Archimedes - Natation of bodies

Archimedes, Natation of bodies, 1662

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ſhall be equal to the Triangle O X K; Therefore (by common
Notion) ſubſtracting thoſe two ſmall Triangles O P K and O X K
from the two others B E K and B R K, the two Remainders ſhall
be equal: one of which Remainders ſhall be the Quadrangle
B E O P, and the other B R X O. And becauſe the whole Quadran-
gle B E O P is full of Liquor, and of the Quadrangle B R X O,
the part B A X O only is full, and the reſidue B R A is wholly void
of Water: It followeth, therefore, that the Quadrangle B E O P
is more ponderous than the Quadrangle B R X O. And if the ſaid
Quadrangle B E O P be more Grave than the Quadrangle
B R X O, much more ſhall the Quadrangle B L O P exceed in Gra-
vity the ſaid Quadrangle B R X O: whence it followeth, that the
part O P is more preſſed than the part O X. But, by the firſt part
of the Suppoſition, the part leſs preſſed ſhould be repulſed by the
part more preſſed: Therefore the part O X muſt be repulſed by
the part O P: But it was preſuppoſed that the Liquid did not
move: Wherefore it would follow that the leſs preſſed would not
be repulſed by the more preſſed: And therefore it followeth of
neceſſity that the Line A B G D is the Circumference of a Circle,
and that the Center of it is the point K. And in like manner ſhall
it be demonſtrated, if the Surface of the Liquid be cut by a Plane
thorow the Center of the Earth, that the Section ſhall be the Cir-
cumference of a Circle, and that the Center of the ſame ſhall be
that very Point which is Center of the Earth. It is therefore mani-
feſt that the Superficies of a Liquid that is conſiſtant and ſetled
ſhall have the Figure of a Sphære, the Center of which ſhall be
the ſame with that of the Earth, by the firſt Propoſition; for it is
ſuch that being ever cut thorow the ſame Point, the Section or Di-
viſion deſcribes the Circumference of a Circle which hath for Cen-
ter the ſelf-ſame Point that is Center of the Earth: Which was to
be demonſtrated.

* i.e. Parallel.

RIC. I do thorowly underſtand theſe your Reaſons, and ſince there is in them
no umbrage of Doubting, let us proceed to his third Propoſition.

PROP. III. THEOR. III.

Solid Magnitudes that being of equal Maſs with the
Liquid are alſo equal to it in Gravity, being demit-

ted into the [^{*} ſetled] Liquid do ſo ſubmerge in the
ſame as that they lie or appear not at all above the
Surface of the Liquid, nor yet do they ſink to the
Bottom.

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