Archimedes - Natation of bodies

Archimedes, Natation of bodies, 1662

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make 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

[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 e-
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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