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-s1411" xml:space="preserve">
              <pb o="[8]" file="0078" n="85"/>
            be uſed, and the point O will fall between E and I, and the point o beyond
              <lb/>
            L, much more beyond I.</s>
            <s xml:id="echoid-s1412" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1413" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s1414" xml:space="preserve">Let the given ratio, EL to LI, be inæqualitatis minoris, i. </s>
            <s xml:id="echoid-s1415" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1416" xml:space="preserve">of a
              <lb/>
            leſs to a greater, and the point O ſought be required to lie between I and the
              <lb/>
            next point to it E; </s>
            <s xml:id="echoid-s1417" xml:space="preserve">or elſe to fall beyond A the other extreme. </s>
            <s xml:id="echoid-s1418" xml:space="preserve">For the ſame
              <lb/>
            conſtruction ſerves for both. </s>
            <s xml:id="echoid-s1419" xml:space="preserve">Here
              <emph style="sc">Case</emph>
            IV. </s>
            <s xml:id="echoid-s1420" xml:space="preserve">of
              <emph style="sc">Problem</emph>
            II. </s>
            <s xml:id="echoid-s1421" xml:space="preserve">is to be uſed, and
              <lb/>
            the point O will fall between E and I, and o beyond A, if we uſe one of the
              <lb/>
            conſtructions there recited: </s>
            <s xml:id="echoid-s1422" xml:space="preserve">but if we uſe the other, the points will ſhift places,
              <lb/>
            as was obſerved under that Caſe, viz. </s>
            <s xml:id="echoid-s1423" xml:space="preserve">O will fall beyond I the other way, and
              <lb/>
            o between L and E.</s>
            <s xml:id="echoid-s1424" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1425" xml:space="preserve">
              <emph style="sc">Case</emph>
            III. </s>
            <s xml:id="echoid-s1426" xml:space="preserve">Let now the point O be ſought between A and E. </s>
            <s xml:id="echoid-s1427" xml:space="preserve">Here ſet off
              <lb/>
            the given ratio in ſuch a manner that EI may be the ſum of the terms, and
              <lb/>
            make uſe of the IIId
              <emph style="sc">Case</emph>
            of
              <emph style="sc">Problem</emph>
            II. </s>
            <s xml:id="echoid-s1428" xml:space="preserve">and the
              <emph style="sc">Limitation</emph>
            here will
              <lb/>
            be evident from the
              <emph style="sc">Limitation</emph>
            there given, viz. </s>
            <s xml:id="echoid-s1429" xml:space="preserve">making EI: </s>
            <s xml:id="echoid-s1430" xml:space="preserve">IL:</s>
            <s xml:id="echoid-s1431" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1432" xml:space="preserve">X,
              <lb/>
            the
              <emph style="sc">Limitation</emph>
            here is that X muſt not be leſs than IE + EL + √4 IEL*.</s>
            <s xml:id="echoid-s1433" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1434" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s1435" xml:space="preserve">
              <emph style="sc">Case</emph>
            IV. </s>
            <s xml:id="echoid-s1436" xml:space="preserve">Here OI the line whoſe ſquare is concerned is
              <lb/>
            to be bounded by I the middle point of the three given ones, and O or o, its
              <lb/>
            other bound is to be ſought between I and either extreine A or E. </s>
            <s xml:id="echoid-s1437" xml:space="preserve">the ſame
              <lb/>
            conſtruction ſerving for both. </s>
            <s xml:id="echoid-s1438" xml:space="preserve">The given ratio muſt here be ſet off in ſuch a
              <lb/>
            manner that EI may be the ſum of the terms of it; </s>
            <s xml:id="echoid-s1439" xml:space="preserve">and make uſe
              <lb/>
            of Iſt
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Problem</emph>
            ; </s>
            <s xml:id="echoid-s1440" xml:space="preserve">with this caution, that of the two ſegments
              <lb/>
            AI, IE, you choſe the leſſer IE whereon to exhibit the given ratio; </s>
            <s xml:id="echoid-s1441" xml:space="preserve">for then
              <lb/>
            it will appear by the work itſelf that O falling between E and L, o will alſo
              <lb/>
            fall between A and I: </s>
            <s xml:id="echoid-s1442" xml:space="preserve">otherwiſe, if AI was leſs than IE, there would want
              <lb/>
            ſome proof of this. </s>
            <s xml:id="echoid-s1443" xml:space="preserve">Therefore of the two extreme given points call that E
              <lb/>
            which bounds the leſſer ſegment, and then the general Demonſtration will fit
              <lb/>
            this Caſe as well as the reſt.</s>
            <s xml:id="echoid-s1444" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1445" xml:space="preserve">
              <emph style="sc">Case</emph>
            V. </s>
            <s xml:id="echoid-s1446" xml:space="preserve">Let the given ratio of EL to LI be inæqualitatis minoris; </s>
            <s xml:id="echoid-s1447" xml:space="preserve">and let
              <lb/>
            the point ſought be required to lie beyond either extreme. </s>
            <s xml:id="echoid-s1448" xml:space="preserve">The ſame con-
              <lb/>
            ſtruction ſerves for both. </s>
            <s xml:id="echoid-s1449" xml:space="preserve">Here we muſt uſe the IVth
              <emph style="sc">Case</emph>
            of the IId
              <emph style="sc">Pro-</emph>
              <lb/>
              <emph style="sc">BLEM</emph>
            , and O being made to fall between E and L, o will fall always beyond
              <lb/>
            A, provided we call that point E which bounds the bigger ſegment. </s>
            <s xml:id="echoid-s1450" xml:space="preserve">I have
              <lb/>
            in the Figure made AI = IE on purpoſe to ſhew that in that caſe the point N
              <lb/>
            will coincide with A. </s>
            <s xml:id="echoid-s1451" xml:space="preserve">But if IE be greater than AI, the point N will
              <lb/>
            always fall beyond A, and conſequently the point o more ſo.</s>
            <s xml:id="echoid-s1452" xml:space="preserve"/>
          </p>
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