(2) The series of positions is supposed to be "an infinite series of successive positions." The moving point must take them one after another. But how can it? _Between any two positions of the point there is an infinite number of intermediate positions_. That is to say, no two of these successive positions must be regarded as _next to_ each other; every position is separated from every other by an infinite number of intermediate ones. How, then, shall the point move? It cannot possibly move from one position to the next, for there is no next. Shall it move first to some position that is not the next? Or shall it in despair refuse to move at all? Evidently there is either something wrong with this doctrine of the infinite divisibility of space, or there is something wrong with our understanding of it, if such absurdities as these refuse to be cleared away. Let us see where the trouble lies. 26. WHAT IS REAL SPACE?--It is plain that men are willing to make a number of statements about space, the ground for making which is not at once apparent. It is a bold man who will undertake to say that the universe of matter is infinite in extent. We feel that we have the right to ask him how he knows that it is. But most men are ready enough to affirm that space is and must be infinite. How do they know that it is? They certainly do not directly perceive all space, and such arguments as the one offered by Hamilton and Spencer are easily seen to be poor proofs. Men are equally ready to affirm that space is infinitely divisible. Has any man ever looked upon a line and perceived directly that it has an infinite number of parts? Did any one ever succeed in dividing a space up infinitely? When we try to make clear to ourselves how a point moves along an infinitely divisible line, do we not seem to land in sheer absurdities? On what sort of evidence does a man base his statements regarding space? They are certainly very bold statements. A careful reflection reveals the fact that men do not speak as they do about space for no reason at all. When they are properly understood, their statements can be seen to be justified, and it can be seen also that the difficulties which we have been considering can be avoided. The subject is a deep one, and it can scarcely be discussed exhaustively in an introductory volume of this sort, but one can, at least, indicate the direction in which it seems most reasonable to look for an answer to the questions which have been raised. How do we come to a knowledge of space, and what do we mean by space? This is the problem to solve; and if we can solve this, we have the key which will unlock many doors.