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

Table of figures

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                <pb xlink:href="040/01/890.jpg" pagenum="197"/>
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              And becauſe in the Quadrilateral Figure I L B H the Sides H B and
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              H I are equal, and the Angles B and I Right Angles, the Side B L ſhall
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              likewiſe be equal to the Side L I: But E I is equal to E F: Therefore the
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              whole Line L E, or N E is equal to the two Lines L B and E F: Let
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              the Common Line E F be taken away, and the remainder F N ſhall be
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              equal to L B: And F B was ſuppoſed equal to B A: Therefore L B ſhall
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              be equal to the two Lines A B and B N. Again, if we ſuppoſe the
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              Time along A B to be the ſaid A B, the Time along E B ſhall be equal to
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              E B; and the Time along the whole E M ſhall be E N, namely, the
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              Mean-proportional betwixt M E and E B: I berefore the Time of the
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              Deſcent of the remaining part B M after E B, or after A B, ſhall be the
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              ſaid B N: But it hath been ſuppoſed, that the Time along A B is A B:
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              Therefore the Time of the Fall along both A B and B M is A B N:
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              And becauſe the Time along E B
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              ex quiete
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              in E is E B, the Time along
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              B M
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              ex quiete
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              in B ſhall be the Mean-proportional between B E and
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              B M; and this is B L: The Time, therefore, along both A B M
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              ex quiete
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              in A is A B N: And the Time along B M only
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              ex quiete
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              in B is B L:
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              But it was proved that B L is equal to the two A B and B N: Therefore
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              the Propoſition is manifeſt.
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              Otherwiſe with more expedition.
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              Let B C be the Inclined Plane, and B A the Perpendicular. </s>
              <s>Continue
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              out C B to E, and unto E C erect a Perpendicular at B, which being
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              prolonged ſuppoſe B H equal to the exceſſe of B E above B A; and to the
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              Angle B H E let the Angle H E L be equal; and let E L continued out
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              meet with B K in L; and from L erect the Perpendicular L M unto E L
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              meeting B C in M. </s>
              <s>I ſay, that
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                <figure id="id.040.01.890.1.jpg" xlink:href="040/01/890/1.jpg" number="135"/>
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              B M is the Space acquired in
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              the Plane B C. </s>
              <s>For becauſe
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              the Angle M L E is a Right­
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              Angle, therefore B L ſhall be
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              a Mean-proportional betwixt
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              M B and B E; and L E a
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              Mean proportional betwixt M
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              E and E B; to which E L let
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              E N be cut equal: And the
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              three Lines N E, E L, and
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              L H ſhall be equal; and H B ſhall be the exceſſe of N E above B L: But
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              the ſaid H B is alſo the exceſſe of N E above N B and B A: Therefore
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              the two Lines N B and B A are equal to B L. </s>
              <s>And if we ſuppoſe E B
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              to be the Time along E B, B L ſhall be the Time along B M
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              ex quiete
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              in
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              B; and B N ſhall be the Time of the ſame B M after E B or after A B;
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              and A B ſhall be the Time along A B: Therefore the Times along A B M,
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              namely, A B N, are equal to the Times along the ſole Line B M
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              ex quiete
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              in B: Which was intended.
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