It is only through accumulated knowledge, especially the work of Gauss, Lobatschewsky, Bolyai, Riemann, and others that modern mathematicians are able to deal easily with space of more than three dimensions. It may be noted that Kant says: "If it be possible that there are developments of other dimensions of space, it is very probable that God has somewhere produced them. For His works have all the grandeur and glory that can be comprised." According to Mr. G.R.S. Mead similar ideas are to be found in certain of the Gnostic cosmogonies. (Fragments of a Faith forgotten, p. 318.) But a detailed historical review would be out of place here and I will therefore proceed at once to a discussion of what is meant by the term "fourth dimension" and will try to explain how it is that we can determine some of the necessary properties of four-dimensional space, even although we cannot picture it to ourselves. At this point I would urge the reader to try to believe that the subject is not one of great difficulty. As a matter of fact it is really exceptionally straightforward if only one faces it and does not allow oneself to be frightened. I know that it is impossible to form any clear mental picture of four-dimensional conditions, but that does not matter. The ideas involved are admittedly unprecedented in our experience, but they are not contrary to reason and I do not ask more than a formal and intellectual assent to the propositions and analogies concerned. Let me start, then, by defining what is meant by a Dimension. The best definition I can think of is to say that, in the sense in which the word is used here, a Dimension means "An independent direction in space." I must amplify this by saying that, "Two directions in space are to be considered as independent when they are so related that no movement, however great, along one of them will result in the slightest movement along, or parallel to, the other. That is to say, at right angles, or perpendicular to one another." Thus in Fig. 1 AOA´ and BOB´ are independent directions. One might move for ever along OA or OA´ and yet one would not have moved in the very least in the direction of OB or of OB´. [Illustration: _Fig. 1_] Now on a flat surface, such as a sheet of paper, it is not possible to draw more than _two_ such directions. Any other line that can be drawn, XOX´ for instance, is in a compound direction, so to speak. That is to say it is partly in the direction AOA´ and partly in the direction BOB´ and it is possible to reach any point in it, Y for example, by moving along OA´ to _a_ and then moving in the direction of OB´ a distance equal to O_b_, or _vice versa_ or by doing the two simultaneously. For the benefit of those who are absolutely ignorant of the rudiments of Geometrical knowledge, I would point out that Parallel lines are said to point, in fact _do_ point, in the same direction. [Illustration: _Fig. 2_]