Discourse on Floating Bodies Part 6

THEOREME. VI.

[Sidenote: The proportion of the greatest thickness of Solids, beyond which encreased they sink.]

_When ever the excess of the Gravity of the Solid above the Gravity of the Water, shall have the same proportion to the Gravity of the Water, that the Alt.i.tude of the Rampart, hath to the thickness of the Solid, that Solid shall not sink, but being never so little thicker it shall._

[Ill.u.s.tration]

Let the Solid I S be superior in Gravity to the water, and of such thickness, that the Alt.i.tude 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 Ma.s.s of water equall to the Ma.s.s I S, is to the Gravity of the Ma.s.s of water equall to the Ma.s.s 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 Ma.s.s of water equall to the Ma.s.s I S, to the Gravity of the said Ma.s.s 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 Ma.s.s of water equall to the Ma.s.s I S: And, converting, as I O is to O A, so shall the Gravity of a Ma.s.s of water equall to the Ma.s.s I S, be to the Gravity of the Solid I S: But as I O is to O A, so is a Ma.s.s of water I S, to a Ma.s.s of water equall to the Ma.s.s A B S O: and so is the Gravity of a Ma.s.s of water I S, to the Gravity of a Ma.s.s of water A S: Therefore as the Gravity of a Ma.s.s of water, equall to the Ma.s.s I S, is to the Gravity of the Solid I S, so is the same Gravity of a Ma.s.s of water I S, to the Gravity of a Ma.s.s of Water A S: Therefore the Gravity of the Solid I S, is equall to the Gravity of a Ma.s.s of water equall to the Ma.s.s 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 Alt.i.tude of the Rampart A I, to maintain the due proportion: But by what hath been supposed, the Alt.i.tude 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 s.p.a.ce 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 princ.i.p.all 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 Alt.i.tude 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 Alt.i.tude 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 Alt.i.tude 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 circ.u.mfused 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 Ma.s.s of that ought to be, that so the a.s.sistance 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 Alt.i.tude hath unto the greatest possible Alt.i.tude 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 Alt.i.tude of the Ramparts of Water, to the Alt.i.tude of the Solid, being const.i.tuted 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 Ma.s.s of water equall to the Ma.s.s 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 pa.s.sions 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 Ma.s.ses are in contrary proportion to their Specifick Gravities are equall in absolute Gravity.]

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

[Ill.u.s.tration]

Let A C and B be two Solids, and let the Ma.s.s A C be to the Ma.s.s 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 Ma.s.s A C be equall to the Ma.s.s B, then, by the a.s.sumption, the Specificall Gravity of B, shall be equall to the Specificall Gravity of A C, and being equall in Ma.s.s, and of the same Specificall Gravity they shall absolutely weigh one as much as another. But if their Ma.s.ses shall be unequall, let the Ma.s.s A C be greater, and in it take the part C, equall to the Ma.s.s B.

And, because the Ma.s.ses 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 a.s.sumption, hath the Ma.s.s C A, to the Ma.s.s B, that is, to the Ma.s.s C: Therefore, the absolute weight of B, to the absolute weight of C, is as the Ma.s.s A C to the Ma.s.s C: But as the Ma.s.s A C, is to the Ma.s.s 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 a.s.signed 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._

[Ill.u.s.tration]

Let the greatest possible Alt.i.tude of the Rampart be the Line D B, and the Diameter of the Base of the Cone to be made of any Matter a.s.signed 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 Alt.i.tude 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 Alt.i.tude D B, hath to the Alt.i.tude B E: But the Cylinder C E, is to the Cone A B C, as the Alt.i.tude E B is to the third part of the Alt.i.tude 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 Alt.i.tude 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 Ma.s.s of the Solid D C, is to the Ma.s.s 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 Ma.s.s of Water equall to the Ma.s.s 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 Alt.i.tude shall sink the least.]

_That making upon the same Basis, a Cone of a less Alt.i.tude, it shall be also less grave, and shall so much the more rest without Submersion._