Discourse on Floating Bodies Part 2

Imagine in the Vessell, as is aforesaid, the Prisme A C D B to be placed, and in the rest of the s.p.a.ce the Water to be diffused as far as the Levell E A: and raising the Solid, let it be transferred to G M, and let the Water be abased from E A to N O: I say, that the descent of the Water, measured by the Line A O, hath the same proportion to the rise of the Prisme, measured by the Line G A, as the Base of the Solid G H hath to the Surface of the Water N O. The which is manifest: because the Ma.s.s of the Solid G A B H, raised above the first Levell E A B, is equall to the Ma.s.s of Water that is abased E N O A. Therefore, E N O A and G A B H are two equall Prismes; for of equall Prismes, the Bases answer contrarily to their heights: Therefore, as the Alt.i.tude A O is to the Alt.i.tude A G, so is the Superficies or Base G H to the Surface of the Water N O. If therefore, for example, a Pillar were erected in a waste Pond full of Water, or else in a Well, capable of little more then the Ma.s.s of the said Pillar, in elevating the said Pillar, and taking it out of the Water, according as it riseth, the Water that invirons it will gradually abate, and the abas.e.m.e.nt of the Water at the instant of lifting out the Pillar, shall have the same proportion, that the thickness of the Pillar hath to the excess of the breadth of the said Pond or Well, above the thickness of the said Pillar: so that if the breadth of the Well were an eighth part larger than the thickness of the Pillar, and the breadth of the Pond twenty five times as great as the said thickness, in the Pillars ascending one foot, the water in the Well shall descend seven foot, and that in the Pond only 1/25 of a foot.

[Sidenote: Why a Solid less grave _in specie_ than water, stayeth not under water, in very small depths:]

This Demonstrated, it will not be difficult to show the true cause, how it comes to pa.s.s, that,

THEOREME III.

_A Prisme or regular Cylinder, of a substance specifically less grave than Water, if it should be totally submerged in Water, stayes not underneath, but riseth, though the Water circ.u.mfused be very little, and in absolute Gravity, never so much inferiour to the Gravity of the said Prisme._

Let then the Prisme A E F B, be put into the Vessell C D F B, the same being less grave _in specie_ than the Water: and let the Water infused rise to the height of the Prisme: I say, that the Prisme left at liberty, it shall rise, being born up by the Water circ.u.mfused C D E A. For the Water C E being specifically more grave than the Solid A F, the absolute weight of the water C E, shall have greater proportion to the absolute weight of the Prisme A F, than the Ma.s.s C E hath to the Ma.s.s A F (in regard the Ma.s.s hath the same proportion to the Ma.s.s, that the weight absolute hath to the weight absolute, in case the Ma.s.ses are of the same Gravity _in specie_.) But the Ma.s.s C E is to the Ma.s.s A F, as the Surface of the water A C, is to the Superficies, or Base of the Prisme A B; which is the same proportion as the ascent of the Prisme when it riseth, hath to the descent of the Water circ.u.mfused C E.

[Ill.u.s.tration]

Therefore, the absolute Gravity of the water C E, hath greater proportion to the absolute Gravity of the Prisme A F; than the Ascent of the Prisme A F, hath to the descent of the said water C E. The Moment, therefore, compounded of the absolute Gravity of the water C E, and of the Velocity of its descent, whilst it forceably repulseth and raiseth the Solid A F, is greater than the Moment compounded of the absolute Gravity of the Prisme A F, and of the Tardity of its ascent, with which Moment it contrasts and resists the repulse and violence done it by the Moment of the water: Therefore, the Prisme shall be raised.

[Sidenote: The Proportion according to which the Submersion & Natation of Solids is made.]

It followes, now, that we proceed forward to demonstrate more particularly, how much such Solids shall be inferiour in Gravity to the water elevated; namely, what part of them shall rest submerged, and what shall be visible above the Surface of the water: but first it is necessary to demonstrate the subsequent Lemma.

LEMMA I.

[Sidenote: The absolute Gravity of Solids, are in a proportion compounded of their Specifick Gravities, and of their Ma.s.ses.]

_The absolute Gravities of Solids, have a proportion compounded of the proportions of their specificall Gravities, and of their Ma.s.ses._

[Ill.u.s.tration]

Let A and B be two Solids. I say, that the Absolute Gravity of A, hath to the Absolute Gravity of B, a proportion compounded of the proportions of the specificall Gravity of A, to the Specificall Gravity of B, and of the Ma.s.s A to the Ma.s.s B. Let the Line D have the same proportion to E, that the specifick Gravity of A, hath to the specifick Gravity of B; and let E be to F, as the Ma.s.s A to the Ma.s.s B: It is manifest, that the proportion of D to F, is compounded of the proportions D and E; and E and F. It is requisite, therefore, to demonstrate, that as D is to F, so the absolute Gravity of A, is to the absolute Gravity of B. Take the Solid C, equall in Ma.s.s to the Solid A, and of the same Gravity _in specie_ with the Solid B.

Because, therefore, A and C are equall in Ma.s.s, the absolute Gravity of A, shall have to the absolute Gravity of C, the same proportion, as the specificall Gravity of A, hath to the specificall Gravity of C, or of B, which is the same _in specie_; that is, as D is to E. And, because, C and B are of the same Gravity _in specie_, it shall be, that as the absolute weight of C, is to the absolute weight of B, so the Ma.s.s C, or the Ma.s.s A, is to the Ma.s.s B; that is, as the Line E to the Line F. As therefore, the absolute Gravity of A, is to the absolute Gravity of C, so is the Line D to the Line E: and, as the absolute Gravity of C, is to the absolute Gravity of B, so is the Line E to the Line F: Therefore, by Equality of proportion, the absolute Gravity of A, is to the absolute Gravity of B, as the Line D to the Line F: which was to be demonstrated. I proceed now to demonstrate, how that,

THEOREME IV.

[Sidenote: The proportion of water requisite to make a Solid swim:]

_If a Solid, Cylinder, or Prisme, lesse grave specifically than the Water, being put into a Vessel, as above, of whatsoever greatnesse, and the Water, be afterwards infused, the Solid shall rest in the bottom, unraised, till the Water arrive to that part of the Alt.i.tude, of the said Prisme, to which its whole Alt.i.tude hath the same proportion, that the Specificall Gravity of the Water, hath to the Specificall Gravity of the said Solid: but infusing more Water, the Solid shall ascend._

[Ill.u.s.tration]

Let the Vessell be M L G N of any bigness, and let there be placed in it the Solid Prisme D F G E, less grave _in specie_ than the water; and look what proportion the Specificall Gravity of the water, hath to that of the Prisme, such let the Alt.i.tude D F, have to the Alt.i.tude F B. I say, that infusing water to the Alt.i.tude F B, the Solid D G shall not float, but shall stand in _Equilibrium_, so, that that every little quant.i.ty of water, that is infused, shall raise it. Let the water, therefore, be infused to the Levell A B C; and; because the Specifick Gravity of the Solid D G, is to the Specifick Gravity of the water, as the alt.i.tude B F is to the alt.i.tude F D; that is, as the Ma.s.s B G to the Ma.s.s G D; as the proportion of the Ma.s.s B G is to the Ma.s.s G D, as the proportion of the Ma.s.s G D is to the Ma.s.s A F, they compose the Proportion of the Ma.s.s B G to the Ma.s.s A F. Therefore, the Ma.s.s B G is to the Ma.s.s A F, in a proportion compounded of the proportions of the Specifick Gravity of the Solid G D, to the Specifick Gravity of the water, and of the Ma.s.s G D to the Ma.s.s A F: But the same proportions of the Specifick Gravity of G D, to the Specifick Gravity of the water, and of the Ma.s.s G D to the Ma.s.s A F, do also by the precedent _Lemma_, compound the proportion of the absolute Gravity of the Solid D G, to the absolute Gravity of the Ma.s.s of the water A F: Therefore, as the Ma.s.s B G is to the Ma.s.s A F, so is the Absolute Gravity of the Solid D G, to the Absolute Gravity of the Ma.s.s of the water A F. But as the Ma.s.s B G is to the Ma.s.s A F; so is the Base of the Prisme D E, to the Surface of the water A B; and so is the descent of the water A B, to the Elevation of the Prisme D G; Therefore, the descent of the water is to the elevation of the Prisme, as the absolute Gravity of the Prisme, is to the absolute Gravity of the water: Therefore, the Moment resulting from the absolute Gravity of the water A F, and the Velocity of the Motion of declination, with which Moment it forceth the Prisme D G, to rise and ascend, is equall to the Moment that results from the absolute Gravity of the Prisme D G, and from the Velocity of the Motion, wherewith being raised, it would ascend: with which Moment it resists its being raised: because, therefore, such Moments are equall, there shall be an _Equilibrium_ between the water and the Solid. And, it is manifest, that putting a little more water unto the other A F, it will increase the Gravity and Moment, whereupon the Prisme D G, shall be overcome, and elevated till that the only part B F remaines submerged. Which is that that was to be demonstrated.

COROLLARY I.

[Sidenote: _H_ow far Solids less grave _in specie_ than water, do submerge.]

_By what hath been demonstrated, it is manifest, that Solids less grave_ in specie _than the water, submerge only so far, that as much water in Ma.s.s, as is the part of the Solid submerged, doth weigh absolutely as much as the whole Solid._

For, it being supposed, that the Specificall Gravity of the water, is to the Specificall Gravity of the Prisme D G, as the Alt.i.tude D F, is to the Alt.i.tude F B; that is, as the Solid D G is to the Solid B G; we might easily demonstrate, that as much water in Ma.s.s as is equall to the Solid B G, doth weigh absolutely as much as the whole Solid D G; For, by the _Lemma_ foregoing, the Absolute Gravity of a Ma.s.s of water, equall to the Ma.s.s B G, hath to the Absolute Gravity of the Prisme D G, a proportion compounded of the proportions, of the Ma.s.s B G to the Ma.s.s G D, and of the Specifick Gravit{y} of the water, to the Specifick Gravity of the Prisme: But the Gravity _in specie_ of the water, to the Gravity _in specie_ of the Prisme, is supposed to be as the Ma.s.s G D to the Ma.s.s G B. Therefore, the Absolute Gravity of a Ma.s.s of water, equall to the Ma.s.s B G, is to the Absolute Gravity of the Solid D G, in a proportion compounded of the proportions, of the Ma.s.s B G to the Ma.s.s G D, and of the Ma.s.s D G to the Ma.s.s G B; which is a proportion of equalitie. The Absolute Gravity, therefore, of a Ma.s.s of Water equall to the part of the Ma.s.s of the Prisme B G, is equall to the Absolute Gravity of the whole Solid D G.

COROLLARY II.

[Sidenote: _A_ Rule to equilibrate Solids in the water.]

_It followes, moreover, that a Solid less grave than the water, being put into a Vessell of any imaginable greatness, and water being circ.u.mfused about it to such a height, that as much water in Ma.s.s, as is the part of the Solid submerged, do weigh absolutely as much as the whole Solid; it shall by that water be justly sustained, be the circ.u.mfused Water in quant.i.ty greater or lesser._

[Ill.u.s.tration]

For, if the Cylinder or Prisme M, less grave than the water, _v. gra._ in Subsequiteriall proportion, shall be put into the capacious Vessell A B C D, and the water raised about it, to three quarters of its height, namely, to its Levell A D: it shall be sustained and exactly poysed in _Equilibrium_. The same will happen; if the Vessell E N S F were very small, so, that between the Vessell and the Solid M, there were but a very narrow s.p.a.ce, and only capable of so much water, as the hundredth part of the Ma.s.s M, by which it should be likewise raised and erected, as before it had been elevated to three fourths of the height of the Solid: which to many at the first sight, may seem a notable Paradox, and beget a conceit, that the Demonstration of these effects, were sophisticall and fallacious: but, for those who so repute it, the Experiment is a means that may fully satisfie them. But he that shall but comprehend of what Importance Velocity of Motion is, and how it exactly compensates the defect and want of Gravity, will cease to wonder, in considering that at the elevation of the Solid M, the great Ma.s.s of water A B C D abateth very little, but the little Ma.s.s of water E N S F decreaseth very much, and in an instant, as the Solid M before did rise, howbeit for a very short s.p.a.ce: Whereupon the Moment, compounded of the small Absolute Gravity of the water E N S F, and of its great Velocity in ebbing, equalizeth the Force and and Moment, that results from the composition of the immense Gravity of the water A B C D, with its great slownesse of ebbing; since that in the Elevation of the Sollid M, the abas.e.m.e.nt of the lesser water E S, is performed just so much more swiftly than the great Ma.s.s of water A C, as this is more in Ma.s.s than that which we thus demonstrate.

[Sidenote: _T_he proportion according to which water riseth and falls in different Vessels at the Immersion and Elevation of Solids.]

In the rising of the Solid M, its elevation hath the same proportion to the circ.u.mfused water E N S F, that the Surface of the said water, hath to the Superficies or Base of the said Solid M; which Base hath the same proportion to the Surface of the water A D, that the abas.e.m.e.nt or ebbing of the water A C, hath to the rise or elevation of the said Solid M. Therefore, by Perturbation of proportion, in the ascent of the said Solid M, the abas.e.m.e.nt of the water A B C D, to the abas.e.m.e.nt of the water E N S F, hath the same proportion, that the Surface of the water E F, hath to the Surface of the water A D; that is, that the whole Ma.s.s of the water E N S F, hath to the whole Ma.s.s A B C D, being equally high: It is manifest, therefore, that in the expulsion and elevation of the Solid M, the water E N S F shall exceed in Velocity of _M_otion the water A B C D, asmuch as it on the other side is exceeded by that in quant.i.ty: whereupon their Moments in such operations, are mutually equall.

[Ill.u.s.tration]

_And, for ampler confirmation, and clearer explication of this, let us consider the present Figure, (which if I be not deceived, may serve to detect the errors of some Practick Mechanitians who upon a false foundation some times attempt impossible enterprizes,) in which, unto the large Vessell E I D F, the narrow Funnell or Pipe I C A B is continued, and suppose water infused into them, unto the Levell L G H, which water shall rest in this position, not without admiration in some, who cannot conceive how it can be, that the heavie charge of the great Ma.s.s of water G D, pressing downwards, should not elevate and repulse the little quant.i.ty of the other, contained in the Funnell or Pipe C L, by which the descent of it is resisted and hindered: But such wonder shall cease, if we begin to suppose the water G D to be abased only to Q D, and shall afterwards consider, what the water C L hath done, which to give place to the other, which is descended from the Levell G H, to the Levell Q O, shall of necessity have ascended in the same time, from the Levell L unto A B. And the ascent L B, shall be so much greater than the descent G Q, by how much the breadth of the Vessell G D, is greater than that of the Funnell I C; which, in summe, is as much as the water G D, is more than the water L C: but in regard that the Moment of the Velocity of the Motion, in one Moveable, compensates that of the Gravity of another what wonder is it, if the swift ascent of the lesser Water C L, shall resist the slow descent of the greater G D?_

The same, therefore, happens in this operation, as in rhe Stilliard, in which a weight of two pounds counterpoyseth an other of 200, asoften as that shall move in the same time, a s.p.a.ce 100 times greater than this: which falleth out when one Arme of the Beam is an hundred times as long as the other. Let the erroneous opinion of those therefore cease, who hold that a s.h.i.+p is better, and easier born up in a great abundance of water, then in a lesser quant.i.ty[13], (_this was believed by_ Aristotle _in his Problems, Sect. 23, Probl. 2._) it being on the contrary true, that its possible, that a s.h.i.+p may as well float in ten Tun of water, as in an Ocean.

[13] A s.h.i.+p flotes as well in ten Tun of Water as in an Ocean.

[Sidenote: A Solid specifiaclly graver than the water, cannot be born up by any quant.i.ty of it.]

But following our matter, I say, that by what hath been hitherto demonstrated, we may understand how, that

COROLLARY III.

_One of the above named Solids, when more grave_ in specie _than the water, can never be sustained, by any whatever quant.i.ty of it._

For having seen how that the Moment wherewith such a Solid, as grave _in specie_ as the water, contrasts with the Moment of any Ma.s.s of water whatsoever, is able to retain it, even to its totall Submersion, without its ever ascending; it remaineth, manifest, that the water is far less able to raise it up, when it exceeds the same _in specie_: so, that though you infuse water till its totall Submersion, it shall still stay at the Bottome, and with such Gravity, and Resistance to Elevation, as is the excess of its Absolute Gravity, above the Absolute Gravity of a Ma.s.s equall to it, made of water, or of a Matter _in specie_ equally grave with the water: and, though you should moreover adde never so much water above the Levell of that which equalizeth the Alt.i.tude of the Solid, it shall not, for all that, encrease the Pression, or Gravitation, of the parts circ.u.mfused about the said Solid, by which greater pression, it might come to be repulsed; because, the Resistance is not made, but only by those parts of the water, which at the Motion of the said Solid do also move, and these are those only, which are comprehended by the two Superficies equidistant to the Horizon, and their parallels, that comprehend the Alt.i.tude of the Solid immerged in the water.

I conceive, I have by this time sufficiently declared and opened the way to the contemplation of the true, intrinsecall and proper Causes of diverse Motions, and of the Rest of many Solid Bodies in diverse _Mediums_, and particularly in the water, shewing how all in effect, depend on the mutuall excesses of the Gravity of the Moveables and of the _Mediums_: and, that which did highly import, removing the Objection, which peradventure would have begotten much doubting, and scruple in some, about the verity of my Conclusion, namely, how that notwithstanding, that the excess of the Gravity of the water, above the Gravity of the Solid, demitted into it, be the cause of its floating and rising from the Bottom to the Surface, yet a quant.i.ty of water, that weighs not ten pounds, can raise a Solid that weighs above 100 pounds: in that we have demonstrated, That it sufficeth, that such difference be found between the Specificall Gravities of the _Mediums_ and Moveables, let the particular and absolute Gravities be what they will: insomuch, that a Solid, provided that it be Specifically less grave than the water, although its absolute weight were 1000 pounds, yet may it be born up and elevated by ten pounds of water, and less: and on the contrary, another Solid, so that it be Specifically more grave than the water, though in absolute Gravity it were not above a pound, yet all the water in the Sea, cannot raise it from the Bottom, or float it. This sufficeth me, for my present occasion, to have, by the above declared Examples, discovered and demonstrated, without extending such matters farther, and, as I might have done, into a long Treatise: yea, but that there was a necessity of resolving the above proposed doubt, I should have contented my self with that only, which is demonstrated by _Archimedes_, in his first _Book De Insidentibus humido_[14]: where in generall termes he infers and confirms the same Conclusions, namely, that Solids (_a_) less grave than water, swim or float upon it, the (_b_) more grave go to the Bottom, and the (_c_) equally grave rest indifferently in all places, yea, though they should be wholly under water.

[14] _Of Natation_ (a) _Lib. 1, Prop. 4._ (b) _Id. Lib. 1. Prop.