A Discourse Presented to the Most Serene Don Cosimo II., Great Duke of Tuscany, Concerning the Natation of Bodies Vpon, and Submersion In, t · Galileo Galilei
PROBL. I.
Let the Solid I S be double in Gravity to the water, and let it be a regular Prisme, or Cylinder, to wit, that hath its two flat Superficies, superiour and inferiour, alike and equall, and at Right Angles with the other laterall Superficies, and let its thickness I O be equall to the greatest Altitude of the Banks of water: I say, that if it be put upon the water, it will not submerge: for the Altitude A I being equall to the Altitude I O, the Mass of the Air A B C I shall be equall to the Mass of the Solid C I O S: and the whole Mass A O S B double to the Mass I S; And since the Mass of the Air A C, neither encreaseth nor diminisheth the Gravity of the Mass I S, and the Solid I S was supposed double in Gravity to the water; Therefore as much water as the Mass submerged A O S B, compounded of the Air A I C B, and of the Solid I O S C, weighs just as much as the same submerged Mass A O S B: but when such a Mass of water, as is the submerged part of the Solid, weighs as much as the said Solid, it descends not farther, but resteth, as by (_a_) _Archimedes_[75], and above by us, hath been demonstrated: Therefore, I S shall descend no farther, but shall rest.
And if the Solid I S shall be Sesquialter in Gravity to the water, it shall float, as long as its thickness be not above twice as much as the greatest Altitude of the Ramparts of water, that is, of A I. For I S being Sesquialter in Gravity to the water, and the Altitude O I, being double to I A, the Solid submerged A O S B, shall be also Sesquialter in Mass to the Solid I S. And because the Air A C, neither increaseth nor diminisheth the ponderosity of the Solid I S: Therefore, as much water in quantity as the submerged Mass A O S B, weighs as much as the said Mass submerged: And, therefore, that Mass shall rest. And briefly in generall. [75] Of Natation Lib. 1. Prop. 3.
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