Page:Harper's New Monthly Magazine - v109.djvu/249



E know that every object has three dimensions, which are usually termed height, breadth, thickness. To take the simplest instances of these three dimensions, and the axes by which they are represented, consider the case of a room (Fig. 1). The lines 1, 2, 3 can be referred to as denning the directions of "up and down," "away or near," "right and left." We can proceed from one corner of the room to another by a straight line, such as A B. But we can equally well pass from A to B by going parallel to each of the axes in turn for a suitable distance. We can pass from any one point in the room to any other point by a combination of movements in these three directions. Since the room indefinitely expanded would occupy the whole of space as we think of it, we ordinarily assert that there is no point in the whole of space which we cannot reach by a combination of movements in three directions.

By means of movements in two directions we can only reach the points in a limited region of space: for instance, by means of movements parallel to axes 2 and 3 we can only reach points on the plane of the floor. But with all three axes and liberty of motion in three directions we can reach any point of space, as we conceive it.

Some thinkers nevertheless have decided that the three dimensions, height, length, breadth, do not exhaust the possibilities of space. They say that just as motion involving the axes 2, 3 will not enable us to reach all points in the space of a room or of the room extended, so motions in the directions of all three axes will not enable us to reach all the points of space as it really is.

Hence the birth and growth of the idea called, for want of a better name, the Fourth Dimension.

Plato, at the beginning of the seventh book of the Republic, describes a set of prisoners who are held in chains before the mouth of a great cavern, bound so that they cannot turn their faces in any other direction than looking straight into the cavern.

On the wall in which the cavern ends they see their shadows projected by the sun. Their only experience of objects is derived by watching these shadows. If passers-by traverse the roadway behind them, all they see is the shadows of these passers-by on the wall. If an object strikes them, what they see is the shadow of that object striking the shadows of themselves.

Plato draws the conclusion that they would identify themselves with their shadows. Since events occurring amongst these shadow forms are the invariable accompaniments of all their sensations, they would think that they themselves were those shadows, and lived and moved in a shadow world.

Now the shadows can only move on the surface of the wall; they cannot approach and recede from it. Hence the prisoners think of themselves as having a two-dimensional existence only. And, says Plato, as these prisoners think of themselves as less than they really are, so we in our turn think of ourselves as less than we really are. His philosophy was an effort to find that Greater which we really are.

Plato turns from the outward image to its inner significance, interrogating his self-consciousness. But in accordance with the modern habit of attending to the record of observation of the outward, let us trace out the objective experience of such prisoners.