Archimedes
,
Natation of bodies
,
1662
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<
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>PROP. V. THEOR. V.</
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The Right Portion of a Right-Angled Conoid lighter
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than the Liquid, when it ſhall have its Axis great
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er than
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Seſquialter
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of the Semi-parameter, if it have
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not greater proportion in Gravity to the Liquid [of
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equal Maſs] than the Exceſſe by which the Square
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made of the Axis is greater than the Square made
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of the Exceſſe by which the Axis is greater than
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ſeſquialter
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of the Semi-Parameter hath to the
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Square made of the Axis being demitted into the Li
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quid, ſo as that its Baſe be wholly within the Liquid,
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and being ſet inclining, it ſhall not remain ſo inclined,
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but ſhall turn about till that its Axis ſhall be accor
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ding to the Perpendicular.
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>For let any Portion be demitted into the Liquid, as hath been
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ſaid; and let its Baſe be wholly within the Liquid, And being
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cut thorow its Axis by a Plain erect upon the Surface of the
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Liquid; its Section ſhall be the Section
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of a Rightangled Cone: Let it be
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A P O L, and let the Axis of the Por
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tion and Diameter of the Section be
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N O; and the Section of the Surface of
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the Liquid I S. </
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<
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>And becauſe the Axis
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is not according to the Perpendicu
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lar, N O will not be at equall angles
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with I S. </
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<
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>Draw K
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touching the Se
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ction A P O L in P, and parallel unto
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I S: and thorow P, draw P F parallel unto N O: and take the
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Centres of Gravity; and of the Solid A P O L let the Centre be
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R; and of that which lyeth above the Liquid let the Centre be B;
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and draw a Line from B to R, prolonging it to G; which let be the
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Centre of Gravity of the Solid demerged within the Liquid: and
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moreover, take R H equall to the Semi-parameter, and let O H be
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double to H M; and do in the reſt as hath been ſaid
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(a)
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above.
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Now foraſmuch as it was ſuppoſed that the Portion hath not greater
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proportion in Gravity to the Liquid, than the Exceſſe by which
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the Square N O is greater than the Square M O, hath to the ſaid
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Square N O: And in regard that whatever proportion in Gravity </
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