Archimedes, Natation of bodies, 1662

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155[Figure 55]
And
from the Point M draw M Y
touching
the Section A M Q L in M;
and
M C perpendicular to B D: and
laſtly
, having drawn A N & prolong­
ed
it to Q, the Lines A N & N Q ſhall
be
equall to each other.
For in
regard
that in the Like Portions

A
M Q L and A X D the Lines A Q
and
A N are drawn from the Baſes
unto
the Portions, which Lines
contain
equall Angles with the ſaid
Baſes
, Q A ſhall have the ſame proportion to A M that L A hath
to
A D: Therefore A N is equall to N Q, and A Q parallel to M Y.

It
is to be demonſtrated that the Portion being demitted into the
Liquid
, and ſo inclined as that its Baſe touch not the Liquid, it
ſhall
continue inclined ſo as that its Baſe ſhall not in the leaſt touch
the
Surface of the Liquid, and its Axis ſhall make an Angle with
the
Liquids Surface greater than the Angle X.
Let it be demitted
into
the Liquid, and let it ſtand, ſo, as that its Baſe do touch the
Surface
of the Liquid in one Point only; and let the Portion be cut
thorow
the Axis by a Plane erect unto the Surface of the Liquid,
56[Figure 56]
and
Let the Section of the Super­
ficies
of the Portion be A P O L,
the
Section of a Rightangled Cone,
and
let the Section of the Liquids
Surface
be A O; And let the Axis
of
the Portion and Diameter of the
Section
be B D: and let B D be

cut
in the Points K and R as hath
been
ſaid; alſo draw P G Parallel to
A
O and touching the Section
A
P O L in P; and from that Point
draw
P T Parallel to B D, and P S perpendicular to the ſame B D.
Now
, foraſmuch as the Portion is unto the Liquid in Gravity, as
the
Square made of the Line ψ is to the Square B D; and ſince that
as
the portion is unto the Liquid in Gravitie, ſo is the part thereof
ſubmerged
unto the whole Portion; and that as the part ſubmerged
is
to the whole, ſo is the Square T P to the Square B D; It follow­
eth
that the Line ψ ſhall be equall to T P: And therefore the Lines
M
N and P T, as alſo the Portions A M Q and A P O ſhall like­
wiſe
be equall to each other.
And ſeeing that in the Equall and
Like
Portions A P O L and A M Q L the Lines A O and A Q

are
drawn from the extremites of their Baſes, ſo, as that the Portions
cut
off do make Equall Angles with their Diameters; as alſo the

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