272 to the first, with a size equal to that of water ao, and let its heaviness be equal to that of water ae; and let the said magnitude be lm: and since size bl is equal to size ae, and size lm is equal to size ao, the size of the combined magnitudes bl, lm is equal to the size of the combined amounts of water ea, ao. But the heaviness of the magnitude of water ae is equal to the heaviness of magnitude lm, while the heaviness of water ao is equal to the heaviness of magnitude bl: hence the total heaviness of both magnitudes bl, lm is equal to the heaviness of water oa, ae. But in addition it has been demonstrated that the size of the magnitudes is equal to the size of water oa, ae; hence, by the first proposition, the magnitudes so combined will be carried neither upward nor downward. Therefore the force of the magnitude bl exerting pressure downward will be as great as the force of magnitude lm, which impels upward: but, by the previous proposition, magnitude lm impels upward with as much force as the quantity of heaviness of water do: therefore magnitude bl will be carried downward with a force as great as the heaviness of water do. Which is what was to be demonstrated.
Now, if this demonstration has been grasped, the answer to these questions can easily be discerned.For it is clear that that the same mobile going down in different media, observes in the swiftness of its motions, the ratio to one another of the excesses of its own heaviness over the heavinesses of the media: thus if the heaviness of the mobile is 8, but the heaviness of a size of one medium, equal to that of the mobile, is 6, then the swiftness of this body will be 2; if the heaviness of an amount of the other medium, equal to the size of the mobile, is 4, then the swiftness of the mobile, in this medium, will be 4. It is therefore evident that these swiftnesses will be to one another as 2 and 4; and not as the thicknesses or the heavinesses of the media, which is what Aristotle wanted, which are to one another as 6 and 4. {1} Similarly the answer to the other question is evident : namely, what ratio the speeds of mobiles equal in size, but unequal in heaviness, observe with one another in the same medium. For the speeds of such mobiles will be to one another as the excesses by which the heavinesses of the mobiles exceed the heaviness of the medium: thus, for example, if two mobiles are equal in size, but unequal in heaviness, the heaviness of one being 8, and of the other 6, but the heaviness of an amount of the medium, equal in size to the size of one of the two mobiles, is 4, then the swiftness of the former mobile will be 4, and that of the latter will be 2. Hence these speeds will observe the ratio of 4 to 2; and not that which is between the heavinesses, namely 8 to 6. {1}And from all the things that have been conveyed here, it will not be difficult to apprehend also the ratio that will be observed by mobiles of different species in different media. For one should scrutinize what ratio the two observe, in swiftness, in the same medium; how this is to be done is evident from the preceding: {1} next, one should investigate what swiftness the other has in the other medium, also by means of what has been conveyed above: and we will have what is sought. Thus, for example, if there are two mobiles, equal in size, but different in heaviness, and the heaviness of one is 12, and of the other 8, and we seek the ratio between the swiftness of the one whose heaviness is 12, going down in water, and the swiftness of the one whose heaviness is 8, going down in air; let us see, first, how much faster 12 goes down in water than 8, next how much more swiftly 8 is carried in air than in water: and we will have what we are aiming at; or, alternatively, let us see how much more swiftly 12 goes down in air than 8, then how much more slowly the 12 is carried in water than in air.
These, then, are the general rules of the ratios of the motions of mobiles, whether of the same species or not, whether in the same medium or in different media, whether moved upward or downward. But it must be noted that a very great difficulty arises here: it will be found that these ratios are not observed by one who has made a test.
For if one takes two different mobiles, which have such properties that one is carried twice as swiftly as the other, and then releases them from the top of a tower, it will certainly not hit the ground faster, twice as swiftly: what is more, if one makes the observation, the one which is lighter at the beginning of the motion will precede the heavier and will be faster.
This is not the place to inquire into how these differences and, so to speak, prodigies come about (for they are accidental): for it is necessary first to examine certain things which have not yet been inspected. For it is necessary, first, to see why natural motion is slower at the beginning.Chapter 9 [274.1-276.23]