Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

Table of figures

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              <s>THEOR. XV. PROP. XXIV.</s>
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              <s>There being given between the ſame Horizontal
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              Parallels a Perpendicular and a
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              P
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              lane eleva­
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              ted from its loweſt term, the Space that a
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              Moveable after the Fall along the
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              P
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              erpendi­
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              cular paſſeth along the Elevated
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              P
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              lane in a
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              Time equal to the Time of the Fall, is greater
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              than that
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              P
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              erpendicular, but leſſe than double
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              the ſame.</s>
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              <s>
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              Between the ſame Horizontal Parallels B C and H G let there
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              be the Perpendicular A E; and let the Elevated Plane be E B,
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              along which after the Fall along the Perpendicular A E out of
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              the Term E let a Reflexion be made towards B. </s>
              <s>I ſay, that the Space,
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              along which the Moveable aſcendeth in a Time equal to the Time of the
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              Deſcent A E, is greater than A E, but leſſe than double the ſame A E.
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              </s>
              <s>Let E D be equal to A E, and as E B is to B D, ſo let D B be to B F. </s>
              <s>It
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              ſhall be proved, firſt that the point F is the Term at which the Moveable
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              with a Reflex Motion along E B arriveth in a Time equal to the Time
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              A E: And then, that E F is greater than E A, but leſſe than double the
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              ſame. </s>
              <s>If we ſuppoſe the Time of the Deſcent along A E to be as A E,
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              the Time of the Deſcent along B E, or Aſcent along E B ſhall be as the
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              ſame Line B E: And D B being a Mean-Proportional betwixt E B
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              and B F, and B E being the Time of Deſcent along the whole B E, B D
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              ſhall be the Time of the Deſcent along B F, and the Remaining part
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              D E the Time of the
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              Deſcent along the Re­
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              maining part F E: But
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              the Time along F E
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              ex
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              quiete
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              in B, and the
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              Time of the Aſcent a­
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              long E F is the ſame, ſince that the Degree of Velocity in E was acqui­
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              red along the Deſcent B E, or A E: Therefore the ſame Time D E ſhall
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              be that in which the Moveable after the Fall out of A along A E,
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              with a Reflex Motion along E B ſhall reach to the Mark F: But it hath
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              been ſuppoſed that E D is equal to the ſaid A E: Which was firſt to be
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              proved. </s>
              <s>And becauſe that as the whole E B is to the whole B D, ſo is the
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              part taken away D B to the part taken away B F, therefore, as the whole
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              E B is to the whole B D, ſo ſhall the Remainder E D be to D F:
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              But E B is greater than B D: Therefore E D is greater than D F, and
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              E F leſſe than double to D E or A E: Which was to be proved.
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              </s>
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