Archimedes
,
Natation of bodies
,
1662
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and touching the Section in P, and T P parallel to B D; and P S perpen
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dicular unto B D. </
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<
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>It is to be demonſtrated that the Portion ſhall
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not ſtand ſo, but ſhall encline until
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that the Baſe touch the Surface of
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the Liquid, in one Point only, for let
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the ſuperior figure ſtand as it was,
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and draw O C, Perpendicular to B D;
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and drawing a
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L
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ine from A to
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X,
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prolong it to Q: A X ſhalbe equall
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to
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X
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Then draw O X parallel
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to A
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And becauſe the Portion
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is ſuppoſed to have the ſame pro
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portion in Gravity to the Liquid
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that the ſquare X O hath to the
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Square B D; the part thereof ſubmerged ſhall alſo have the ſame
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proportion to the whole; that is, the Square T P to the Square
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B D; and ſo T P ſhall be equal to
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X
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O: And ſince that of the
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P
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ortions
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I P M and A O Q the Diameters are equall, the portions ſhall alſo be
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equall.
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A
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gain, becauſe that in the Equall and
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L
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ike
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P
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ortions A O Q L
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and AP ML the Lines A Q and I M, which cut off equall
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P
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or
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tions, are drawn, that, from the Extremity of the
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B
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aſe, and this
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not from the Extremity; it appeareth that that which is drawn from
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the end or Extremity of the
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B
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aſe, ſhall make the Acute Angle with
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the Diameter of the whole
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P
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ortion leſset.
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A
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nd the Angle at
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X
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being leſſe than the Angle at N, B C ſhall be greater than B S; and
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C R leſſer than S R:
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A
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nd, therfore O G ſhall be leſſer than P Z;
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and G
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X
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greater than Z T: Therfore P Z is greater than double of
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Z T; being that O G is double of G X. </
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<
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>Let P H be double to H T;
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and drawing a Line from H to K, prolong it to
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The Center of
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Gravity of the whole Portion ſhall be K; the Center of the part
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which is within the Liquid H, and that of the part which is above
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the Liquid in the Line K
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; which ſuppoſed to be
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Therefore it
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ſhall be demonſtrated, both, that K H is perpendicular to the Surface
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of the Liquid, and thoſe Lines alſo that are drawn thorow the Points
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Hand
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parallel to K H: And therfore the Portion ſhall not reſt, but
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ſhall encline untill that its Baſe do touch the Surface of the Liquid
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in one Point; and ſo it ſhall continue. </
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<
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>For in the Equall Portions
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A O Q L and A P M L, the
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Lines A Q and A M, that cut off
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equall Portions, ſhall be dawn
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from the Ends or Terms of the Baſes;
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and A O Q and A P M ſhall be
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demonſtrated, as in the former, to
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be equall: Therfore A Q and A M,
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do make equall Acute Angles with
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the Diameters of the Portions; and </
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