[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 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._
[Illustration]
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.