The Complete Aristotle (eng.)

of the quicker can
occupy neither an equal time nor more time. It can only be, then,
that it occupies less time, and thus we get the necessary
consequence that the quicker will pass over an equal magnitude (as
well as a greater) in less time than the slower.
    And since every motion is in time and a motion may occupy any
time, and the motion of everything that is in motion may be either
quicker or slower, both quicker motion and slower motion may occupy
any time: and this being so, it necessarily follows that time also
is continuous. By continuous I mean that which is divisible into
divisibles that are infinitely divisible: and if we take this as
the definition of continuous, it follows necessarily that time is
continuous. For since it has been shown that the quicker will pass
over an equal magnitude in less time than the slower, suppose that
A is quicker and B slower, and that the slower has traversed the
magnitude GD in the time ZH. Now it is clear that the quicker will
traverse the same magnitude in less time than this: let us say in
the time ZO. Again, since the quicker has passed over the whole D
in the time ZO, the slower will in the same time pass over GK, say,
which is less than GD. And since B, the slower, has passed over GK
in the time ZO, the quicker will pass over it in less time: so that
the time ZO will again be divided. And if this is divided the
magnitude GK will also be divided just as GD was: and again, if the
magnitude is divided, the time will also be divided. And we can
carry on this process for ever, taking the slower after the quicker
and the quicker after the slower alternately, and using what has
been demonstrated at each stage as a new point of departure: for
the quicker will divide the time and the slower will divide the
length. If, then, this alternation always holds good, and at every
turn involves a division, it is evident that all time must be
continuous. And at the same time it is clear that all magnitude is
also continuous; for the divisions of which time and magnitude
respectively are susceptible are the same and equal.
    Moreover, the current popular arguments make it plain that, if
time is continuous, magnitude is continuous also, inasmuch as a
thing asses over half a given magnitude in half the time taken to
cover the whole: in fact without qualification it passes over a
less magnitude in less time; for the divisions of time and of
magnitude will be the same. And if either is infinite, so is the
other, and the one is so in the same way as the other; i.e. if time
is infinite in respect of its extremities, length is also infinite
in respect of its extremities: if time is infinite in respect of
divisibility, length is also infinite in respect of divisibility:
and if time is infinite in both respects, magnitude is also
infinite in both respects.
    Hence Zeno’s argument makes a false assumption in asserting that
it is impossible for a thing to pass over or severally to come in
contact with infinite things in a finite time. For there are two
senses in which length and time and generally anything continuous
are called ‘infinite’: they are called so either in respect of
divisibility or in respect of their extremities. So while a thing
in a finite time cannot come in contact with things quantitatively
infinite, it can come in contact with things infinite in respect of
divisibility: for in this sense the time itself is also infinite:
and so we find that the time occupied by the passage over the
infinite is not a finite but an infinite time, and the contact with
the infinites is made by means of moments not finite but infinite
in number.
    The passage over the infinite, then, cannot occupy a finite
time, and the passage over the finite cannot occupy an infinite
time: if the time is infinite the magnitude must be infinite also,
and if the magnitude is infinite, so also is the time. This may be
shown as follows. Let AB be a finite magnitude, and let us suppose
that it is traversed in infinite time G, and let a finite period GD
of the time be taken. Now in this period the thing in motion will
pass over a certain segment of the magnitude: let BE be the segment
that it has thus passed over. (This will be either an exact measure
of AB or less or greater than an exact measure: it makes no
difference which it is.) Then, since a magnitude equal to BE will
always be passed over in an equal time, and BE measures the whole
magnitude, the whole time occupied in passing over AB will be
finite: for it