Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667
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              <s>PROBL.XV. PROP. XXXVII.</s>
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              <s>A Perpendicular and Inclined Plane of the ſame
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              Elevation being given, to find a part in the In­
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              clined Plane that is equal to the Perpendicu­
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              lar, and paſſed in the ſame Time as the ſaid
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              Perpendicular.</s>
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              <s>
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              LET A B be the Perpendicular, and A C the Inclined Plane. </s>
              <s>It is
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              required in the Inclined to find a part equal to the Perpendicular
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              A B, that after Reſt in A may be paſſed in a Time equal to the
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              Time in which the Perpendicular is paſſed. </s>
              <s>Let A D be equal to A B,
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              and cut the Remainder B C in two equal parts in I; and as A C is to
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              C I, ſo let C I be to another Line
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              A E; to which let D G be equal: It
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              is manifeſt that E G is equal to A D
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              and to A B. </s>
              <s>I ſay moreover, that
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              this ſame E G is the ſame that is
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              paſſed by the Moveable coming out
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              of Reſt in A in a Time equal to the
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              Time in which the Moveable fall eth along A B. </s>
              <s>For becauſe that as
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              A C is to C I, ſo is C I to A E, or I D to D G; Therefore by Converſion
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              of the proportion, as C A is to A I, ſo is D I to I G. </s>
              <s>And becauſe as the
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              whole C A is to the whole A I, ſo is the part taken away C I to the part
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              I G; therefore the Remaining part I A ſhall be to the Remainder A G,
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              as the whole C A is to the whole A I: Therefore A I is a Mean-propor­
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              tional betwixt C A and A G; and C I a Mean-proportional betwixt
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              C A and A E: If therefore we ſuppoſe the Time along A B to be as A B;
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              A C ſhall be the Time along A C, and C I or I D the Time along A E:
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              And becauſe A I is a Mean-proportional betwixt C A and A G; and
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              C A is the Time along the whole A C: Therefore A I ſhall be the Time
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              along. </s>
              <s>A G; and the Remainder I C that along the Remainder G C: But
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              D I was the Time along A E: Therefore D I and I C are the Times
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              along both the Spaces A E and C G: Therefore the Remainder D A ſhall
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              be the Time along E G, to wit, equal to the Time along A B. </s>
              <s>Which was
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              to be done.
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              <s>COROLLARIE.</s>
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              <s>Hence it is manifeſt, that the Space required is an intermedial be­
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              tween the upper and lower parts that are paſt in equal
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              Times.</s>
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