Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667
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              * Or Angle of
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              45.</s>
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              Of the Triangle M C B, about the Right-Angle C, let the Ho­
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              rizontal Line B C and the Perpendicular C M be equal; for
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              ſo the Angle M B C ſhall be Semi-right; and prolonging C M
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              to D, let there be conſtituted in B two equal Angles above and below
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              the Diagonal M B,
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              viz.
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              M B E, and M B D. </s>
              <s>It is to be demonſtrated
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              that the Amplitudes of the Parabola's deſcribed by the Projects be­
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              ing emitted
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              [or ſhot off]
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              with the ſame
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              Impetus
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              out of the Term B,
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              according to the Elevations of the Angles E B C and D B C, are equal.
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              <s>For in regard that the extern Angle B M C, is equal to the two intern
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              M D B and M B D, the Angle M B C ſhall alſo be equal to them. </s>
              <s>And if
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              we ſuppoſe M B E inſtead of the Angle M B D,
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              the ſaid Angle M B C ſhall be equal to the two
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              Angles M B E and B D C: And taking away
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              the common Angle M B E, the remaining An­
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              gle B D C ſhall be equal to the remaining An­
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              gle E B C: Therefore the Triangles D C B
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              and B C E are alike. </s>
              <s>Let the Right Lines
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              D C and E C be divided in the midſt in H and
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              F; and draw H I and F G parallel to the Ho­
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              rizontal Line C B; and as D H is to H I, ſo
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              let I H be to H L: the Triangle I H L ſhall be
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              like to the Triangle I H D, like to which alſo is E G F. </s>
              <s>And ſeeing
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              that I H and G F are equal (to wit, halves of the ſame B C:) There­
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              fore F E, that is F C, ſhall be equal to H L: And, adding the common
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              Line F H, C H ſhall be equal to F L. </s>
              <s>If therefore we underſtand the Se­
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              miparabola to be deſcribed along by H and B, whoſe Altitude ſhall be
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              H C, and Sublimity H L, its Amplitude ſhall be C B, which is double
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              to HI, that is, the Mean betwixt D H, or C H, and HL: And D B
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              ſhall be a Tangent to it, the Lines C H and H D being equal. </s>
              <s>And if,
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              again, we conceive the Parabola to be deſcribed along by F and B from
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              the Sublimity FL, with the Altitude F C, betwixt which the Mean­
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              proportional is F G, whoſe double is the Horizontal Line C B: C B, as
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              before, ſhall be its Amplitude; and E B a Tangent to it, ſince E F and
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              F C are equal: But the Angles D B C and E B C
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              (ſcilicet,
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              their Eleva­
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              tions) ſhall be equidiſtant from the Semi-Right Angle: Therefore the
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              Propoſition is demonſtrated.
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              <s>THEOR. VI.
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              P
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              RO
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              P.
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              IX.</s>
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              <s>The Amplitudes of Parabola's, whoſe Altitudes
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              and Sublimities anſwer to each other
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              è contra­
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              rio,
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              are equall.</s>
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