y converge, whatever be the original interval
or the contrasted speeds, toward infinitesimal shortness. This
proportionality of the shortness of the times to that of the spaces
required frees us, it is claimed, from the sophism which the word
'never' suggests.

But this criticism misses Zeno's point entirely. Zeno would have been
perfectly willing to grant that if the tortoise can be overtaken at
all, he can be overtaken in (say) twenty seconds, but he would still
have insisted that he can't be overtaken at all. Leave Achilles and
the tortoise out of the account altogether, he would have said--they
complicate the case unnecessarily. Take any single process of change
whatever, take the twenty seconds themselves elapsing. If time be
infinitely divisible, and it must be so on intellectualist principles,
they simply cannot elapse, their end cannot be reached; for no matter
how much of them has already elapsed, before the remainder, however
minute, can have wholly elapsed, the earlier half of it must first
have elapsed. And this ever re-arising need of making the earlier half
elapse _first_ leaves time with always something to do _before_ the
last thing is done, so that the last thing never gets done. Expressed
in bare numbers, it is like the convergent series 1/2 plus 1/4 plus
1/8..., of which the limit is one. But this limit, simply because it
is a limit, stands outside the series, the value of which approaches
it indefinitely but never touches it. If in the natural world there
were no other way of getting things save by such successive addition
of their logically involved fractions, no complete units or whole
things would ever come into being, for the fractions' sum would always
leave a remainder. But in point of fact nature doesn't make eggs by
making first half an egg, then a quarter, then an eighth, etc., and
adding them together. She either makes a whole egg at once or none
at all, and so of all her other units. It is only in the sphere of
change, then, where one phase of a thing must needs come into being
before another phase can come that Zeno's paradox gives trouble.

And it gives trouble then only if the succession of steps of change
be infinitely divisible. If a bottle had to be emptied by an infinite
number of successive decrements, it is mathematically impossible that
the emptying should ever positively terminate. In point of fact,
however, bottles and coffee-pots empty themselves by a finite number
of decrem