that enable him to grasp certain laws of chemical combination. That suggestion might have sufficed in the early days of the atomic theory; but now the existence of atoms as entities in the real world of physics is fully demonstrated. This confident assertion is not inconsistent with philosophic doubts as to the meaning of ultimate reality.
When therefore we are asked whether the four-dimensional world may not be regarded merely as an illustration of mathematical processes, we must bear in mind that our questioner has probably an ulterior motive. He has already a belief in a real world of three Euclidean dimensions, and he hopes to be allowed to continue in this belief undisturbed. In that case our answer must be definite; the real three-dimensional world is obsolete, and must be replaced by the four-dimensional space-time with non-Euclidean properties. In this book we have sometimes employed illustrations which certainly do not correspond to any physical reality—imaginary time, and an unperceived fifth dimension. But the four-dimensional world is no mere illustration; it is the real world of physics, arrived at in the recognised way by which physics has always (rightly or wrongly) sought for reality.
I hold a certain object before me, and see an outline of the figure of Britannia; another observer on the other side sees a picture of a monarch; a third observer sees only a thin rectangle. Am I to say that the figure of Britannia is the real object; and that the crude impressions of the other observers must be corrected to make allowance for their positions? All the appearances can be accounted for if we are all looking at a three-dimensional object—a penny—and no reasonable person can doubt that the penny is the corresponding physical reality. Similarly, an observer on the earth sees and measures an oblong block; an observer on another star contemplating the same object finds it to be a cube. Shall we say that the oblong block is the real thing, and that the other observer must correct his measures to make allowance for his motion? All the appearances are accounted for if the real object is four-dimensional, and the observers are merely measuring different three-dimensional appearances or sections; and it seems impossible to doubt that this is the true explanation. He who doubts the reality of the
