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

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        <div xml:id="echoid-div425" type="section" level="1" n="178">
          <head xml:id="echoid-head183" xml:space="preserve">PROBL. XXXIII. PROP. LXXXIV.</head>
          <p>
            <s xml:id="echoid-s4359" xml:space="preserve">Datæ Ellipſi, vel circulo, per terminos cuiuſcunque in ipſo ap-
              <lb/>
            plicatę MINIMAM Ellipſim circumſcribere, cuius tranſuerſum la-
              <lb/>
            tus æquale ſit dato, quod tamen maius ſit tranſuerſa diametro datæ
              <lb/>
            Ellipſis.</s>
            <s xml:id="echoid-s4360" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4361" xml:space="preserve">SIt data Ellipſis, vel circulus ABCO, cuius tranſuerſa diameter BO, & </s>
            <s xml:id="echoid-s4362" xml:space="preserve">
              <lb/>
            quædam ad eam applicata AC: </s>
            <s xml:id="echoid-s4363" xml:space="preserve">oportet per terminos A, C, cum tranſ-
              <lb/>
            uerſo DE, quod excedat BO _MINIMAM_ Ellipſim circumſcribere.</s>
            <s xml:id="echoid-s4364" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4365" xml:space="preserve">Ducatur ex A contingens AK productæ
              <lb/>
              <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a" number="121">
                <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0154-01"/>
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            diametro occurrens in K, & </s>
            <s xml:id="echoid-s4366" xml:space="preserve">KF bifariam
              <lb/>
            ſecetur in puncto G, quod cadet iuter B, & </s>
            <s xml:id="echoid-s4367" xml:space="preserve">
              <lb/>
            K, vt in 83. </s>
            <s xml:id="echoid-s4368" xml:space="preserve">h. </s>
            <s xml:id="echoid-s4369" xml:space="preserve">oſtenſum fuit; </s>
            <s xml:id="echoid-s4370" xml:space="preserve">& </s>
            <s xml:id="echoid-s4371" xml:space="preserve">ad datam
              <lb/>
            lineam DE applicetur parallelogrammum,
              <lb/>
            æquale quadrato GF, excedens figura
              <lb/>
            quadrata, ſitque rectangulum DHE, & </s>
            <s xml:id="echoid-s4372" xml:space="preserve">
              <lb/>
            ſumpta HI media proportionali inter DH,
              <lb/>
            HE, erit rectangulum DHI, ſiue quadra-
              <lb/>
            tum GF, ęquale quadrato HI, hoc eſt linea
              <lb/>
            GF æqualis HI: </s>
            <s xml:id="echoid-s4373" xml:space="preserve">ſumpta ergo GL æquali
              <lb/>
            HE, erit reliqua LF æqualis EI, & </s>
            <s xml:id="echoid-s4374" xml:space="preserve">pũctum
              <lb/>
            L cadet extra B: </s>
            <s xml:id="echoid-s4375" xml:space="preserve">quoniam cum ſit OK ad
              <lb/>
            KB, vt AF ad FB, ſitque KF bifariam
              <note symbol="a" position="left" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve">36. pri-
                <lb/>
              mi conic.</note>
            cta in G, erit rectangulum OGB æquale quadrato GF, ſiue quadrato
              <note symbol="b" position="left" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve">79. h.</note>
            ſiue rectangulo DHE; </s>
            <s xml:id="echoid-s4376" xml:space="preserve">ſed eſt OB minor DE, ex conſtructione, quare GB
              <lb/>
            erit maior HE, ſiue maior GL; </s>
            <s xml:id="echoid-s4377" xml:space="preserve">itaque punctum L cadet extra Ellipſim
              <note symbol="c" position="left" xlink:label="note-0154-03" xlink:href="note-0154-03a" xml:space="preserve">80. h.</note>
            CO. </s>
            <s xml:id="echoid-s4378" xml:space="preserve">Sumatur ampliùs FN æqualis ID, & </s>
            <s xml:id="echoid-s4379" xml:space="preserve">erit tota LN æqualis datæ ED,
              <lb/>
            itemque punctum N cadet extra Ellipſim ABCO: </s>
            <s xml:id="echoid-s4380" xml:space="preserve">Nam cum ſit rectangulum
              <lb/>
            DHE, ſiue NGL æquale quadrato HI, ſiue GF, ſitque rectangulum OGB,
              <lb/>
            æquale eidem quadrato GF, vt ſupra oſtendimus, erunt rectangula OGB,
              <lb/>
            NGL inter ſe æqualia, & </s>
            <s xml:id="echoid-s4381" xml:space="preserve">ideo vt OG ad GN, ita LG ad GB, ſed eſt LG mi-
              <lb/>
            nor GB, vt ſuperiùs demonſtrauimus, vnde, & </s>
            <s xml:id="echoid-s4382" xml:space="preserve">OG minor erit GN, nempe
              <lb/>
            punctum N cadet extra Ellipſim ABCO. </s>
            <s xml:id="echoid-s4383" xml:space="preserve">Poſtremò cum tranſuerſo latere
              <lb/>
            NL, quod æquale eſt datæ lineæ DE, circa applicatam AC
              <note symbol="d" position="left" xlink:label="note-0154-04" xlink:href="note-0154-04a" xml:space="preserve">Coroll.
                <lb/>
              57. h.</note>
            Ellipſis ALCN. </s>
            <s xml:id="echoid-s4384" xml:space="preserve">Dico hanc eſſe _MINIMAM_ circumſcriptam quæſitam.</s>
            <s xml:id="echoid-s4385" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4386" xml:space="preserve">Quoniam cum ſit rectangulum NGL æquale quadrato GF, erit NG ad
              <lb/>
            GF, vt GF ad GL, & </s>
            <s xml:id="echoid-s4387" xml:space="preserve">componendo, NG cum GF, ſiue NK, erit ad GF, vt
              <lb/>
            FG cum GL, ſiue vt KL ad GL, & </s>
            <s xml:id="echoid-s4388" xml:space="preserve">permutando NK ad KL, vt GF ad GL,
              <lb/>
            vel vt NG ad GF, vel vt NF ad FL, ergo recta KAM Ellipſim ALCN
              <note symbol="e" position="left" xlink:label="note-0154-05" xlink:href="note-0154-05a" xml:space="preserve">Coroll.
                <lb/>
              12. h.</note>
            tingit in A, ſed eadem KAM contingit quoque ad idem punctum A
              <note symbol="f" position="left" xlink:label="note-0154-06" xlink:href="note-0154-06a" xml:space="preserve">4. h.</note>
            pſim ABCO: </s>
            <s xml:id="echoid-s4389" xml:space="preserve">quapropter Ellipſis ALCN datæ ABCO erit circumſcripta.</s>
            <s xml:id="echoid-s4390" xml:space="preserve">
              <note symbol="g" position="left" xlink:label="note-0154-07" xlink:href="note-0154-07a" xml:space="preserve">61. h.</note>
            At ipſa erit _MINIMA_: </s>
            <s xml:id="echoid-s4391" xml:space="preserve">nam quælibet alia, quæ ipſi adſcribitur per eoſdem
              <lb/>
            terminos communis applicatæ AC, & </s>
            <s xml:id="echoid-s4392" xml:space="preserve">cum tranſuerſa diametro æquali ipſi
              <lb/>
            LN, _licet maior ſuerit eadem ALCN_, inſcriptam ABCO omnino ſecat; </s>
            <s xml:id="echoid-s4393" xml:space="preserve">
              <note symbol="b" position="left" xlink:label="note-0154-08" xlink:href="note-0154-08a" xml:space="preserve">83. h.</note>
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