Archimedes - Natation of bodies

Archimedes, Natation of bodies, 1662

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1Angles at Y and G being equall; therefore the Lines Y B and G B,
and B C and B S ſhall alſo be equall: And therefore C R and S R,
and M V and P Z, and V N and Z T, ſhall be equall likewiſe.

Since therefore M V is Leſſer than double of V N, it is manifeſt that
P Z is leſſer than double of Z T. Let P ω be double of ω T; and
drawing a Line from ω to K, prolong it to E. Now the Centre of
Gravity of the whole Portion ſhall be the point K; and the Centre
of that part which is in the Liquid ſhall be ω, and of that which is
above the Liquid ſhall be in the Line K E, which let be E: But the
Line K Z ſhall be perpendicular unto the Surface of the Liquid:
And therefore alſo the Lines drawn thorow the Points E and ω parall

lell unto K Z, ſhall be perpendicular sunto the ſame: Therefore the
Portion ſhall not abide, but ſhall turn about ſo, as that its Baſe
do not in the leaſt touch the Surface of the Liquid; in regard that
now when it toucheth in but one Point only, it moveth upwards, on

the part towards A: It is therefore perſpicuous, that the Portion
ſhall conſiſt ſo, as that its Axis ſhall make an Angle with the Liquids
Surface greater than the Angle X.

COMMANDINE.

If the Portion have leſſer proportion in Gravity to the Liquid,
than the Square S B hath to the Square B D, but greater than the
Square X O hath to the Square B D.] This is the ſecond part of the Tenth
propoſition; and the other pat is with their Demonſtrations, ſhall hereafter follow in the ſame Order.

Ψ ſhall be greater than X O, but leſſer than the Exceſs by

which the Axis is greater than Seſquialter of the Semi-parameter,
that is than S B.] This followeth from the 10 of the fifth Book of Euclids Elements.

It ſhall be demonſtrated, that M H is double to H N, like as it
was demonſtrated, that O G is double to G X.] As in the firſt Concluſion
of this Propoſition, and from what we have but even now written, thereupon appeareth:

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 N, that L A hath to A D.]
This we have demonstrated above.

Therefore A N is equall to N Q] For ſince that Q A is to A N, as L A to
A D; Dividing and Converting, A N ſhall be to N Q as A D to D L: But A D
is equall to D L; for that D B is ſuppoſed to be the Diameter of the Portion: Therefore

alſo (a) A N is equall to N que

(a) By 14 of the
fifth.

And A Q parallel to M Y.] By the fifth of the ſecond Book of Apollonius his Conicks.

And let B D be cut in the Points K and R as hath been ſaid.]

In the firſt Conciuſion of this Propoſition: And let it be cut in K, ſo, as that B K be double to
K D, and in R ſo, as that K R may be equall to the Semi-parameter.

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 Extremities
of their Baſes, ſo, as that the Portions cut off, do make equall Angles

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