Now, it is quite true that the motion of the point can be described in a number of different ways; but the important thing to remark here is that, if the motion really is uniform, and if the line really is infinitely divisible, this series must, as satisfactorily as any other, describe the motion of the point.  And it would be absurd to maintain that a part of the series can describe the whole motion.  We cannot say, for example, that, when the point has moved over one half, one fourth, and one eighth of the line, it has completed its motion.  If even a single member of the series is left out, the whole line has not been passed over; and this is equally true whether the omitted member represent a large bit of line or a small one.

The whole series, then, represents the whole line, as definite parts of the series represent definite parts of the line.  The line can only be completed when the series is completed.  But when and how can this series be completed?  In general, a series is completed when we reach the final term, but here there appears to be no final term.  We cannot make zero the final term, for it does not belong to the series at all.  It does not obey the law of the series, for it is not one half as large as the term preceding it—­what space is so small that dividing it by 2 gives us [omicron]?  On the other hand, some term just before zero cannot be the final term; for if it really represents a little bit of the line, however small, it must, by hypothesis, be made up of lesser bits, and a smaller term must be conceivable.  There can, then, be no last term to the series; i.e. what the point is doing at the very last is absolutely indescribable; it is inconceivable that there should be a very last.

It was pointed out many centuries ago that it is equally inconceivable that there should be a very first.  How can a point even begin to move along an infinitely divisible line?  Must it not before it can move over any distance, however short, first move over half that distance?  And before it can move over that half, must it not move over the half of that?  Can it find something to move over that has no halves?  And if not, how shall it even start to move?  To move at all, it must begin somewhere; it cannot begin with what has no halves, for then it is not moving over any part of the line, as all parts have halves; and it cannot begin with what has halves, for that is not the beginning. What does the point do first? that is the question.  Those who tell us about points and lines usually leave us to call upon gentle echo for an answer.

The perplexities of this moving point seem to grow worse and worse the longer one reflects upon them.  They do not harass it merely at the beginning and at the end of its journey.  This is admirably brought out by Professor W. K. Clifford (1845-1879), an excellent mathematician, who never had the faintest intention of denying the possibility of motion, and who did not desire to magnify the perplexities in the path of a moving point.  He writes:—­