_Weights absolutely unequall, do alternately counterpoyse and become of equall Moments, as oft as their Gravities, with contrary proportion, answer to the Velocity of their Motions._
That is to say, that by how much the one is less grave than the other, by so much is it in a constitution of moving more swiftly than that.
Having prefatically explicated these things, we may begin to enquire, what Bodyes those are which totally submerge in Water, and go to the Bottom, and which those that by constraint float on the top, so that being thrust by violence under Water, they return to swim, with one part of their Mass visible above the Surface of the Water: and this we will do by considering the respective operation of the said Solids, and of Water: Which operation followes the Submersion and sinking; and this it is[6], That in the Submersion that the Solid maketh, being depressed downwards by its proper Gravity, it comes to drive away the water from the place where it successively subenters, and the water repulsed riseth and ascends above its first levell, to which Ascent on the other side it, as being a grave Body of its own nature, resists: And because the descending Solid more and more immerging, greater and greater quantity of Water ascends, till the whole Sollid be submerged; its necessary to compare the Moments of the Resistance of the water to Ascension, with the Moments of the pressive Gravity of the Solid: And if the Moments of the Resistance of the water, shall equalize the Moments of the Solid, before its totall Immersion[7]; in this case doubtless there shall be made an _Equilibrium_, nor shall the Body sink any farther. But if the Moment of the Solid, shall alwayes exceed the Moments wherewith the repulsed water successively makes Resistance[8], that Solid shall not only wholly submerge under water, but shall descend to the Bottom. But if, lastly, in the instant of totall Submersion, the equality shall be made between the Moments of the prement Solid, and the resisting Water[9]; then shall rest, ensue, and the said Solid shall be able to rest indifferently, in whatsoever part of the water. By this time is manifest the necessity of comparing the Gravity of the water, and of the Solid[10]; and this comparison might at first sight seem sufficient to conclude and determine which are the Solids that float a-top, and which those that sink to the Bottom in the water, asserting that those shall float which are lesse grave _in specie_ than the water, and those submerge, which are _in specie_ more grave. For it seems in appearance, that the Sollid in sinking continually, raiseth so much Water in Mass, as answers to the parts of its own Bulk submerged: whereupon it is impossible, that a Solid less grave _in specie_, than water, should wholly sink, as being unable to raise a weight greater than its own, and such would a Mass of water equall to its own Mass be. And likewise it seems necessary, that the graver Solids do go to the Bottom, as being of a Force more than sufficient for the raising a Masse of water, equall to its own, though inferiour in weight. Nevertheless the business succeeds otherwise: and though the Conclusions are true, yet are the Causes thus assigned deficient, nor is it true, that the Solid in submerging, raiseth and repulseth Masses of Water, equall to the parts of it self submerged; but the Water repulsed, is alwayes less than the parts of the Solid submerged[11]: and so much the more by how much the Vessell in which the Water is contained is narrower: in such manner that it hinders not, but that a Solid may submerge all under Water, without raising so much Water in Mass, as would equall the tenth or twentieth part of its own Bulk: like as on the contrary, a very small quantity of Water, may raise a very great Solid Mass[12], though such Solid should weigh absolutely a hundred times as much, or more, than the said Water, if so be that the Matter of that same Solid be _in specie_ less grave than the Water. And thus a great Beam, as suppose of a 1000 weight, may be raised and born afloat by Water, which weighs not 50: and this happens when the Moment of the Water is compensated by the Velocity of its Motion.
[6] How the submersion of Solids in the Water, is effected.
[7] What Solids shall float on the Water.
[8] What Solids shall sinke to the botome.
[9] What Solids shall rest in all places of the Water.
[10] The Gravitie of the Water and Solid must be compared in all Problems, of Natation of Bodies.
[11] The water repelled is ever less than the parts of the Sollid submerged.
[12] _A_ small quantity of water, may float a very great Solid Mass.
But because such things, propounded thus in abstract, are somewhat difficult to be comprehended, it would be good to demonstrate them by particular examples; and for facility of demonstration, we will suppose the Vessels in which we are to put the Water, and place the Solids, to be inviron'd and included with sides erected perpendicular to the Plane of the Horizon, and the Solid that is to be put into such vessell to be either a streight Cylinder, or else an upright Prisme.
_The which proposed and declared, I proceed to demonstrate the truth of what hath been hinted, forming the ensuing Theoreme._
_THEOREME I._
[Sidenote: The Proportion of the water raised to the Solid submerged.]
The Mass of the Water which ascends in the submerging of a Solid, Prisme or Cylinder, or that abaseth in taking it out, is less than the Mass of the said Solid, so depressed or advanced: and hath to it the same proportion, that the Surface of the Water circumfusing the Solid, hath to the same circumfused Surface, together with the Base of the Solid.
_Let the Vessell be A B C D, and in it the Water raised up to the Levell E F G, before the Solid Prisme H I K be therein immerged; but after that it is depressed under Water, let the Water be raised as high as the Levell L M, the Solid H I K shall then be all under Water, and the Mass of the elevated Water shall be L G, which is less than the Masse of the Solid depressed, namely of H I K, being equall to the only part E I K, which is contained under the first Levell E F G. Which is manifest, because if the Solid H I K be taken out, the Water I G shall return into the place occupied by the Mass E I K, where it was continuate before the submersion of the Prisme. And the Mass L G being equall to the Mass E K: adde thereto the Mass E N, and it shall be the whole Mass E M, composed of the parts of the Prisme E N, and of the Water N F, equall to the whole Solid H I K: And, therefore, the Mass L G shall have the same proportion to E M, as to the Mass H I K: But the Mass L G hath the same proportion to the Mass E M, as the Surface L M hath to the Surface M H: Therefore it is manifest, that the Mass of Water repulsed L G, is in proportion to the Mass of the Solid submerged H I K; as the Surface L M, namely, that of the Water ambient about the Sollid, to the whole Surface H M, compounded of the said ambient water, and the Base of the Prisme H N. But if we suppose the first Levell of the Water the according to the Surface H M, and the Prisme allready submerged H I K; and after to be taken out and raised to E A O, and the Water to be faln from the first Levell H L M as low as E F G; It is manifest, that the Prisme E A O being the same with H I K, its superiour part H O, shall be equall to the inferiour E I K: and remove the common part E N, and, consequently, the Mass of the Water L G is equall to the Mass H O; and, therefore, less than the Solid, which is without the Water, namely, the whole Prisme E A O, to which likewise, the said Mass of Water abated L G, hath the same proportion, that the Surface of the Waters circumfused L M hath to the same circumfused Surface, together with the Base of the Prisme A O: which hath the same demonstration with the former case above._
[Illustration]
_And from hence is inferred, that the Mass of the Water, that riseth in the immersion of the Solid, or that ebbeth in elevating it, is not equall to all the Mass of the Solid, which is submerged or elevated, but to that part only, which in the immersion is under the first Levell of the Water, and in the elevation remaines above the first Levell: Which is that which was to be demonstrated. We will now pursue the things that remain._
And first we will demonstrate that,
