Archimedes
,
Natation of bodies
,
1662
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the Portion hath to the Liquid of equall Maſſe, the ſame hath the
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Magnitude of the Portion ſubmerged unto the whole Portion; as
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hath been demonſtrated in the firſt Propoſition; The Magnitude
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ſubmerged, therefore, ſhall not have greater proportion to the
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whole
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(b)
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Portion, than that which hath been mentioned: ^{*}And
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therefore the whole Portion hath not greater proportion unto that
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which is above the Liquid, than the Square N O hath to the Square
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M O: But the
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(c)
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whole Portion hath the ſame proportion unto
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that which is above the Liquid that the Square N O hath to the
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Square P F: Therefore the Square N O hath not greater propor
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tion unto the Square P F, than it hath unto the Square M O: ^{*}And
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hence it followeth that P F is not leſſe than O M, nor P B than O
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H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O
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ſhall meet with B P betwixt P and B: Let it be in T: And be
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cauſe that in the Section of the Rectangled Cone P F is parallel unto
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the Diameter N O; and H T perpendicular unto the ſaid Diame
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ter; and R H equall to the Semi-parameter: It is manifeſt that
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R T prolonged doth make Right Angles with K P
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: And there
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fore doth alſo make Right Angles with I S: Therefore R T is per
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pendicular unto the Surface of the Liquid; And if thorow the
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Points B and G Lines be drawn parallel unto R T, they ſhall be
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perpendicular unto the Liquids Surface. </
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>The Portion, therefore,
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which is above the Liquid ſhall move downwards in the Liquid ac
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cording to the Perpendicular drawn thorow B; and that part
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which is within the Liquid ſhall move upwards according to the
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Perpendicular drawn thorow G; and the Solid Portion A P O L
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ſhall not continue ſo inclined, [
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as it was at its demerſion
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], but ſhall
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move within the Liquid untill ſuch time that N O do ſtand accor
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ding to the Perpendicular.</
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(a)
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In 4. Prop. of
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this.
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(a)
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By 11. of the
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fifth.
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A</
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(b)
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By 26. of the
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Book
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De Conoid.
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<
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>& Sphæroid.</
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B</
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C</
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<
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>COMMANDINE.
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A</
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>And therefore the whole Portion hath not greater proportion
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unto that which is above the Liquid, than the Square N O hath to
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the Square M O.]
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For in regard that the Magnitude of the Portion demerged
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within the Liquid hath not greater proportion unto the whole Portion than the Exceſſe by which
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the Square N O is greater than the Square M O hath to the ſaid Square N O; Converting of
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the Proportion, by the 26. of the fifth of
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Euclid,
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of
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Campanus
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his Tranſlation, the whole
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Portion ſhall not have leſſer proportion unto the Magnitude ſubmerged, than the Square N O
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hath unto the Exceſſe by which N O is greater than the Square M O. </
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<
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>Let a Portion be taken;
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and let that part of it which is above the Liquid be the firſt Magnitude; the part of it which
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is ſubmerged the ſecond: and let the third Magnitude be the Square M O; and let the Exceſſe
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by which the Square N O is greater than the Square M O be the fourth. </
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<
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>Now of theſe Mag
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nitudes, the proportion of the firſt and ſecond, unto the ſecond, is not leſſe than that of the third &
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fourth unto the fourth: For the Square M O together with the Exceſſe by which the Square
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N O exceedeth the Square M O is equall unto the ſaid Square N O: Wherefore, by Converſi
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on of Proportion, by 30 of the ſaid fifth Book, the proportion of the firſt and ſecond unto the
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firſt, ſhall not be greater than that of the third and fourth unto the third: And, for the ſame
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