270 is carried upward with a force as great as the heaviness of water tf (for it is by this heaviness that water eb is heavier than the heaviness of water tb, that is, than the heaviness of magnitude cd). Now since the heaviness of water tb is equal to the heaviness of cd, water tb will exert an upward pressure so as to raise the magnitude with as much force as that with which the magnitude will resist being raised. And thus the heaviness of the part of water exerting pressure, namely tb, is equal to the resistance of the solid magnitude: but the heaviness of all the water that exerts pressure, eb, surpasses the heaviness of water tb by the heaviness of water tf: hence the heaviness of all the water eb will surpass the resistance of solid cd by the heaviness of water tf. And so the heaviness of all the water exerting pressure will impel the solid magnitude upward with as much force as the heaviness of part tf of the water: which was to be demontrated.
From this demonstration it is evident, first, how, as was said above, upward motion comes about from heaviness, but the heaviness of the medium, not of the mobile: second, the aim of our research is concluded.
For since we are investigating how much more swiftly the same mobile goes up through one medium than another, whenever we know with how much swiftness it is carried through each one of them, we will also know the interval between the swiftnesses: and that is what we seek.
If then this piece of wood, for example, whose heaviness is 4, is carried upward in water, and the heaviness of an amount of water as great in size as the size of the wood is 6, then the wood will be carried with a swiftness of 2: but if, on the contrary, this same piece of wood is carried upward in a medium heavier than water, such that the heaviness of an amount of this second medium as great in size as the size of the wood is 10, then the wood will be carried upward in it with a swiftness of 6. But it was carried in the other medium with a swiftness of 2: hence these two swiftnesses will be to one another as 6 and 2, not as the heavinesses or the thicknesses of the media, as Aristotle wanted it, which are to one another as 10 and 6. Thus it is evident that, in all cases, the swiftnesses of the upward motions are to one another as the excess of heaviness of one medium over the heaviness of the mobile is to the excess of heaviness of the other medium over the heaviness of the same mobile. For that reason, if on the spot we want to know the swiftnesses of the same mobile in two media, let us take from each medium two amounts equal in size to the size of the mobile, and let the heaviness of the mobile be substracted from the heaviness of each one of the media; the remaining numbers will be to one another {2} as