en 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