Return to Vignettes of Ancient Mathematics
Definition of faster (Physics Z 2.232a23-7)
Since every magnitude is divisible into magnitudes (for it was proved that it is impossible for a continuous magnitude to be from indivisible ones, but every magnitude is continuoous), then it is necessary that (a) the faster traverse a greater distance in the equal time and (b) an equal distance in the lesser time and (c) more distance in the lesser time, just as some define the faster.
Argument for a (Physics Z 2.232a27-31):
(figure 1: moving) or (figure 2: still)
For let A be faster than B. Then
since that which changes first is faster, in the time in which A has changed
from to , e.g. ZH, B will not yet be at , but it will fall short, so that
in the equal time the faster will traverse more in an equal time.
Argument for c (Physics Z 2.232a31-b5):
1) In fact, it will also move more in less time. For in the
time in which A has come to be at , let
B be at E, since it is the slower. (figure
2) Accordingly since A has come to be at in
the whole time ZH, it will be at in
a smaller time than this. (figure
3) And let it be in time ZK. And so , which A has traversed is
larger than E, and time ZK
is smaller than the whole time ZH, so that it will traverse a
larger amount in a smaller time.
Argument 1 for b (Physics Z 2.232b5-14):
1) It is also obvious from these that the faster traverses
an equal amount in a smaller time. For since it traverses a greater
in less time than the slower, while taken by itself it will traverse
more than the lesser amount in more time, e.g. M larger
than , the
time P in which it traverses
M would be more than , the
time in which it traverses . (figure
2) Thus if the time P is
smaller than X, the time in which the slower traverses , and will be smaller than X,
since it is smaller than P, as
that which is smaller than a smaller is itself smaller. Thus it
will move the equal amount in less time.
Argument 2 for b (Physics Z 2.232b14-29):
Furthermore, if everything must move in equal or less or in more time, then that which moves in more is slower, that in equal equally fast, while the faster is neither equally fast nor slower, the faster neither move in equal nor in more time. And so, it remains that it moves in less time, so that it is also necessary that the faster traverse an equal magnitude in less time.