280 double the lightness of b); however the swiftness 14 is less than double the swiftness 8.But swiftness 14 certainly has the same ratio, arithmetically, to swiftness 8 as the lightness of d to the lightness of b; since the excess of 14 over 8 is 6, and 6 is also the excess of the lightness of d, 12, over the lightness of b, 6. But if medium de should be lighter, in such a way that the heaviness of this d is 5, then the swiftness f will be 15 (for the excess of the heaviness of mobile a over the heaviness of medium d will be 15); and, again, swiftness 15 will be to swiftness 8 in the same ratio as the heaviness of medium b, 12, to the heaviness of medium d, 5, that is, the lightness of d to the lightness of b: for the excess in both cases will be 7. If, on the other hand, the heaviness of d is only 4, the swiftness f will be 16; and swiftness 16 will be to swiftness 8 (whose excess is 8) similarly in the same arithmetic ratio as the heaviness of b, 12, to the heaviness of d, 4, that is, the lightness of d to the lightness of b, whose excess is similarly 8. Now if, again, medium de is lighter and the heaviness of d is only 3, then the swiftness f will be 17; and the swiftness f, 17, will be to swiftness 8 (whose excess is 9) in the same arithmetic ratio as the heaviness of b, 12, to the heaviness of d, 3, that is, the lightness of d to the lightness of b. And if, again, medium de is lighter and the heaviness of d is only 2, then the swiftness f will be 18; and its arithmetic ratio to the swiftness 8 will be the same as that of the heaviness of b, 12, to the heaviness of d, 2, that is, of the lightness of d to the lightness of b: for the excess in both cases will be 10. If, again, medium de is lighter and the heaviness of d is only 1, then the swiftness f will be 19; it will have the same arithmetic ratio to swiftness 8, as the heaviness of b, 12, has to the heaviness of d, 1, that is, the lightness of d to the lightness of b: for the excess in both cases will be 11. And if, finally, the heaviness of d is 0, so that the excess of the heaviness of mobile a over medium d is 20, the swiftness f will be 20; and swiftness f, 20, will be to swiftness 8 in the same ratio, arithmetically, as that of the heaviness of b, 12, over the heaviness of d, 0: for in both cases the excess will be 12.
It is therefore evident how it is not geometrically but arithmetically that swiftness observes to swiftness the ratio that the lightness of the medium has to the lightness of the medium: and since it is not absurd, in an arithmetic ratio, that a quantity be to a quantity as a quantity to nothing, it will