Archimedes, Natation of bodies, 1662

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1
PROP. II. THEOR. II.
Let us ſuppoſe a Liquid that is of ſuch a conſiſtance as that it
is
not moved, and that its Superficies be cut by a Plane along
by
the Center of the Earth, and let the Center of the Earth
be
the Point K: and let the Section of the Superficies be the Line
A
B G D.
I ſay that the Line A B G D is the Circumference of a
3[Figure 3]
Circle
, and that the Center
thereof
is the Point K And
if
it be poſſible that it may
not
be the Circumference
of
a Circle, the Right­

Lines
drawn ^{*} by the Point
K
to the ſaid Line A B G D
ſhall
not be equal.
There­
fore
let a Right-Line be
taken
greater than ſome of thoſe produced from the Point K unto
the
ſaid Line A B G D, and leſſer than ſome other; and upon the
Point
K let a Circle be deſcribed at the length of that Line,
Now
the Circumference of this Circle ſhall fall part without the
ſaid
Line A B G D, and part within: it having been preſuppoſed
that
its Semidiameter is greater than ſome of thoſe Lines that may
be
drawn from the ſaid Point K unto the ſaid Line A B G D, and
leſſer
than ſome other.
Let the Circumference of the deſcribed
Circle
be R B G H, and from B to K draw the Right-Line B K: and
drawn
alſo the two Lines K R, and K E L which make a Right­
Angle
in the Point K: and upon the Center K deſcribe the Circum­
ference
X O P in the Plane and in the Liquid.
The parts, there­
fore
, of the Liquid that are ^{*} according to the Circumference

X
O P, for the reaſons alledged upon the firſt Suppoſition, are equi­
jacent
, or equipoſited, and contiguous to each other; and both
theſe
parts are preſt or thruſt, according to the ſecond part of the
Suppoſition, by the Liquor which is above them. And becauſe the
two
Angles E K B and B K R are ſuppoſed equal [by the 26. of 3.
of Euclid,] the two Circumferences or Arches B E and B R ſhall
be
equal (foraſmuch as R B G H was a Circle deſcribed for ſatis­
faction
of the Oponent, and K its Center:) And in like manner
the
whole Triangle B E K ſhall be equal to the whole Triangle
B
R K.
And becauſe alſo the Triangle O P K for the ſame reaſon

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