Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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Mechanicks: That the Moment of the Weight elevated upon the Plane
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according to the Line A B C, is
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to its total Moment, as B E to B A;
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And that the Moment of the ſame
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Weight upon the Elevation A D,
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is to its total Moment, as D F to
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D A or B A: Therefore the Mo
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ment of the ſaid Weight upon the
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Plane inclined according to D A,
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is to the Moment upon the Plane
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inclined according to A B C, as
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the Line D F to the Line B E:
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Therefore the Spaces which the
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ſaid Weight ſhall paſſe in equal
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Times along the Inclined Planes C A and D A, ſhall be to each other as
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the Line B E to D F; by the ſecond Propoſition of the Firſt Book:
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But as B E is to D F, ſo A C is demonſtrated to be to D A:
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Therefore the ſame Moveable will in equal Times paſſe the Lines
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C A and D A.
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And that C A is to D A as B E is to D F, is thus demonſtrated.
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Draw a Line from C to D; and by D and B draw the Lines
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D G L, (cutting C A in the point I) and B H, Parallels to A F:
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And the Angle A D I ſhall be equal to the Angle D C A, for that
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the parts L A and A D of the Circumference ſubtending them, are
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equal, and the Angle D A C common to them both: Therefore of
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the equiangled Triangles C A D and D A I, the ſides about the
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equal Angles ſhall be proportional: And as C A is to A D, ſo is
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D A to A I, that is B A to A I, or H A to A G; that is, B E to
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D F: Which was to be proved.
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Or elſe the ſame ſhall be demonſtrated more ſpeedily thus.
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Vnto the Horizon A B, let a Circle be erect, whoſe Diameter is
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perpendicular to the Horizon: and
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from the higheſt Term D let a Plane
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at pleaſure D F, be inclined to the
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Circumference. </
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>I ſay that the De
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ſcent along the Plane D F, and the
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Fall along the Diameter B C, will
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be paſſed by the ſame Moveable in
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equal Times. </
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<
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>For let F G be drawn
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parallel to the Horizon A B, which
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ſhall be perpendicular to the Diameter
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D C, and let a Line conjoyn F and
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C: and becauſe the Time of the Fall
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along D C, is to the Time of the Fall along D G, as the Mean
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Proportional between C D and D G, is to the ſaid D G; and the
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