* * * * "When a point moves along a line, we know that between any two positions of it there is an infinite number (in this new sense[73]) of intermediate positions. That is because the motion is continuous. Each of those positions is where the point was at some instant or other. Between the two end positions on the line, the point where the motion began and the point where it stopped, there is no point of the line which does not belong to that series. We have thus an infinite series of successive positions of a continuously moving point, and in that series are included all the points of a certain piece of line-room. May we say then that the line is made up of that infinite series of points? [73] Professor Clifford has used the word _number_ in two senses, a quantitative and a qualitative. By number in the latter sense he means simply _unlimited units_. "Yes; if we mean no more than that the series makes up the _points_ of the line. But _no_, if we mean that the line is made up of those points in the same way that it is made up of a great many very small pieces of line. A point is not to be regarded as a _part_ of a line, in any sense whatever. It is the boundary between two parts." These extracts suffice, I think, to show what the common doctrine is, and to show also the unavoidable difficulties connected with it. These were clearly seen long ago. Motion, argues Zeno of Elea,[74] cannot begin, because a body in motion must pass through an infinite number of intermediate places before it can arrive at any other place. Achilles can never overtake the tortoise, for by the time that he has reached the place where it was, it has always moved a little beyond. If Professor Clifford could not move a thing from one position to another, without making it go though an infinite number of intermediate positions, if these positions must be gone through with successively, and if infinite really mean _without any end_, then the final member of the series could never have been reached, for the plain reason that there is no final member to an endless series. If the new position is reached without passing through every member of the series and leaving none farther to pass through, it is not reached by passing through an infinite number of intermediate positions. The difficulty here is a hopeless one; either the series has a final member, _and then it is not infinite_; or it has not, _and then one cannot come to the end_. [74] Ueberweg, Hist. of Philos., Vol. I, § 20. N. Y., 1877, pp. 57-58.