Archimedes
,
Natation of bodies
,
1662
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with their Diameters; as alſo, the Angles at Y and G being equall;
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Therefore, the Lines Y B and G B, & B C & B S, ſhall alſo be equall.]
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Let the Line A Q cut the Diameter D B in
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and let it cut A O in
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Now becauſe that in
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the equall and like Portions A P O L & A M Q L,
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from the Extremities of their Baſes, A O and
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A Q are drawn, that contain equall Angles with
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thoſe Baſes; and ſince the Angles at D, are both
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Right; Therefore, the Remaining Angles A
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D
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and A
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D
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ſhall be equall to one another: But
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the Line P G is parallel unto the Line A O; alſo
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M Y is parallel to A
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and P S and M C to
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A D: Therefore the Triangles P G S and M Y C,
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as alſo the Triangles A
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D and A
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D, are all
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alike to each other
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: (b)
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And as A D is to A
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ſo is A D to A
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: and,
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Permutando,
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the Lines
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A D and A D are equall to each other: Therefore,
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A
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and A
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are alſo equall: But A O and
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A Q are equall to each other; as alſo their halves
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A T and A N: Therefore the Remainders T
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and N
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; that is, TG and MY, are alſo
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equall. </
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>And, as
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(c)
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P G is to G S, ſo is M Y to
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Y C: and
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Permutando,
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as P G is to M Y, ſo is
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G S to Y C: And, therefore, G S and Y C are
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equall; as alſo their halves B S and B C: From
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whence it followeth, that the Remainders S R and C R
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are alſo equall: And, conſequently, that P Z and
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M V, and V N and Z T, are lkiewiſe equall to one
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another.
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H</
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(b)
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By 4. of the
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ſixth.
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(c)
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By 34 of the
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firſt,
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>Since, therefore, that N V is leſſer
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than double of V N.]
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For M H is double of
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H N, and M V is leſſer than M H: Therefore, M V
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is leſſer than double of H N, and much leſſer than
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double of V N.
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K</
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>Therefore, the Portion ſhall not abide, but ſhall turn about,
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ſo, as that its Baſe do not in the leaſt touch the Surface of
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the Liquid; in regard that now when it toucheth in but one Point
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only, it moveth upwards on the part towards A.] Tartaglia's
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his Tranſla
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tion hath it thus,
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Non ergo manet Portio ſed inclinabitur ut Baſis ipſius, nec ſecundum
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unum tangat Superficiem Humidi, quon am nunc ſecundum unum tacta ipſa reclina
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tur
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: Which we have thought fit in this manner to correct, from other Places of
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Archimedes,
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that the ſenſe might be the more perſpicuous. </
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>For in the ſixth Propoſition of this,
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he thus writeth (as we alſo have it in the Tranſlation,)
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The Solid A P O L, therefore, ſhall
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turn about, and its Baſe ſhall not in the leaſt touch the Surface of the Liquid.
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Again,
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in the ſeventh Propoſition
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; From whence it is manifeſt, that its Baſe ſhall turn about in
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ſuch manner, a that its Baſe doth in no wiſe touch the Surface of the Liquid; For
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that now when it toucheth but in one Point only, it moveth downwards on the part
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towards L.
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And that the Portion moveth upwards, on the part towards A, doth plainly ap
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pear: For ſince that the Perpendiculars unto the Surface of the Liquid, that paſs thorow
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, de
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fall on the part towards A, and thoſe that paſs thorow E, on the part towards L; it is neceſſary
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that the Centre
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do move upwards, and the Centre E downwards.
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L</
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<
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>It is therefore perſpicuous, that the Portion ſhall conſiſt, ſo, as that
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its Axis ſhall make an Angle with the Liquids Surface greater than
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the Angle
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X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter
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