Let, therefore, the Cone _A B D_ be of any supposed greatnesse, and alike in specificall Gravity to the water. It is manifest, that being put lightly into the water, it shall rest without descending; and it shall advance above water, the Point _A S T_, tripple in height to the height of the Rampart _E S_: Now, suppose the Cone _A B D_ more depressed, so that it advance above water, only the Point _A I R_, higher by half than the Point _A S T_, with the Rampart about it _C I R N_. And, because, the Cone _A B D_ is to the Cone _A I R_, as the cube of the Line _S T_ is to the cube of the Line _I R_, but the Cylinder _E S T O_, is to the Cylinder _C I R N_, as the Square of _S T_ to the Square of _I R_, the Cone _A S T_ shall be Octuple to the Cone _A I R_, and the Cylinder _E S T O_, quadruple to the Cylinder _C I R N_: But the Cone _A S T_, is equall to the Cylinder _E S T O_: Therefore, the Cylinder _C I R N_, shall be double to the Cone _A I R_: and the water which might be contained in the Rampart _C I R N_, would be double in Mass and in Weight to the Cone _A I R_, and, therefore, would be able to sustain the double of the Weight of the Cone _A I R_: Therefore, if to the whole Cone _A B D_, there be added as much Weight as the Gravity of the Cone _A I R_, that is to say, the eighth part of the weight of the Cone _A S T_, it also shall be sustained by the Rampart _C I R N_, but without that it shall go to the bottome: the Cone _A B D_, being, by the addition of the eighth part of the weight of the Cone _A S T_, made specifically more grave than the water. But if the Altitude of the Cone _A I R_, were two thirds of the Altitude of the Cone _A S T_, the Cone _A S T_ would be to the Cone _A I R_, as twenty seven to eight; and the Cylinder _E S T O_, to the Cylinder _C I R N_, as nine to four, that is, as twenty seven to twelve; and, therefore, the Cylinder _C I R N_, to the Cone _A I R_, as twelve to eight; and the excess of the Cylinder _C I R N_, above the Cone _A I R_, to the Cone _A S T_, as four to twenty seven: therefore if to the Cone _A B D_ be added so much weight as is the four twenty sevenths of the weight of the Cone _A S T_, which is a little more then its seventh part, it also shall continue to swimme, and the height of the emergent Point shall be double to the height of the Rampart. This that hath been demonstrated in Cones, exactly holds in Piramides, although the one or the other should be very sharp in their Point or Cuspis[77]: From whence we conclude, that the same Accident shall so much the more easily happen in all other Figures, by how much the less sharp the Tops shall be, in which they determine, being assisted by more spacious Ramparts. [77] Natatio{n} easiest effected in Figures broad toward the top. THEOREME XIII. [Sidenote: All Figures sink or swim, upon bathing or not bathing of their tops.]