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.
top


Argument for c (Physics Z 2.232a31-b5):

(figure 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.
top


Argument 1 for b (Physics Z 2.232b5-14):

(figure 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.
top


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.

top