THEOREME X.

There may be Cones and Piramides of any Matter, which demitted into the water, rest only their Bases.

That it is possible of any assigned Matter, to form a Piramide or Cone upon any Base, which being put upon the Water shall not submerge, nor wet any more than its Base.

Let the greatest possible Altitude of the Rampart be the Line D B, and the Diameter of the Base of the Cone to be made of any Matter assigned B C, at right angles to D B: And as the Specificall Gravity of the Matter of the Piramide or Cone to be made, is to the Specificall Gravity of the water, so let the Altitude of the Rampart D B, be to the third part of the Piramide or Cone A B C, described upon the Base, whose Diameter is B C: I say, that the said Cone A B C, and any other Cone, lower then the same, shall rest upon the Surface of the water B C without sinking. Draw D F parallel to B C, and suppose the Prisme or Cylinder E C, which shall be tripple to the Cone A B C. And, because the Cylinder D C hath the same proportion to the Cylinder C E, that the Altitude D B, hath to the Altitude B E: But the Cylinder C E, is to the Cone A B C, as the Altitude E B is to the third part of the Altitude of the Cone: Therefore, by Equality of proportion, the Cylinder D C is to the Cone A B C, as D B is to the third part of the Altitude B E: But as D B is to the third part of B E, so is the Specificall Gravity of the Cone A B C, to the Specificall Gravity of the water: Therefore, as the Mass of the Solid D C, is to the Mass of the Cone A B C, so is the Specificall Gravity of the said Cone, to the Specificall Gravity of the water: Therefore, by the precedent Lemma, the Cone A B C weighs in absolute Gravity, as much as a Mass of Water equall to the Mass D C: But the water which by the imposition of the Cone A B C, is driven out of its place, is as much as would precisely lie in the place D C, and is equall in weight to the Cone that displaceth it: Therefore, there shall be an Equilibrium, and the Cone shall rest without farther submerging. And its manifest,

COROLARY I.

Amongst Cones of the same Base, those of least Altitude shall sink the least.

That making upon the same Basis, a Cone of a less Altitude, it shall be also less grave, and shall so much the more rest without Submersion.

COROLARY II.

There may be Cones and Piramides of any Matter, which demitted with the Point downwards do float atop.

It is manifest, also, that one may make Cones and Piramids of any Matter whatsoever, more grave than the water, which being put into the water, with the Apix or Point downwards, rest without Submersion.

Because if we reassume what hath been above demonstrated, of Prisms and Cylinders, and that on Bases equall to those of the said Cylinders, we make Cones of the same Matter, and three times as high as the Cylinders, they shall rest afloat, for that in Mass and Gravity they shall be equall to those Cylinders, and by having their Bases equall to those of the Cylinders, they shall leave equall Masses of Air included within the Ramparts. This, which for Example sake hath been demonstrated, in Prisms, Cylinders, Cones and Piramids, might be proved in all other Solid Figures, but it would require a whole Volume (such is the multitude and variety of their Symptoms and Accidents) to comprehend the particuler demonstration of them all, and of their severall Segments: but I will to avoid prolixity in the present Discourse, content my self, that by what I have declared every one of ordinary Capacity may comprehend, that there is not any Matter so grave, no not Gold it self, of which one may not form all sorts of Figures, which by vertue of the superiour Air adherent to them, and not by the Waters Resistance of Penetration, do remain afloat, so that they sink not. Nay, farther, I will shew, for removing that Error, that,