THEOREME IV.
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 Altitude, of the said Prisme, to which its whole Altitude 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.
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 Altitude D F, have to the Altitude F B. I say, that infusing water to the Altitude F B, the Solid D G shall not float, but shall stand in Equilibrium, so, that that every little quantity 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 altitude B F is to the altitude F D; that is, as the Mass B G to the Mass G D; as the proportion of the Mass B G is to the Mass G D, as the proportion of the Mass G D is to the Mass A F, they compose the Proportion of the Mass B G to the Mass A F. Therefore, the Mass B G is to the Mass 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 Mass G D to the Mass A F: But the same proportions of the Specifick Gravity of G D, to the Specifick Gravity of the water, and of the Mass G D to the Mass 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 Mass of the water A F: Therefore, as the Mass B G is to the Mass A F, so is the Absolute Gravity of the Solid D G, to the Absolute Gravity of the Mass of the water A F. But as the Mass B G is to the Mass 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.
How 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 Mass, as is the part of the Solid submerged, doth weigh absolutely as much as the whole Solid.