Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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<
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>THEOR. VI. PROP. VI.</
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>If from the higheſt or loweſt part of a Circle,
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erect upon the Horizon, certain Planes be
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drawn inclined towards the Circumference,
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the Times of the Deſcents along the ſame
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ſhall be equal.</
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Let the Circle be erect upon the Horizon G H, whoſe Diameter
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recited upon the loweſt point, that is upon the contact with the
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Horizon, let be F A, and from the higheſt point A let certain
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Planes A B and A C incline towards the Circumference: I ſay that the
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Times of the Deſcents along the ſame are equal. </
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>Let B D and C E be
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two Perpendiculars let fall unto the Diameter; and let A I be a Mean
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Proportional between the Altitudes
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of the Planes E A and A D. </
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>And
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becauſe the Rectangles F A E and
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F A D are equal to the Squares of
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A C and A B; And alſo becauſe
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that as the Rectangle F A E, is to
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the Rectangle F A D, ſo is E A to
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A D. </
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>Therefore as the Square of
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C A is to the Square of B A,
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ſo is the Line E A to the Line
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A D. </
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>But as the Line E A is to
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D A, ſo is the Square of I A to the Square of A D: Therefore
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the Squares of the Lines C A and A B are to each other as the Squares
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of the Lines I A and A D: And therefore as the Line C A is to A B,
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ſo is I A to A D: But in the precedent Propoſition it hath been demon
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ſtrated that the proportion of the Time of the Deſcent along A C to the
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Time of the Deſcent by A B, is compounded of the proportions of C A
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to A B, and of D A to A I, which is the ſame with the proportion of
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B A to A C: Therefore the proportion of the Time of the Deſcent along
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A C, to the Time of the Deſcent along A B, is compounded of the pro
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portions of C A to A B, and of B A to A C: Therefore the proporti
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on of thoſe Times is a proportion of equality: Therefore the Propoſition
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is evident.
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The ſame is another way demonſtrated from the Mechanicks: Name
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ly that in the enſuing Figure the Moveable paſſeth in equal Times along
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C A and D A. </
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>For let B A be equal to the ſaid D A, ond let fall the
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Perpendiculars B E and D F: It is manifeſt by the Elements of the
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