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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          <p>
            <s xml:id="echoid-s2081" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s2082" xml:space="preserve">This Problem alſo hath three Epitagmas; </s>
            <s xml:id="echoid-s2083" xml:space="preserve">ſirſt, when O is
              <lb/>
            ſought between I, the point which bounds the ſegment whoſe ſquare is
              <lb/>
            concerned, and either of the other given ones; </s>
            <s xml:id="echoid-s2084" xml:space="preserve">ſecondly, the ſaid point
              <lb/>
            being an extreme one, when O is ſought beyond it, or beyond both the
              <lb/>
            other given points with reſpect to it; </s>
            <s xml:id="echoid-s2085" xml:space="preserve">and thirdly, when O is required be-
              <lb/>
            yond the next in order to the abovementioned point I: </s>
            <s xml:id="echoid-s2086" xml:space="preserve">theſe are each of
              <lb/>
            them ſubdiviſible into other more particular caſes.</s>
            <s xml:id="echoid-s2087" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2088" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s2089" xml:space="preserve">Here O is ſought between I, the point which bounds the
              <lb/>
            ſegment whoſe ſquare is concerned, and the next in order to it: </s>
            <s xml:id="echoid-s2090" xml:space="preserve">and there
              <lb/>
            are four caſes, viz. </s>
            <s xml:id="echoid-s2091" xml:space="preserve">when I is an extreme point; </s>
            <s xml:id="echoid-s2092" xml:space="preserve">and the given ratio of R
              <lb/>
            to S the ratio of a greater to a leſs; </s>
            <s xml:id="echoid-s2093" xml:space="preserve">when I, remaining as before, the given
              <lb/>
            ratio is of a leſs to a greater; </s>
            <s xml:id="echoid-s2094" xml:space="preserve">again when I is the middle point, and O
              <lb/>
            ſought between it and either of the other given ones.</s>
            <s xml:id="echoid-s2095" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2096" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s2097" xml:space="preserve">Here the points B and C are both made to fall beyond I
              <lb/>
            (Fig. </s>
            <s xml:id="echoid-s2098" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s2099" xml:space="preserve">and DH is drawn through the center of the circle on BC, and O
              <lb/>
            will fall between the point I, and the next in order thereto; </s>
            <s xml:id="echoid-s2100" xml:space="preserve">becauſe by
              <lb/>
            conſtruction, EC is to CO as BO is to IB, and therefore when EC is greater
              <lb/>
            than CO, BO will be greater than IB, and when leſs, leſs; </s>
            <s xml:id="echoid-s2101" xml:space="preserve">but it is plain
              <lb/>
            that if O ſhould fall either beyond E or I, this could not be the Caſe. </s>
            <s xml:id="echoid-s2102" xml:space="preserve">It
              <lb/>
            is farther manifeſt that ſhould the points A and E change places, the con-
              <lb/>
            ſtruction would be no otherwiſe altered than that AB would then be greater
              <lb/>
            than IC.</s>
            <s xml:id="echoid-s2103" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2104" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2105" xml:space="preserve">If the given points retain their poſition, but the ratio be made
              <lb/>
            of a leſs to a greater, the conſtruction will then be by Fig. </s>
            <s xml:id="echoid-s2106" xml:space="preserve">18, where B
              <lb/>
            muſt be made to fall beyond A, and C beyond E with reſpect to I; </s>
            <s xml:id="echoid-s2107" xml:space="preserve">but
              <lb/>
            DH is ſtill drawn through the center of the circle on BC: </s>
            <s xml:id="echoid-s2108" xml:space="preserve">and that O will
              <lb/>
            fall as required may be made appear by reaſonings ſimilar to thoſe uſed in
              <lb/>
            Caſe I. </s>
            <s xml:id="echoid-s2109" xml:space="preserve">Moreover no change will enſue in the conſtruction when the Points
              <lb/>
            A and E change places, except that B and C will change ſituations alſo.</s>
            <s xml:id="echoid-s2110" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2111" xml:space="preserve">
              <emph style="sc">Cases</emph>
            III and IV, Are conſtructed at once by Fig. </s>
            <s xml:id="echoid-s2112" xml:space="preserve">19, when B muſt
              <lb/>
            fall between A and I, C between I and E, and DH be drawn as before:
              <lb/>
            </s>
            <s xml:id="echoid-s2113" xml:space="preserve">and it is here evident that the conſtruction will be the ſame let the given
              <lb/>
            ratio be what it will. </s>
            <s xml:id="echoid-s2114" xml:space="preserve">None of thoſe Caſes admit of any Limitations.</s>
            <s xml:id="echoid-s2115" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2116" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s2117" xml:space="preserve">There are here only two Caſes, viz. </s>
            <s xml:id="echoid-s2118" xml:space="preserve">when O is required
              <lb/>
            beyond I, the point which bounds the ſegment whoſe ſquare is concerned;</s>
            <s xml:id="echoid-s2119" xml:space="preserve"/>
          </p>
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