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

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Maſs I S. I ſay, that the Solid I S ſhall not
ſinke, but being never ſo little thicker it ſhall
go to the bottom: For being that as A I is

Figure: /permanent/archimedes/galil_natat_074_en_1663/figures not scanned
[Figure 11]

to I O, ſo is the Exceſs of the Gravity of the
Solid I S, above the Gravity of a Maſs of water
equall to the Maſs I S, to the Gravity of the
ſaid Maſs of water: Therefore, compounding, as A O is to O I, ſo
ſhall the Gravity of the Solid I S, be to the Gravity of a Maſs of water
equall to the Maſs I S: And, converting, as I O is to O A, ſo ſhall the
Gravity of a Maſs of water equall to the Maſs I S, be to the Gravity
of the Solid I S: But as I O is to O A, ſo is a Maſs of water I S, to a
Maſs of water equall to the Maſs A B S O: and ſo is the Gravity of
a Maſs of water I S, to the Gravity of a Maſs of water A S: Therefore
as the Gravity of a Maſs of water, equall to the Maſs I S, is to the
Gravity of the Solid I S, ſo is the ſame Gravity of a Maſs of water
I S, to the Gravity of a Maſs of Water A S: Therefore the Gra­
vity of the Solid I S, is equall to the Gravity of a Maſs of water
quall to the Maſs A S: But the Gravity of the Solid I S, is the ſame
with the Gravity of the Solid A S, compounded of the Solid I S,
and of the Air A B C I.
Therefore the whole compounded Solid
A O S B, weighs as much as the water that would be compriſed in the
place of the ſaid Compound A O S B: And, therefore, it ſhall make
an Equilibrium and reſt, and that ſame Solid I O S C ſhall ſinke no
farther.
But if its thickneſs I O ſhould be increaſed, it would be ne­
ceſſary alſo to encreaſe the Altitude of the Rampart A I, to main­
tain the due proportion: But by what hath been ſuppoſed, the Alti­
tude of the Rampart A I, is the greateſt that the Nature of the
Water and Air do admit, without the waters repulſing the Air ad­
herent to the Superficies of the Solid I C, and poſſeſſing the ſpace
A I C B: Therefore, a Solid of greater thickneſs than I O, and of the
ſame Matter with the Solid I S, ſhall not reſt without ſubmerging,
but ſhall deſcend to the bottome: which was to be demonſtrated.
In conſequence of this that hath been demonſtrated, ſundry and va­
rious Concluſions may be gathered, by which the truth of my prin­
cipall Propoſition comes to be more and more confirmed, and the
imperfection of all former Argumentations touching the preſent
Queſtion cometh to be diſcovered.

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