Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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        <div xml:id="echoid-div275" type="section" level="1" n="123">
          <head xml:id="echoid-head128" xml:space="preserve">THEOR. XXVII. PROP. XXXXIX.</head>
          <p>
            <s xml:id="echoid-s2970" xml:space="preserve">Si binæ Parabolæ, aut binæ concentricæ Hyperbolæ fuerint per
              <lb/>
            diuerſos vertices ſimul adſcriptæ, ipſæ, vel ad neutram partem ſe
              <lb/>
            vnquam ſecabunt, vel ſi ad alteram partem occurrant, occurrent
              <lb/>
            quoque ad aliam, punctaque occurſuum erunt extrema eiuſdem
              <lb/>
            communis applicatæ, ac in ijſdem occurſibus ſe mutuò ſecabunt.</s>
            <s xml:id="echoid-s2971" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2972" xml:space="preserve">SInt duæ Parabolæ ABC, DEF, vel duæ concentricæ Hyperbolæ per di-
              <lb/>
            uerſos vertices B, E ſimul adſcriptæ. </s>
            <s xml:id="echoid-s2973" xml:space="preserve">Dico primùm, ſi huiuſmodi ſectio-
              <lb/>
            nes ad alteram partium, vt ad A nunquam conueniunt, ad aliam quoque C
              <lb/>
            nunquam conuenire.</s>
            <s xml:id="echoid-s2974" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2975" xml:space="preserve">Nam ſumpto in ſectione DEF, ad
              <lb/>
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                <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0112-01"/>
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            partes C, quocunque puncto F, per ip-
              <lb/>
            sũ ordinatim applicetur recta CFGDA,
              <lb/>
            quæ vtrinque producta, vtrique ſectio-
              <lb/>
            ni occurret (cum ipſæ ob Hypoteſim,
              <lb/>
            ſint ſectiones in infinitam diſtantiam
              <lb/>
            abeuntes ad inferiores partes) cumque
              <lb/>
            in ſectione ABC ſit ſemi-applicata AG,
              <lb/>
            æqualis GC, & </s>
            <s xml:id="echoid-s2976" xml:space="preserve">in ſectione DEF, ſemi-
              <lb/>
            applicata DG, æqualis GF, ſitque ante-
              <lb/>
            cedens AG maior antecedente DG (cum ad partes A nunquam cõueniant)
              <lb/>
            erit etiam conſequens GC, maior conſequenti GF, quare punctum F, ſe-
              <lb/>
            ctionis DEF cadit intra ABC, & </s>
            <s xml:id="echoid-s2977" xml:space="preserve">ſic de reliquis. </s>
            <s xml:id="echoid-s2978" xml:space="preserve">Quod primò, &</s>
            <s xml:id="echoid-s2979" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2980" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2981" xml:space="preserve">Si verò ſectiones ad alteras partes, veluti ad A, & </s>
            <s xml:id="echoid-s2982" xml:space="preserve">D conueniant vt in H.
              <lb/>
            </s>
            <s xml:id="echoid-s2983" xml:space="preserve">Dico ipſas ad alias quoque partes ſimul occurrere ad extrema puncta eiuſ-
              <lb/>
            de m applicatæ.</s>
            <s xml:id="echoid-s2984" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2985" xml:space="preserve">Nam, ducta per H communi applicata HI, ipſa producatur ſecans ſectio-
              <lb/>
            nem BC in L; </s>
            <s xml:id="echoid-s2986" xml:space="preserve">& </s>
            <s xml:id="echoid-s2987" xml:space="preserve">EF in M. </s>
            <s xml:id="echoid-s2988" xml:space="preserve">Erit in ſectione ABC ſemi- applicata HI æqualis
              <lb/>
            IL, & </s>
            <s xml:id="echoid-s2989" xml:space="preserve">in ſectione DEF eadem HI æqualis IM; </s>
            <s xml:id="echoid-s2990" xml:space="preserve">ergo IL, & </s>
            <s xml:id="echoid-s2991" xml:space="preserve">IM æquales, ideo-
              <lb/>
            que ſectionum puncta L, M in vnum conueniunt; </s>
            <s xml:id="echoid-s2992" xml:space="preserve">Quare cum ſectiones
              <lb/>
            ABC, DEF non per vertices ſimul adſcriptæ ad alteram partem occurrunt,
              <lb/>
            occurrent quoque ad aliam, punctaque occurſuum erunt extrema eiuſdem
              <lb/>
            communis applicatæ.</s>
            <s xml:id="echoid-s2993" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2994" xml:space="preserve">Quod autem in occurſibus H, & </s>
            <s xml:id="echoid-s2995" xml:space="preserve">L ſe mutuò ſecent, ſic demonſtratur. </s>
            <s xml:id="echoid-s2996" xml:space="preserve">Nam
              <lb/>
            ſi huiuſmodi ſectiones tangerent ſe mutuò in occurſu H, ita vt ſectionis, ver-
              <lb/>
            bi gratia, ED partes HO, HN cadant totæ intra ſectionem ABC: </s>
            <s xml:id="echoid-s2997" xml:space="preserve">ſumpto in
              <lb/>
            altera ipſarum partium, vt puta OH, quolibet puncto D, & </s>
            <s xml:id="echoid-s2998" xml:space="preserve">per ipſum ducta
              <lb/>
            communi applicata ADGFG vtranque ſectionem ſecane
              <unsure/>
            , erunt in ſectione
              <lb/>
            ABC rectæ AG, GC æquales, & </s>
            <s xml:id="echoid-s2999" xml:space="preserve">in ſectione DEF rectæ DG, GF itẽ ę;</s>
            <s xml:id="echoid-s3000" xml:space="preserve">qua-
              <lb/>
            les, ſed eſt AG maior GD cum ponatur peripheria OH cadere intra BH, vn-
              <lb/>
            de & </s>
            <s xml:id="echoid-s3001" xml:space="preserve">CG maior erit ipſa FG, hoc eſt punctum F cadet intra. </s>
            <s xml:id="echoid-s3002" xml:space="preserve">Idemque de-
              <lb/>
            monſtrabitur de quolibet alio extremo puncto cuiuſcunque applicatæ inter
              <lb/>
            O, & </s>
            <s xml:id="echoid-s3003" xml:space="preserve">N, tùm ſupra, tùm infra occurſum L: </s>
            <s xml:id="echoid-s3004" xml:space="preserve">quare ſectio DEF </s>
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