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

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11[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
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,
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
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
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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