are familiarly acquainted with right and left handed shapes. The right hand itself and the left-hand image it meets in a mirror are examples of these configurations—they are alike, one another, on opposite sides of a plane. One cannot be turned into another in our space. Now in a plane, rotation takes place round a point; in our space, round an axis; and hence we should conclude by analogy that in four-dimensional space rotation took place round a plane. This conclusion is found to be justified if one looks into the matter. In fact, in a plane two triangles, such as shown in Fig. 5, are incapable of being turned into one another by any motion in the plane. One cannot be made to occupy the space of the other however it is turned about in the plane. Such figures correspond to our right and left handed shapes, and the rotation round a line by which they would be turned into one another is just as inconceivable to a plane being as rotation round a plane is to us. Our right and left handed shapes are, on the hypothesis of a fourth dimension, shapes turned half-way round.

Now there are two substances, two varieties of tartaric acid, which are alike in all physical and chemical properties, save in their behavior with regard to polarized light. One turns the plane of polarization in one direction, the other in the opposite direction. This is due to the molecules of the one being of exactly