Apollonius <Pergaeus>; Lawson, John, The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies
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          <head xml:id="echoid-head74" xml:space="preserve">PROBLEMS
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          CONCERNING
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          DETERMINATE SECTION.</head>
          <head xml:id="echoid-head75" xml:space="preserve">PROBLEM I.</head>
          <p>
            <s xml:id="echoid-s1043" xml:space="preserve">TO cut a given indefinite right line in one point, ſo that of the ſegments
              <lb/>
            intercepted between that point and two other points given in the inde-
              <lb/>
            finite right line, the ſquare of one of them may be to the rectangle under the
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            other and a given external right line, in a given ratio.</s>
            <s xml:id="echoid-s1044" xml:space="preserve"/>
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            <s xml:id="echoid-s1045" xml:space="preserve">In the given indefinite right line let be aſſigned the points A and E, it is then
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            required to cut it in the point O, ſo that
              <emph style="ol">AO</emph>
              <emph style="sub">2</emph>
            may be to OE into a given
              <lb/>
            line AU in the ratio of R to S; </s>
            <s xml:id="echoid-s1046" xml:space="preserve">which ratio let be expreſſed by AI to AU,
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            ſetting off AI from A either way, either towards E or the contrary; </s>
            <s xml:id="echoid-s1047" xml:space="preserve">and
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            then from A and I erect two perpendiculars AY equal to AE, and IR
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            equal to AI, and theſe on the ſame ſide of the given indefinite line, if AI
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            was ſet off towards E; </s>
            <s xml:id="echoid-s1048" xml:space="preserve">but on oppoſite ſides, if AI was ſet off the other way.
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            </s>
            <s xml:id="echoid-s1049" xml:space="preserve">The former conſtruction I will beg leave to call Homotactical, and the latter
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            Antitactical. </s>
            <s xml:id="echoid-s1050" xml:space="preserve">Let now the extremities of theſe perpendiculars Y and R be
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            joined, and upon YR as a diameter let a circle be deſcribed, I ſay that the
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            interſection of this circle with the given indefinite line ſolves the Problem. </s>
            <s xml:id="echoid-s1051" xml:space="preserve">
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            If it interſects the line in two places, the Problem admits of two Solutions;</s>
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