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

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Figure: /permanent/archimedes/galil_natat_074_en_1663/figures not scanned
[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.

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