Galilei, Galileo - Discourse concerning the natation of bodies

Galilei, Galileo, Discourse concerning the natation of bodies, 1663

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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.

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.

How far Solids
leſs grave in ſpe
cie than water,
do ſubmerge.

For, it being ſuppoſed, that the Specificall Gravity of the water,
is to the Specificall Gravity of the Priſme D G, as the Altitude
D F, is to the Altitude F B; that is, as the Solid D G is to the
Solid B G; we might eaſily demonſtrate, that as much water in Maſs
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 Maſs of water, equall to the Maſs B G, hath to the Ab
ſolute Gravity of the Priſme D G, a proportion compounded of the
proportions, of the Maſs B G to the Maſs G D, and of the Specifick
Gravit 7 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 Maſs G D to the Maſs G B. There
fore, the Abſolute Gravity of a Maſs of water, equall to the Maſs
B G, is to the Abſolute Gravity of the Solid D G, in a proportion
compounded of the proportions, of the Maſs B G to the Maſs G D,
and of the Maſs D G to the Maſs G B; which is a proportion of
equalitie. The Abſolute Gravity, therefore, of a Maſs of Water
equall to the part of the Maſs of the Priſme B G, is equall to the Ab
ſolute Gravity of the whole Solid D G.

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