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

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Cone weighing equally with the water, the part ſubmerged S B D T,
becomes indifferent to move downwards or upwards; and the Cone
A S T, being equall in Maſs to the water that would be contained in
the concave of the Rampart E S T O, ſhall be alſo equall unto it in
Gravity: and, therefore, there ſhall be a perfect Equilibrium, and,
conſequently, a Reſt.
Now here ariſeth a doubt, whether the
Cone A B D may be made heavier, in ſuch ſort, that when it is put
wholly under water, it goes to the bottom, but yet not in ſuch ſort,
as to take from the Rampart the vertue of ſuſtaining it that it ſink not,
and, the reaſon of the doubt is this: that although at ſuch time as
the Cone A B D is ſpecifically as grave as the water, the Rampart
E S T O ſuſtaines it, not only when the point A S T is tripple in
height to the Altitude of the Rampart E S, but alſo when a leſſer
part is above water; [for although in the Deſcent of the Cone the
Point A S T by little and little diminiſheth, and ſo likewiſe the
Rampart E S T O, yet the Point diminiſheth in

Figure: /permanent/archimedes/galil_natat_074_en_1663/figures not scanned
[Figure 17]

greater proportion than the Rampart, in that
it diminiſheth according to all the three Di­
menſions, but the Rampart according to two
only, the Altitude ſtill remaining the ſame;
or, if you will, becauſe the Cone S T goes di­
miniſhing, according to the proportion of the
cubes of the Lines that do ſucceſſively become
the Diameters of the Baſes of emergent Cones,
and the Ramparts diminiſh according to the proportion of the
Squares of the ſame Lines; whereupon the proportions of the Points
are alwayes Seſquialter of the proportions of the Cylinders, con­
tained within the Rampart; ſo that if, for Example, the height of
the emergent Point were double, or equall to the height of the
Rampart, in theſe caſes, the Cylinder contained within the Ram­
part, would be much greater than the ſaid Point, becauſe it would be
either ſeſquialter or tripple, by reaſon of which it would perhaps
ſerve over and above to fuſtain the whole Cone, ſince the part ſub­
merged would no longer weigh any thing;] yet, nevertheleſs, when
any Gravity is added to the whole Maſs of the Cone, ſo that alſo the
part ſubmerged is not without ſome exceſſe of Gravity above the
Gravity of the water, it is not manifeſt, whether the Cylinder con­
tained within the Rampart, in the deſcent that the Cone ſhall make,
can be reduced to ſuch a proportion unto the emergent Point, and to
ſuch an exceſſe of Maſs above the Maſs of it, as to compenſate the
exceſſe of the Cones Specificall Gravity above the Gravity of the wa­
ter: and the Scruple ariſeth, becauſe that howbeit in the deſcent
made by the Cone, the emergent Point A S T diminiſheth, whereby
there is alſo a diminution of the exceſs of the Cones Gravity above

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