That it is possible [to fo{r}m Solid Bodies,] of what Figure and greatness soever, that of their own Nature goe to the Bottome; But by the help of the Air contained in the Rampart, rest without submerging.
The truth of this Proposition is sufficiently manifest in all those Solid Figures, that determine in their uppermost part in a plane Superficies: for making such Figures of some Matter specifically as grave as the water, putting them into the water, so that the whole Mass be covered, it is manifest, that they shall rest in all places, provided, that such a Matter equall in weight to the water, may be exactly adjusted: and they shall by consequence, rest or lie even with the Levell of the water, without making any Rampart. If, therefore, in respect of the Matter, such Figures are apt to rest without submerging, though deprived of the help of the Rampart, it is manifest, that they may admit so much encrease of Gravity, (without encreasing their Masses) as is the weight of as much water as would be contained within the Rampart, that is made about their upper plane Surface: by the help of which being sustained, they shall rest afloat, but being bathed, they shall descend, having been made graver than the water. 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. Now here ariseth a doubt, whether the Cone A B D may be made heavier, in such sort, that when it is put wholly under water, it goes to the bottom, but yet not in such sort, as to take from the Rampart the vertue of sustaining it that it sink not, and, the reason of the doubt is this: that although at such time as the Cone A B D is specifically as grave as the water, the Rampart E S T O sustaines it, not only when the point A S T is tripple in height to the Altitude of the Rampart E S, but also when a lesser part is above water; [for although in the Descent of the Cone the Point A S T by little and little
diminisheth, and so likewise the Rampart E S T O, yet the Point diminisheth in greater proportion than the Rampart, in that it diminisheth according to all the three Dimensions, but the Rampart according to two only, the Altitude still remaining the same; or, if you will, because the [Cone S {A} T] goes diminishing, according to the proportion of the cubes of the Lines that do successively become the Diameters of the Bases of emergent Cones, and the Ramparts diminish according to the proportion of the Squares of the same Lines; whereupon the proportions of the Points are alwayes Sesquialter of the proportions of the Cylinders, contained within the Rampart; so that if, for Example, the height of the emergent Point were double, or equall to the height of the Rampart, in these cases, the Cylinder contained within the Rampart, would be much greater than the said Point, because it would be either sesquialter or tripple, by reason of which it would perhaps serve over and above to sustain the whole Cone, since the part submerged would no longer weigh any thing;] yet, nevertheless, when any Gravity is added to the whole Mass of the Cone, so that also the part submerged is not without some excesse of Gravity above the Gravity of the water, it is not manifest, whether the Cylinder contained within the Rampart, in the descent that the Cone shall make, can be reduced to such a proportion unto the emergent Point, and to such an excesse of Mass above the Mass of it, as to compensate the excesse of the Cones Specificall Gravity above the Gravity of the water: and the Scruple ariseth, because that howbeit in the descent made by the Cone, the emergent Point A S T diminisheth, whereby there is also a diminution of the excess of the Cones Gravity above the Gravity of the water, yet the case stands so, that the Rampart doth also contract it self, and the Cylinder contained in it doth deminish. 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: From whence we conclude, that the same Accident [Natatio{n} easiest effected] in Figures broad toward the top. 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.