Archimedes
,
Natation of bodies
,
1662
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the ſame manner we might demon
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ſtrate the
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L
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ine T H to be perpendi
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cular unto the Surface of the Liquid:
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and that the Portion demerged with
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in the
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L
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iquid moveth or aſcend
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eth out of the
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L
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iquid according to
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the Perpendicular that ſhall be
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drawn thorow Z unto the Surface
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of the Liquid; and that the part
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that is above the Liquid deſcendeth
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into the Liquid according to that
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drawn thorow G: therefore the Portion will not continue ſo inclined
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as was ſuppoſed: But neither ſhall it return to Rectitude or Per
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pendicularity; For that of the Perpendiculars drawn thorow Z and
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G, that paſſing thorow Z doth fall on thoſe parts which are to
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wards L; and that that paſſeth thorow G on thoſe towards A:
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Wherefore it followeth that the Centre Z do move upwards,
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and G downwards: Therefore the parts of the whole Solid which
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are towards A ſhall move downwards, and thoſe towards L up
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wards. </
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>Again let the Propoſition run in other termes; and let
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the Axis of the Portion make an Angle with the Surface of the
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Liquid leſſe than that which is at B. </
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<
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>Therefore the Square P I
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hath leſſer Proportion unto the Square
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I Y, than the Square E
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hath to the
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Square
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B: Wherefore K R hath
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leſſer proportion to I Y, than the half
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of K R hath to
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B: And, for the
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ſame reaſon, I Y is greater than dou
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ble of
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B: but it is double of O I:
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Therefore O I ſhall be greater than
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B: But the Totall O
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is equall
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to R B, and the Remainder
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I leſſe
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than
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R: Wherefore P H ſhall alſo
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be leſſe than F. And, in regard that
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M P is equall to F Q, it is manifeſt that P M is greater than ſeſqui
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alter of P H; and that P H is leſſe than double of
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H
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M.
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L
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et P Z
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be double to Z M. </
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<
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>The Centre of Gravity of the whole Solid ſhall
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again be T; that of the part which is within the Liquid Z; and
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drawing a Line from Z to T, the Centre of Gravity of that which
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is above the Liquid ſhall be found in that Line portracted, that is
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in G: Therefore, Perpendiculars being drawn thorow Z and G
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unto the Surface of the Liquid that are parallel to T H, it followeth
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that the ſaid Portion ſhall not ſtay, but ſhall turn about till
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that its Axis do make an Angle with the Waters Surface greater than
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that which it now maketh. </
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<
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>And becauſe that when before we </
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