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
THEOREME XII.
In Figures, therefore, that determine above in a plane, we may cleerly comprehend, that the Rampart added or removed, may prohibit or permit the descent: but in those Figures that go lessening upwards towards the top, some Persons may, and that not without much seeming Reason, doubt whether the same may be done, and especially by those which terminate in a very acute Point, such as are your Cones and small Piramids. Touching these, therefore, as more dubious than the rest, I will endeavour to demonstrate, that they also lie under the same Accident of going, or not going to the Bottom, be they of any whatever bigness. Let therefore the Cone be A B D, made of a matter specifically as grave as the water; it is manifest that being put all under water, it shall rest in all places (alwayes provided, that it shall weigh exactly as much as the water, which is almost impossible to effect) and that any small weight being added to it, it shall sink to the bottom: but if it shall descend downwards gently, I say, that it shall make the Rampart E S T O, and that there shall stay out of the water the point A S T, tripple in height to the Rampart E S: which is manifest, for the Matter of the Cone weighing equally with the water, the part submerged _S B D T_, becomes indifferent to move downwards or upwards; and the Cone _A S T_, being equall in Mass to the water that would be contained in the concave of the Rampart _E S T O_, shall be also equall unto it in Gravity: and, therefore, there shall be a perfect _Equilibrium_, and, consequently, a Rest.
Nevertheless it shall be demonstrated, how that the Cone _A B D_ being of any supposed bignesse, and made at the first of a Matter exactly equall in Gravity to the Water, if there may be affixed to it some Weight, by means of which i{t} may descend to the bottom, when submerged under water, it may also by vertue of the Rampart stay above without sinking. 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.
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