Discourse on Floating Bodies Part 7

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 rea.s.sume 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 Ma.s.s and Gravity they shall be equall to those Cylinders, and by having their Bases equall to those of the Cylinders, they shall leave equall Ma.s.ses 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 mult.i.tude 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 s.p.a.cious 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 Ma.s.s to the Cone _A B C_; so that the whole Ma.s.s 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 Ma.s.se _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.

[Ill.u.s.tration]

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 Ma.s.s of water equall to the Cylinder _L B D P_: Therefore it shall not rest in this state, but shall descend.

[Ill.u.s.tration]

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 Alt.i.tude of the Cone F N S, shall either be more than half the whole Alt.i.tude 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 Alt.i.tude of the Cone F N S, shall be more than Sesquialter of the Alt.i.tude 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 Alt.i.tude of the Point F N S, shall be half the Alt.i.tude of the whole Cone F T O, the same Alt.i.tude of the said Cone F N S shall be Sesquialter to the Alt.i.tude E N: and, therefore, E N S C shall be double to the Cone F N S; and as much water in Ma.s.s 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 Ma.s.s 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 Ma.s.s 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 Ma.s.s of water equall to the Ma.s.s, 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 Ma.s.s of Air contained in the Rampart diminis.h.i.+ng, 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 pa.s.sive 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._

[Ill.u.s.tration]

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 Ma.s.s 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 Ma.s.ses) 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 Ma.s.s 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 Alt.i.tude 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 Alt.i.tude still remaining the same; or, if you will, because the Cone _S {A} T_ goes diminis.h.i.+ng, 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 Ma.s.s 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 Ma.s.s above the Ma.s.s 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.

[Ill.u.s.tration]

[Ill.u.s.tration]

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 Ma.s.s 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 Alt.i.tude of the Cone _A I R_, were two thirds of the Alt.i.tude 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 a.s.sisted by more s.p.a.cious Ramparts.

[77] Natatio{n} easiest effected in Figures broad toward the top.

THEOREME XIII.

[Sidenote: All Figures sink or swim, upon bathing or not bathing of their tops.]

_All Figures, therefore, of whatever greatnesse, may go, and not go, to the Bottom, according as their Sumities or Tops shall be bathed or not bathed._

And this Accident being common to all sorts of Figures, without exception of so much as one. Figure hath, therefore, no part in the production of this Effect, of sometimes sinking, and sometimes again not sinking, but only the being sometimes conjoyned to, and sometimes seperated from, the supereminent Air: which cause, in fine, who so shall rightly, and, as we say, with both his Eyes, consider this business, will find that it is reduced to, yea, that it really is the same with, the true, Naturall and primary cause of Natation or Submersion; to wit, the excess or deficiency of the Gravity of the water, in relation to the Gravity of that Solid Magnitude, that is demitted into the water. For like as a Plate of Lead, as thick as the back of a Knife, which being put into the water by it self alone goes to the bottom, if upon it you fasten a piece of Cork four fingers thick, doth continue afloat, for that now the Solid that is demitted in the water, is not, as before, more grave than the water, but less, so the Board of Ebony, of its own nature more grave than water; and, therefore, descending to the bottom, when it is demitted by it self alone into the water, if it shall be put upon the water, conjoyned with an Expanded vail of Air, that together with the Ebony doth descend, and that it be such, as that it doth make with it a compound less grave than so much water in Ma.s.s, as equalleth the Ma.s.s already submerged and depressed beneath the Levell of the waters Surface, it shall not descend any farther, but shall rest, for no other than the universall and most common cause, which is that Solid Magnitudes, less grave _in specie_ than the water, go not to the bottom.

So that if one should take a Plate of Lead, as for Example, a finger thick, and an handfull broad every way, and should attempt to make it swimme, with putting it lightly on the water, he would lose his Labour, because that if it should be depressed an Hairs breadth beyond the possible Alt.i.tude of the Ramparts of water, it would dive and sink; but if whilst it is going downwards, one should make certain Banks or Ramparts about it, that should hinder the defusion of the water upon the said Plate, the which Banks should rise so high, as that they might be able to contain as much water, as should weigh equally with the said Plate, it would, witho{u}t all Question, descend no lower, but would rest, as being sustained by vertue of the Air contained within the aforesaid Ramparts: and, in short, there would be a Vessell by this means formed with the bottom of Lead. But if the thinness of the Lead shall be such, that a very small height of Rampart would suffice to contain so much Air, as might keep it afloat, it shall also rest without the Artificiall Banks or Ramparts, but yet not without the Air, because the Air by it self makes Banks sufficient for a small height, to resist the Superfusion of the water: so that that which in this case swimmes, is as it were a Vessell filled with Air, by vertue of which it continueth afloat.

I will, in the last place, with an other Experime{n}t, attempt to remove all difficulties, if so be there should yet be any doubt left in any one, touching the opperation of this [E]Continuity of the Air, with the thin Plate which swims, and afterwards put an end to this part of my discourse.

[E] Or rather Contiguity,

I suppose my self to be questioning with some of my Oponents.