Archimedes, Natation of bodies, 1662

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1make equall Angles; and that in the Triangles N F S and G ω C
the
Angles at F and ω are equall; as alſo, that S B and B C, and
S
R and C R are equall to one another: And, therefore, N X and
G
Y are alſo equall; and X T and Y I.
And ſince G H is double
to
H I, N X ſhall be leſſer than double of X T.
Let N M therefore
be
double to M T; and drawing a Line from M to K, prolong it
unto
E.
Now the Centre of Gravity of the whole ſhall be the
Point
K; of the part which is in the Liquid the Point M; and
that
of the part which is above the Liquid in the Line prolonged
as
ſuppoſe in E. Therefore, by what was even now demonſtrated
it
is manifeſt that the Portion ſhall not ſtay thus, but ſhall incline, ſo
as
that its Baſe do in no wiſe touch the Surface of the Liquid
And
that the Portion will ſtand, ſo, as to make an Angle with the
Surface
of the Liquid leſſer than
75[Figure 75]
the
Angle φ, ſhall thus be demon
ſtrated
.
Let it, if poſſible, ſtand,
ſo
, as that it do not make an Angle
leſſer
than the Angle φ; and diſpoſe
all
things elſe in the ſame manner a
before
; as is done in the preſet
Figure
.
We are to demonſtrat
in
the ſame method, that N T is
quall
to ψ; and by the ſame reaſor
equall
alſo to G I.
And ſince that in
the
Triangles P φ C and N F S, the Angle F is not leſſer than the
Angle
φ, B F ſhall not be greater than B C: And, therefore, neither
ſhall
S R be leſſer than C R; nor N X than P Y: But ſince P F is
greater
than N T, let P F be Seſquialter of P Y: N T ſhall be leſſer
than
Seſquialter of N X: And, therefore, N X ſhall be greate
than
double of X T.
Let N M be double of M T; and drawing
Line
from M to K prolong it.
It is manifeſt, now, by what hath
been
ſaid, that the Portion ſhall not continue in this poſition, but ſhall
turn
about, ſo, as that its Axis do make an Angle with the Surface
of
the Liquid, leſſer than the Angle φ.

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