Archimedes
,
Natation of bodies
,
1662
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remaining Figure I S L A. </
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<
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>Becauſe now that N O is
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Seſquialter
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of R O, but leſs than
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Seſquialter ejus quæ uſque ad Axem
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(or of its
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Semi-parameter
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;) ^{*} R O ſhall be leſſe than
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quæ uſque ad Axem
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(or
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than the
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Semi-parameter
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;) ^{*} whereupon the Angle R P
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ſhall be
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acute. </
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>For ſince the Line
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quæ uſque ad Axem
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(or
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Semi-parameter
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)
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is greater than R O, that Line which is drawn from the Point R,
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and perpendicular to K
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namely RT, meeteth with the line F P
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without the Section, and for that cauſe muſt of neceſſity fall be
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tween the Points
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P
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and
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Therefore if
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L
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ines be drawn through
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B and G, parallel unto R T, they ſhall contain Right Angles with
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the Surface of the Liquid: ^{*} and the part that is within the Li
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quid ſhall move upwards according to the Perpendicular that is
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drawn thorow B, parallel to R T, and the part that is above the Li
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quid ſhall move downwards according to that which is drawn tho
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row G; and the Solid A P O L ſhall not abide in this Poſition; for
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that the parts towards A will move upwards, and thoſe towards
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B downwards; Wherefore N O ſhall be conſtituted according to
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the Perpendicular.]</
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*
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Supplied by
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Fe
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derico Comman
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dino.</
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B</
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C</
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D</
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E</
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F</
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G</
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<
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>COMMANDINE.</
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The Demonſtration of this propoſition hath been much deſired; which we have (in like man
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ner as the 8 Prop. </
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>of the firſt Book) reſtored according to
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Archimedes
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his own Schemes, and
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illustrated it with Commentaries.
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>The Right Portion of a Rightangled Conoid, when it ſhall
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have its Axis leſſe than
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Seſquialter ejus quæ uſque ad Axem
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(or of
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its
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Semi-parameter] In the Tranſlation of
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Nicolo Tartaglia
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it is falſlyread
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great
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er then Seſquialter,
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and ſo its rendered in the following Propoſition; but it is the Right
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Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we ſay
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that Conoids are then conſtituted erect when the cutting Plane, that is to ſay, the Plane of the
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Baſe, ſhall be parallel to the Surface of the Liquid.
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A</
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>Which ſhall be the Diameter of the Section I P O S, and the
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Axis of the Portion demerged in the Liquid.]
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By the 46 of the firſt of
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the Conicks of
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Apollonious,
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or by the Corol
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lary of the 51 of the ſame.
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B</
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>And of the Solid Magnitude A P
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O L, let the Centre of Gravity be R;
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and of I P O S let the Centre be B.]
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For the Centre of Gravity of the Portion of a Right
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angled Conoid is in its Axis, which it ſo divideth
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as that the part thereof terminating in the vertex,
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be double to the other part terminating in the Baſe; as
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in our Book
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De Centro Gravitatis Solidorum Propo. </
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<
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>29.
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we have demonſtrated. </
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<
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>And
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ſince the Centre of Gravity of the Portion A P O L is R, O R ſhall be double to RN and there
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fore N O ſhall be Seſquialter of O R. </
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<
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>And for the ſame reaſon, B the Centre of Gravity of the Por
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tion I P O S is in the Axis P F, ſo dividing it as that P B is double to B F;
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C</
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<
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>And draw a Line from B to R prolonged unto G; which let
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be the Centre of Gravity of the remaining Eigure I S L A.] </
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