401 From the things that have been set out above, it follows that swiftness to swiftness always observes the ratio of the lightness of one medium to another arithmetically, not geometrically. {1}For if, again, the heaviness of c is 2, so that its lightness is four times the lightness of b, the swiftness of e certainly will not be four times the swiftness of d, but will be 3/2; for the swiftness e will be as 18: and yet it will have the same arithmetic ratio [to d] as c has to b {1}, since the excesses are equal, namely 6. And if, again, the heaviness of c is 1, so that the lightness of c is eight times the heaviness {1} of b, the swiftness of e certainly will not be eight times that of d, but by far smaller than eight times, namely 19/12; for the swiftness of e will be as 19: and the arithmetic ratio [of the swftnesses] will be the same as that of the lightness [of one medium] to the lightness [of the other], since the excess is the same. namely 7. Now if the heaviness of c is zero, so that the lightness of c has no ratio to the lightness of b, the swiftness of e will be as 20, having the same arithmetic ratio to d as 8 to 0: for the excess of swiftness 20 to swiftness 12 is the same as that by which 8 exceeds 0, namely 8.And thus, contrary to what Aristotle says, it is not unacceptable that a number has to another number the same ratio as a third number has to zero, provided that we speak of an arithmetic ratio: 20 to 12 has the same ratio as 8 has to 0; for the excess of 20 over 12 is the same as the excess of 8 to 0.
DO. Oh! what a subtle discovery, oh! how beautifully imagined!Let them remain silent, silent, those who assert that they can pursue philosophy without a knowledge of divine mathematics. And will anyone ever deny that only with it as guide can the true be distinguished from the false, that with its aid keenness of mind is stimulated, and that, finally, with it as guide whatever is really known among us mortals can be apprehended and understood? {1}
AL. Listen, I pray you. On the basis of his hypotheses, Aristotle drew a second argument: namely, that if motion in a void took place in time, then lighter and heavier things would be moved with the same swiftness, since for both lighter and heavier things there would be no resistance from the medium {1}; which is unacceptable. In this argument Aristotle has been deceived in a similar way, in that he assumed that the swiftness and the slowness of motion arise only from the resistence of the medium, whereas in fact the whole affair depends