Part 7

Let the Solid I S be superior in Gravity to the water, and of such thickness, that the Altitude of the Rampart A I, be in proportion to the thickness of the Solid I O, as the excess of the Gravity of the said Solid I S, above the Gravity of a Mass of water equall to the Mass I S, is to the Gravity of the Mass of water equall to the Mass I S. I say, that the Solid I S shall not sinke, but being never so little thicker it shall go to the bottom: For being that as A I is to I O, so is the Excess of the Gravity of the Solid I S, above the Gravity of a Mass of water equall to the Mass I S, to the Gravity of the said Mass of water: Therefore, compounding, as A O is to O I, so shall the Gravity of the Solid I S, be to the Gravity of a Mass of water equall to the Mass I S: And, converting, as I O is to O A, so shall the Gravity of a Mass of water equall to the Mass I S, be to the Gravity of the Solid I S: But as I O is to O A, so is a Mass of water I S, to a Mass of water equall to the Mass A B S O: and so is the Gravity of a Mass of water I S, to the Gravity of a Mass of water A S: Therefore as the Gravity of a Mass of water, equall to the Mass I S, is to the Gravity of the Solid I S, so is the same Gravity of a Mass of water I S, to the Gravity of a Mass of Water A S: Therefore the Gravity of the Solid I S, is equall to the Gravity of a Mass of water equall to the Mass A S: But the Gravity of the Solid I S, is the same with the Gravity of the Solid A S, compounded of the Solid I S, and of the Air A B C I. Therefore the whole compounded Solid A O S B, weighs as much as the water that would be comprised in the place of the said Compound A O S B: And, therefore, it shall make an _Equilibrium_ and rest, and that same Solid I O S C shall sinke no farther. But if its thickness I O should be increased, it would be necessary also to encrease the Altitude of the Rampart A I, to maintain the due proportion: But by what hath been supposed, the Altitude of the Rampart A I, is the greatest that the Nature of the Water and Air do admit, without the waters repulsing the Air adherent to the Superficies of the Solid I C, and possessing the space A I C B: Therefore, a Solid of greater thickness than I O, and of the same Matter with the Solid I S, shall not rest without submerging, but shall descend to the bottome: which was to be demonstrated. In consequence of this that hath been demonstrated, sundry and various Conclusions may be gathered, by which the truth of my principall Proposition comes to be more and more confirmed, and the imperfection of all former Argumentations touching the present Question cometh to be discovered.

_And first we gather from the things demonstrated, that,_

THEOREME VII.

[Sidenote: The heaviest Bodies may swimme.]

_All Matters, how heavy soever, even to Gold it self, the heaviest of all Bodies, known by us, may float upon the Water._

Because its Gravity being considered to be almost twenty times greater than that of the water, and, moreover, the greatest Altitude that the Rampart of water can be extended to, without breaking the Contiguity of the Air, adherent to the Surface of the Solid, that is put upon the water being predetermined, if we should make a Plate of Gold so thin, that it exceeds not the nineteenth part of the Altitude of the said Rampart, this put lightly upon the water shall rest, without going to the bottom: and if Ebony shall chance to be in sesquiseptimall proportion more grave than the water, the greatest thickness that can be allowed to a Board of Ebony, so that it may be able to stay above water without sinking, would be seaven times more than the height of the Rampart Tinn, _v. gr._ eight times more grave than water, shall swimm as oft as the thickness of its Plate, exceeds not the 7th part of the Altitude of the Rampart.

[Sidenote: _He elsewhere cites this as a Proposition, therefore I make it of that number._]

And here I will not omit to note, as a second Corrollary dependent upon the things demonstrated, that,

THEOREME VIII.

[Sidenote: Natation and Submersion, collected from the thickness, excluding the length and breadth of Plates.]

_The Expansion of Figure not only is not the Cause of the Natation of those grave Bodies, which otherwise do submerge, but also the determining what be those Boards of Ebony, or Plates of Iron or Gold that will swimme, depends not on it, rather that same determination is to be collected from the only thickness of those Figures of Ebony or Gold, wholly excluding the consideration of length and breadth, as having no wayes any share in this Effect._

It hath already been manifested, that the only cause of the Natation of the said Plates, is the reduction of them to be less grave than the water, by means of the connexion of that Air, which descendeth together with them, and possesseth place in the water; which place so occupyed, if before the circumfused water diffuseth it self to fill it, it be capable of as much water, as shall weigh equall with the Plate, the Plate shall remain suspended, and sinke no farther.

Now let us see on which of these three dimensions of the Solid depends the terminating, what and how much the Mass of that ought to be, that so the assistance of the Air contiguous unto it, may suffice to render it specifically less grave than the water, whereupon it may rest without Submersion. It shall undoubtedly be found, that the length and breadth have not any thing to do in the said determination, but only the height, or if you will the thickness: for, if we take a Plate or Board, as for Example, of Ebony, whose Altitude hath unto the greatest possible Altitude of the Rampart, the proportion above declared, for which cause it swims indeed, but yet not if we never so little increase its thickness; I say, that retaining its thickness, and encreasing its Superficies to twice, four times, or ten times its bigness, or dminishing it by dividing it into four, or six, or twenty, or a hundred parts, it shall still in the same manner continue to float: but encreasing its thickness only a Hairs breadth, it will alwaies submerge, although we should multiply the Superficies a hundred and a hundred times. Now forasmuch as that this is a Cause, which being added, we adde also the Effect, and being removed, it is removed; and by augmenting or lessening the length or breadth in any manner, the effect of going, or not going to the bottom, is not added or removed: I conclude, that the greatness and smalness of the Superficies hath no influence upon the Natation or Submersion. And that the proportion of the Altitude of the Ramparts of Water, to the Altitude of the Solid, being constituted in the manner aforesaid, the greatness or smalness of the Superficies, makes not any variation, is manifest from that which hath been above demonstrated, and from this, that, _The Prisms and Cylinders which have the same Base, are in proportion to one another as their heights._ Whence Cylinders or Prismes[76], namely, the Board, be they great or little, so that they be all of equall thickness, have the same proportion to their Conterminall Air, which hath for Base the said Superficies of the Board, and for height the Ramparts of water; so that alwayes of that Air, and of the Board, Solids, are compounded, that in Gravity equall a Mass of water equall to the Mass of the Solids, compounded of Air, and of the Board: whereupon all the said Solids do in the same manner continue afloat. We will conclude in the third place, that,

[76] Prismes and Cylinders having the same Base, are to one another as their heights.

THEOREME. IX.

[Sidenote: All Figures of all Matters, float by hep of the Rampart replenished with Air, and some but only touch the water.]

_All sorts of Figures of whatsoever Matter, albeit more grave than the Water, do by Benefit of the said Rampart, not only float, but some Figures, though of the gravest Matter, do stay wholly above Water, wetting only the inferiour Surface that toucheth the Water._

And these shall be all Figures, which from the inferiour Base upwards, grow lesser and lesser; the which we shall exemplifie for this time in Piramides or Cones, of which Figures the passions are common. We will demonstrate therefore, that,

_It is possible to form a Piramide, of any whatsoever Matter preposed, which being put with its Base upon the Water, rests not only without submerging, but without wetting it more then its Base._

For the explication of which it is requisite, that we first demonstrate the subsequent Lemma, namely, that,

LEMMA II.

[Sidenote: Solids whose Masses are in contrary proportion to their Specifick Gravities are equall in absolute Gravity.]

_Solids whose Masses answer in proportion contrarily to their Specificall Gravities, are equall in Absolute Gravities._

Let A C and B be two Solids, and let the Mass A C be to the Mass B, as the Specificall Gravity of the Solid B, is to the Specificall Gravity of the Solid A C: I say, the Solids A C and B are equall in absolute weight, that is, equally grave. For if the Mass A C be equall to the Mass B, then, by the Assumption, the Specificall Gravity of B, shall be equall to the Specificall Gravity of A C, and being equall in Mass, and of the same Specificall Gravity they shall absolutely weigh one as much as another. But if their Masses shall be unequall, let the Mass A C be greater, and in it take the part C, equall to the Mass B. And, because the Masses B and C are equall; the Absolute weight of B, shall have the same proportion to the Absolute weight of C, that the Specificall Gravity of B, hath to the Specificall Gravity of C; or of C A, which is the same _in specie_: But look what proportion the Specificall Gravity of B, hath to the Specificall Gravity of C A, the like proportion, by the Assumption, hath the Mass C A, to the Mass B, that is, to the Mass C: Therefore, the absolute weight of B, to the absolute weight of C, is as the Mass A C to the Mass C: But as the Mass A C, is to the Mass C, so is the absolute weight of A C, to the absolute weight of C: Therefore the absolute weight of B, hath the same proportion to the absolute weight of C, that the absolute weight of A C, hath to the absolute weight of C: Therefore, the two Solids A C and B are equall in absolute Gravity: which was to be demonstrated. Having demonstrated this, I say,

THEOREME X.

[Sidenote: 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.

[Sidenote: 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.

[Sidenote: 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,

THEOREME XI.

[Sidenote: A Piramide or Cone, demitted with the Point downwards shal swim, with its Base downward shall sink.]

_A Piramide or Cone put into the Water, with the Point downward shall swimme, and the same put with the Base downwards shall sinke, and it shall be impossible to make it float._

Now the quite contrary would happen, if the difficulty of Penetrating the water, were that which had hindred the descent, for that the said Cone is far apter to pierce and penetrate with its sharp Point, than with its broad and spacious Base.

And, to demonstrate this, let the Cone be _A B C_, twice as grave as the water, and let its height be tripple to the height of the Rampart _D A E C_: I say, first, that being put lightly into the water with the Point downwards, it shall not descend to the bottom: for the Aeriall Cylinder contained betwixt the Ramparts _D A C E_, is equall in Mass to the Cone _A B C_; so that the whole Mass of the Solid compounded of the Air _D A C E_, and of the Cone _A B C_, shall be double to the Cone _A C B_: And, because the Cone _A B C_ is supposed to be of Matter double in Gravity to the water, therefore as much water as the whole Masse _D A B C E_, placed beneath the Levell of the water, weighs as much as the Cone _A B C_: and, therefore, there shall be an _Equilibrium_, and the Cone _A B C_ shall descend no lower. Now, I say farther, that the same Cone placed with the Base downwards, shall sink to the bottom, without any possibility of returning again, by any means to swimme.

Let, therefore, the Cone be _A B D_, double in Gravity to the water, and let its height be tripple the height of the Rampart of water L B: It is already manifest, that it shall not stay wholly out of the water, because the Cylinder being comprehended betwixt the Ramparts _L B D P_, equall to the Cone _A B D_, and the Matter of the Cone, beig double in Gravity to the water, it is evident that the weight of the said Cone shall be double to the weight of the Mass of water equall to the Cylinder _L B D P_: Therefore it shall not rest in this state, but shall descend.

COROLARY I.

[Sidenote: 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 _C_one 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 _C_ylinder E N S C, would weigh as much as the part of the _C_one 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 _C_ylinder E N S C, and of the Solid N T O S, shall weigh less than the _C_one F T O, by as much as the weight of a Mass of water equall to the Solid N T O S: Therefore, the _C_one sha{l}l also descend. Again, because the Solid N T O S, is septuple to the Cone F N S, to which the _C_ylinder E S is double, the proportion of the Solid N T O S, shall be to the _C_ylinder E N S C, as seaven to two: Therefore, the whole Solid compounded of the _C_ylinder 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 _C_ylinder E N S C, and of N T O S.

COROLARY II.

[Sidenote: Part of the Cones towards the Cuspis removed, it shall still sink.]

_From whence it followeth, that though one should remove and take away the part of the Cone F N S, the sole remainder N T O S would go to the bottom._

COROLARY III.

[Sidenote: The more the Cone is immerged, the more impossible is its floating.]

_And if we should more depress the Cone F T O, it would be so much the more impossible that it should sustain it self afloat, the part submerged N T O S still encreasing, and the Mass of Air contained in the Rampart diminishing, which ever grows less, the more the Cone submergeth._

That Cone, therefore, that with its Base upwards, and its _Cuspis_ downwards doth swimme, being dimitted with its Base downward must of necessity sinke. They have argued farre from the truth, therefore, who have ascribed the cause of Natation to waters resistance of Division, as to a passive principle, and to the breadth of the Figure, with which the division is to be made, as the Efficient.

I come in the fourth place, to collect and conclude the reason of that which I have proposed to the Adversaries, namely,

THEOREME XII.

[Sidenote: Solids of any Figure & greatnesse, that naturally sink, may by help of the Air in the Rampart swimme.]

_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._