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

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THEOREME IV.
Let the Veſſell be M L G N of any bigneſs, and let there be pla­
ced in it the Solid Priſme D F G E, leſs grave in ſpecie than the
water; and look what proportion the Specificall Gravity of
the water, hath to that of the Priſme, ſuch let the Altitude D F, have
to the Altitude F B.
I ſay, that infuſing water to the Altitude F B,
the Solid D G ſhall not float, but ſhall ſtand in Equilibrium, ſo, that
that every little quantity of water, that is infuſed, ſhall raiſe it.
Let
the water, therefore, be infuſed to the Levell A B C, and, becauſe
the Specifick Gravity of the Solid D G, is to the Specifick Gravity of
the water, as the altitude B F is to the altitude F D; that is, as the Maſs
B G to the Maſs G D; as the proportion of the Maſs B G is to the
Maſs G D, as the proportion of the Maſs G D is to the Maſs A F, they
compoſe the Proportion of the Maſs B G to the Maſs A F. Therefore,
the Maſs B G is to the Maſs A F, in a proportion compounded of the
proportions of the Specifick Gravity of the Solid G D, to the Speci­
fick Gravity of the water, and of the Maſs G D
to the Maſs A F: But the ſame proportions

Figure: /permanent/archimedes/galil_natat_074_en_1663/figures not scanned
[Figure 5]

of the Specifick Gravity of G D, to the Specifick
Gravity of the water, and of the Maſs G D to
the Maſs A F, do alſo by the precedent Lemma,
compound the proportion of the abſolute Gra­
vity of the Solid D G, to the abſolute Gravity
of the Maſs of the water A F: Therefore,
as the Maſs B G is to the Maſs A F, ſo is the
Abſolute Gravity of the Solid D G, to the Ab­
ſolute Gravity of the Maſs of the water A F.
But as the Maſs B G
is to the Maſs A F; ſo is the Baſe of the Priſme D E, to the Surface
of the water AB; and ſo is the deſcent of the water A B, to the
Elevation of the Priſme D G; Therefore, the deſcent of the

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