Archimedes
,
Natation of bodies
,
1662
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Portion demitted into the Liquid, like as hath been ſaid, ſhall ſtand
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enclined ſo as that its Axis do make an Angle with the Surface of
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the Liquid equall unto the Angle E B
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For demit any Portion
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into the Liquid ſo as that its Baſe
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touch not the Liquids Surface;
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and, if it can be done, let the
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Axis not make an Angle with the
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Liquids Surface equall to the
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Angle E B
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; but firſt, let it be
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greater: and the Portion being
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cut thorow the Axis by a Plane e
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rect unto [
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or upon
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] the Surface of
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the Liquid, let the Section be A P
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O L the Section of a Rightangled
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Cone; the Section of the Surface of the Liquid X S; and let the
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Axis of the Portion and Diameter of the Section be N O: and
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draw P Y parallel to X S, and touching the Section A P O L in P;
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and P M parallel to N O; and P I perpendicular to N O: and
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moreover, let B R be equall to O
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and R K to T
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and let
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H
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be perpendicular to the Axis. </
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<
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>Now becauſe it hath been ſuppoſed
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that the Axis of the Portion doth make an Angle with the Surface
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of the Liquid greater than the Angle B, the Angle P Y I ſhall be
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greater than the Angle B: Therefore the Square P I hath greater
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proportion to the Square Y I, than the Square E
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hath to the
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Square
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B: But as the Square P I is to the Square Y I, ſo is the
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Line K R unto the Line I Y; and as the Square E
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is to the Square
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<
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B, ſo is half of the Line K R unto the Line
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B: Wherefore
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(a)
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K R hath greater proportion to I Y, than the half of K R hath
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to
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B: And, conſequently, I Y isleſſe than the double of
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B,
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and is the double of O I: Therefore O I is leſſe than
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B; and I
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greater than
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R: but
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R is equall to F: Therefore I
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is greater
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than F. </
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>And becauſe that the Portion is ſuppoſed to be in Gra
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vity unto the Liquid, as the Square F Q is to the Square B D; and
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ſince that as the Portion is to the Liquid in Gravity, ſo is the part
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thereof ſubmerged unto the whole Portion; and in regard that as
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the part thereof ſubmerged is to the whole, ſo is the Square P M to
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the Square O N; It followeth, that the Square P M is to the Square
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N O, as the Square F Q is to the Square B D: And therefore F
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Q is equall to P M: But it hath been demonſtrated that P H is
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greater than F: It is manifeſt, therefore, that P M is leſſe than
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ſeſquialter of P H: And conſequently that P H is greater than
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the double of H M. </
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<
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>Let P Z be double to Z M: T ſhall be the Cen
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tre of Gravity of the whole Solid; the Centre of that part of it
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which is within the Liquid, the Point Z; and of the remaining
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part the Centre ſhall be in the Line Z T prolonged unto G. </
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<
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