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

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THEOREME X.
There may be
Cones and Pira­
mides of any
Matter, which
demittedinto the
water, reſt only
their Baſes.
Let the greateſt poſſible Altitude of the Rampart be the Line D B,
and the Diameter of the Baſe of the Cone to be made of any Mat­
ter aſſigned B C, at right angles to D B: And as the Specificall Gravity
of the Matter of the Piramide or Cone to be made, is to the Specificall
Gravity of the water, ſo let the Altitude of the

Figure: /permanent/archimedes/galil_natat_074_en_1663/figures not scanned
[Figure 13]

Rampart D B, be to the third part of the Piramide
or Cone A B C, deſcribed upon the Baſe, whoſe
Diameter is B C: I ſay, that the ſaid Cone A B C,
and any other Cone, lower then the ſame, ſhall reſt
upon the Surface of the water B C without ſinking.
Draw D F parallel to B C, and ſuppoſe the Priſme
or Cylinder E C, which ſhall be tripple to the Cone
A B C. And, becauſe the Cylinder D C hath the ſame proportion
to the Cylinder C E, that the Altitude D B, hath to the Altitude B E:
But the Cylinder C E, is to the Cone A B C, as the Altitude E B is to
the third part of the Altitude of the Cone: Therefore, by Equality of
proportion, the Cylinder D C is to the Cone A B C, as D B is to the
third part of the Altitude B E: But as D B is to the third part of B E,
ſo is the Specificall Gravity of the Cone A B C, to the Specificall Gra­
vity of the water: Therefore, as the Maſs of the Solid D C, is to the
Maſs of the Cone A B C, ſo is the Specificall Gravity of the ſaid Cone,
to the Specificall Gravity of the water: Therefore, by the precedent
Lemma, the Cone A B C weighs in abſolute Gravity as much as a
Maſs of Water equall to the Maſs D C: But the water which by the
impoſition of the Cone A B C, is driven out of its place, is as much
as would preciſely lie in the place D C, and is equall in weight to the
Cone that diſplaceth it: Therefore, there ſhall be an Equilibrium,
and the Cone ſhall reſt without farther ſubmerging.
And its ma­
nifeſt,

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