Archimedes, Natation of bodies, 1662

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1ſaid K ω in H, and A S is parallel unto the Line that toucheth in
P
; It is neceſſary that P I hath unto P H either the ſame proportion
that
N ω hath to ω O, or greater; for this hath already been de­
monſtrated
: But N ω is ſeſquialter of ω O; and P I, therefore, is
either
Seſquialter of H P, or more than ſeſquialter: Wherefore

P
H is to H I either double, or leſſe than double. Let P T be
double
to T I: the Centre of Gravity of the part which is within
the
Liquid ſhall be the Point T. Therefore draw a Line from T
to
F prolonging it; and let the Centre of
30[Figure 30]
Gravity
of the part which is above the Liquid
be
G: and from the Point B at Right Angles
unto
N O draw B R. And ſeeing that P I is
parallel
unto the Diameter N O, and B R
perpendicular
unto the ſaid Diameter, and F
B
equall to the Semi-parameter; It is mani­
feſt
that the Line drawn thorow the Points
F
and R being prolonged, maketh equall
Angles
with that which toucheth the Section
A
P O L in the Point P: and therefore doth alſo make Right An­
gles
with A S, and with the Surface of the Liquid: and the Lines
drawn
thorow T and G parallel unto F R ſhall be alſo perpendicu­
lar
to the Surface of the Liquid: and of the Solid Magnitude A P
O
L, the part which is within the Liquid moveth upwards according
to
the Perpendicular drawn thorow T; and the part which is above
the
Liquid moveth downwards according to that drawn thorow G:

The
Solid A P O L, therefore, ſhall turn about, and its Baſe ſhall
not
in the leaſt touch the Surface of the Liquid, And if P I do not
cut
the Line K ω, as in the ſecond Figure, it is manifeſt that the
Point T, which is the Centre of Gravity of the ſubmerged Portion,
falleth
betwixt P and I: And for the other particulars remaining,
they
are demonſtrated like as before.
COMMANDINE.
(a) By 10. of the
fifth
.

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