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is let down into this water, be abcd; and let the magnitude ef, when it has been let down into the water, not be completely submerged, if this can be done, but let a certain part protrude, namely e; and let only part f be submerged.Then it is necessary that, while the magnitude f is being submerged, the water should be raised: accordingly, let the surface of the water ao be raised up to the surface st.It is consequently manifest that the size of water so is as great as the size of the submerged part of the magnitude, namely f: for it is necessary that the place, into which the magnitude enters, should be evacuated of water, and that an amount of water should be removed that is as great in size as the size of the magnitude that is being submerged.And so the size of water so is equal to the size of the submerged magnitude, namely this f; hence also the heaviness of this same f will be equal to the heaviness of water so.And since water so strives by its heaviness to return to its former position, but it cannot achieve this unless solid ef is first removed from the water and raised by the water; and the solid, so as not to be raised, resists with all its proper heaviness; and both the solid magnitude and the water are assumed to be standing still in this position; therefore it is necessary that the heaviness of water so, by which it strives to raise the solid upward, be equal to the heaviness with which the solid resists and exerts pressure downward (for if the heaviness of water so were greater than the heaviness of solid ef, ef would be raised and expelled by the water; but if the heaviness of the solid ef were greater, the water, on the other hand, would be raised: yet all these things are assumed to be standing still as they are.Consequently the heaviness of water so is equal to the heaviness of the whole of ef: which is unacceptable; for the heaviness of the same so is equal to the heaviness of the part f.It is consequently manifest that no part of the solid magnitude ef will protrude, but that it will be completely submerged.

This is the complete demonstration, which I have explained by means of a rather lengthy account in this way in order that those who have come upon it for the first time may be able to understand it more easily; but it could also have been better explained by means of a briefer exposition, in such a way that the complete core of the demonstration would be as follows.It must be demonstrated that the magnitude ef, which is assumed to be equally as heavy as water, is completely submerged.For, if it is not completely submerged, let a certain part of it protrude: let e protrude; and let the water be raised up to the surface st; and, if such a thing can be done, let both the water and the magnitude remain in this position. Since, consequently, the magnitude ef by its heaviness exerts pressure on and raises water so; and water so, so as not to be raised further, resists with its heaviness; it is necessary that the heaviness of ef that exerts pressure be as great as