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

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Let, therefore, the Cone A B D be of any ſuppoſed greatneſſe,
and alike in ſpecificall Gravity to the water.
It is manifeſt, that
being put lightly into the water, it ſhall reſt without deſcending;
and it ſhall advance above water, the Point

Figure: /permanent/archimedes/galil_natat_074_en_1663/figures not scanned
[Figure 18]

AS T, tripple in height to the height of the
Rampart E S: Now, ſuppoſe the Cone A B D
more depreſſed, ſo that it advance above wa­
ter, only the Point A I R, higher by half than
the Point A S T, with the Rampart about it
C I R N. And, becauſe, the Cone A B D is
to the Cone A I R, as the cube of the Line S T
is to the cube of the Line I R, but the Cylin­
der E S T O, is to the Cylinder C I R N, as the Square of S T to
the Square of I R, the Cone A S T ſhall be Octuple to the Cone
A I R, and the Cylinder E S T O, quadruple to the Cylinder C I R N:
But the Cone A S T, is equall to the Cylinder E S T O: Therefore,
the Cylinder C I R N, ſhall be double to the Cone A I R: and the
water which might be contained in the Rampart C I R N, would be
double in Maſs and in Weight to the Cone A I R, and, therefore,
would be able to ſuſtain the double of the Weight of the Cone AIR:
Therefore, if to the whole Cone A B D, there be added as much
Weight as the Gravity of the Cone A I R, that is to ſay, the eighth
part of the weight of the Cone A S T, it alſo ſhall be ſuſtained by
the Rampart C I R N, but without that it ſhall go to the bottome:
the Cone A B D, being, by the addition of the eighth part of the
weight of the Cone A S T, made ſpecifically more grave than the
water.
But if the Altitude of the Cone A I R, were two thirds
of the Altitude of the Cone A S T, the Cone A S T would be to the
Cone A I R, as twenty ſeven to eight; and the Cylinder E S T O, to
the Cylinder C I R N, as nine to four, that is, as twenty ſeven to
twelve; and, therefore, the Cylinder C I R N, to the Cone A I R,
as twelve to eight; and the exceſs of the Cylinder C I R N, above
the Cone A I R, to the Cone A S T, as four to twenty ſeven: there­
fore if to the Cone A B D be added ſo much weight as is the four
twenty ſevenths of the weight of the Cone A S T, which is a little
more then its ſeventh part, it alſo ſhall continue to ſwimme, and

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