Archimedes
,
Natation of bodies
,
1662
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Therefore B and R are equall. </
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<
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>And becauſe that of the Magni
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tude FA the
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G
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ravity is B: Therefore of the Liquid Body
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N
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I the
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Gravity is O R. </
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>As F A is to N I, ſo is B to O R, or, ſo is R to
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O R: But as R is to O R, ſo is I to N I, and A to F A: Therefore
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I is to N I, as F A to N I: And as I to N I ſo is
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(b)
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A to F A.
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<
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>Therefore F A is to N I, as A is to F A: Which was to be demon
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ſtrated.</
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(a)
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By 5. of the
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firſt of this.
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(b)
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By 11. of the
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fifth of
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Eucl.</
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<
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>PROP. II. THEOR. II.
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A</
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>^{*}
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The Right Portion of a Right angled Conoide, when it
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ſhall have its Axis leſſe than
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ſeſquialter ejus quæ ad
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Axem (
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or of its
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Semi-parameter)
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having any what
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ever proportion to the Liquid in Gravity, being de
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mitted into the Liquid ſo as that its Baſe touch not
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the ſaid Liquid, and being ſet ſtooping, it ſhall not
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remain ſtooping, but ſhall be restored to uprightneſſe.
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</
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<
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>I ſay that the ſaid Portion ſhall ſtand upright when
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the Plane that cuts it ſhall be parallel unto the Sur
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face of the Liquid.
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<
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>Let there be a Portion of a Rightangled Conoid, as hath been
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ſaid; and let it lye ſtooping or inclining: It is to be demon
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ſtrated that it will not ſo continue but ſhall be reſtored to re
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ctitude. </
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>For let it be cut through the Axis by a plane erect upon
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the Surface of the Liquid, and let the Section of the Portion be
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A PO L, the Section of a Rightangled Cone, and let the Axis
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of the Portion and Diameter of the
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Section be N O: And let the Sect
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ion of the Surface of the Liquid be
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I S. </
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<
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>If now the Portion be not
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erect, then neither ſhall A L be Pa
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rallel to I S: Wherefore N O will
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not be at Right Angles with I S. </
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Draw therefore K
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touching the Section of the Cone I, in the
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Point P [that is parallel to I S: and from the Point P unto I S
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draw P F parallel unto O N, ^{*} which ſhall be the Diameter of the
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Section I P O S, and the Axis of the Portion demerged in the
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L
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i
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quid. </
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<
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>In the next place take the Centres of Gravity: ^{*} and of
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the Solid Magnitude A P O L, let the Centre of Gravity be R; and
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of I P O S let the Centre be B: ^{*} and draw a Line from B to R
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prolonged unto G; which let be the Centre of Gravity of the </
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