Galilei, Galileo - Discourse concerning the natation of bodies

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

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[Figure 1]

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 be
fore 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, there
fore, 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 ma
nifeſt, 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 propor
tion, 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.

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,

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