A Discourse Presented to the Most Serene Don Cosimo II., Great Duke of Tuscany, Concerning the Natation of Bodies Vpon, and Submersion In, the Water.

by Galileo Galilei

COROLARY I.

Much less shall the said Cone swim, if one immerge a part thereof.

I say farther; that much lesse shall the said Cone stay afloat, if one immerge a part thereof.

Which you may see, comparing with the water as well the part that shall immerge as the other above water. Let us therefore of the Cone A B D, submergeth part N T O S, and advance the Point N S F above water. The Altitude of the Cone F N S, shall either be more than half the whole Altitude of the Cone F T O, or it shall not be more: if it shall be more than half, the Cone F N S shall be more than half of the Cylinder E N S C: for the Altitude of the Cone F N S, shall be more than Sesquialter of the Altitude of the Cylinder E N S C: And, because the Matter of the Cone is supposed to be double in Specificall Gravity to the water, the water which would be contained within the Rampart E N S C, would be less grave absolutely than the Cone F N S; so that the whole Cone F N S cannot be sustained by the Rampart: But the part immerged N T O S, by being double in Specificall Gravity to the water, shall tend to the bottom: Therefore, the whole Cone F T O, as well in respect of the part submerged, as the part above water shall descend to the bottom. But if the Altitude of the Point F N S, shall be half the Altitude of the whole Cone F T O, the same Altitude of the said Cone F N S shall be Sesquialter to the Altitude E N: and, therefore, E N S C shall be double to the Cone F N S; and as much water in Mass as the Cylinder E N S C, would weigh as much as the part of the Cone F N S. But, because the other immerged part N T O S, is double in Gravity to the water, a Mass of water equall to that compounded of the Cylinder E N S C, and of the Solid N T O S, shall weigh less than the Cone F T O, by as much as the weight of a Mass of water equall to the Solid N T O S: Therefore, the Cone [sha{l}l also descend.] Again, because the Solid N T O S, is septuple to the Cone F N S, to which the Cylinder E S is double, the proportion of the Solid N T O S, shall be to the Cylinder E N S C, as seaven to two: Therefore, the whole Solid compounded of the Cylinder E N S C, and of the Solid N T O S, is much less than double the Solid N T O S: Therefore, the single Solid N T O S, is much graver than a Mass of water equall to the Mass, compounded of the Cylinder E N S C, and of N T O S.