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

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        <div xml:id="echoid-div398" type="section" level="1" n="168">
          <p>
            <s xml:id="echoid-s4140" xml:space="preserve">
              <pb o="123" file="0147" n="147" rhead=""/>
            _MI_ circuli, erunt, vt quadrata eorumdem numerorum diſparium ab vnita
              <lb/>
            te. </s>
            <s xml:id="echoid-s4141" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s4142" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div400" type="section" level="1" n="169">
          <head xml:id="echoid-head174" xml:space="preserve">PROBL. XXXI. PROP. LXXVIII.</head>
          <p>
            <s xml:id="echoid-s4143" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM
              <lb/>
            Parabolen inſcribere; </s>
            <s xml:id="echoid-s4144" xml:space="preserve">& </s>
            <s xml:id="echoid-s4145" xml:space="preserve">è contra.</s>
            <s xml:id="echoid-s4146" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4147" xml:space="preserve">Datæ Parabolæ, per punctum extra ipſam datum cum dato ſe-
              <lb/>
            mi- tranſuerſo latere MINIMAM Hyperbolen circumſcribere.</s>
            <s xml:id="echoid-s4148" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4149" xml:space="preserve">SIt data Hyperbole ABC, cuius centrum E, & </s>
            <s xml:id="echoid-s4150" xml:space="preserve">punctum intra ipſam da-
              <lb/>
            tum ſit G. </s>
            <s xml:id="echoid-s4151" xml:space="preserve">Oportet per G _MAXIMAM_ Parabolen inſcribere.</s>
            <s xml:id="echoid-s4152" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4153" xml:space="preserve">Iungatur EG ſecans Hyperbolen in B, & </s>
            <s xml:id="echoid-s4154" xml:space="preserve">concipiatur EG eſſe mediam
              <lb/>
            arithmeticam, EB verò mediam geometricam inter eaſdem ignotas extre-
              <lb/>
            mas, quæ reperiantur, & </s>
            <s xml:id="echoid-s4155" xml:space="preserve">ſint EH, EF, & </s>
            <s xml:id="echoid-s4156" xml:space="preserve">per F applicetur AFC, & </s>
            <s xml:id="echoid-s4157" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-0147-01" xlink:href="note-0147-01a" xml:space="preserve">74. h.</note>
            diametrum GF adſcribatur ipſi Hyperbolæ ABC, Parabole
              <note symbol="b" position="right" xlink:label="note-0147-02" xlink:href="note-0147-02a" xml:space="preserve">57. h.</note>
            quarum communis applicata ſit AC. </s>
            <s xml:id="echoid-s4158" xml:space="preserve">Dico ipſam Parabolen eſſe quæſitam.</s>
            <s xml:id="echoid-s4159" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4160" xml:space="preserve">Cum enim ſit FE ad EB, vt EB ad EH, erit
              <lb/>
              <figure xlink:label="fig-0147-01" xlink:href="fig-0147-01a" number="113">
                <image file="0147-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0147-01"/>
              </figure>
            rectangulum FEH æquale quadrato EB, qua-
              <lb/>
            re AH Hyperbolen continget. </s>
            <s xml:id="echoid-s4161" xml:space="preserve">Cumque
              <note symbol="c" position="right" xlink:label="note-0147-03" xlink:href="note-0147-03a" xml:space="preserve">conuerſ.
                <lb/>
              37. primi
                <lb/>
              conic.
                <lb/>
              Comand.</note>
            EG media arithmetica inter FE, EH, erunt
              <lb/>
            ipſarum diſferentiæ FG, GH æquales, vnde
              <lb/>
              <note symbol="d" position="right" xlink:label="note-0147-04" xlink:href="note-0147-04a" xml:space="preserve">2. h.</note>
            eadem AH Parabolen quoque continget:</s>
            <s xml:id="echoid-s4162" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-0147-05" xlink:href="note-0147-05a" xml:space="preserve">61. h.</note>
            quare Parabole D G M Hyperbolæ ABC erit inſcripta. </s>
            <s xml:id="echoid-s4163" xml:space="preserve">Quod autem ſit _MAXIMA_,
              <lb/>
            patet; </s>
            <s xml:id="echoid-s4164" xml:space="preserve">cum quælibet alia per G adſcripta cum
              <lb/>
            recto minori, minor eſt AGC, quę verò cum
              <lb/>
            maiori, eſt quidem maior, ſed omninò ſecat
              <lb/>
            Hyperbolen ABC, cum ſectio Parabole in in-
              <lb/>
            finitum abeat, & </s>
            <s xml:id="echoid-s4165" xml:space="preserve">ſuperficies ABCGA vndi-
              <lb/>
            que ſit clauſa. </s>
            <s xml:id="echoid-s4166" xml:space="preserve">Quod primò, &</s>
            <s xml:id="echoid-s4167" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4168" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4169" xml:space="preserve">IAM ſit data Parabole AGC, & </s>
            <s xml:id="echoid-s4170" xml:space="preserve">datum extra ipſam pũctum ſit B, per quod
              <lb/>
            oporteat, cum dato quolibet ſemi-tranſuerſo D, _MINIMAM_ Hyperbo-
              <lb/>
            len circumſcribere.</s>
            <s xml:id="echoid-s4171" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4172" xml:space="preserve">Ducatur per B diameter Parabolæ DGF, quæ vltra B producatur, ſuma-
              <lb/>
            turque BE ipſi D æqualis, & </s>
            <s xml:id="echoid-s4173" xml:space="preserve">facta EB media geometrica, & </s>
            <s xml:id="echoid-s4174" xml:space="preserve">EG media ari-
              <lb/>
            thmetica inter eaſdem ignotas extremas, reperiantur ipſæ extremæ,
              <note symbol="f" position="right" xlink:label="note-0147-06" xlink:href="note-0147-06a" xml:space="preserve">74. h.</note>
            ſint EH, EF; </s>
            <s xml:id="echoid-s4175" xml:space="preserve">& </s>
            <s xml:id="echoid-s4176" xml:space="preserve">per F applicetur in Parabola AFC; </s>
            <s xml:id="echoid-s4177" xml:space="preserve">& </s>
            <s xml:id="echoid-s4178" xml:space="preserve">ſuper AC ad diame-
              <lb/>
            tri ſegmentum BF, cum ſemi- tranſuerſo BE deſcribatur Hyperbole ABC.</s>
            <s xml:id="echoid-s4179" xml:space="preserve">
              <note symbol="g" position="right" xlink:label="note-0147-07" xlink:href="note-0147-07a" xml:space="preserve">57. h.</note>
            Dico ipſam eſſe _MINIMAM_ quæſitam.</s>
            <s xml:id="echoid-s4180" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4181" xml:space="preserve">Si quidem iuncta HA, ijſdem omnino argumentis, ac ſupra, demonſtra-
              <lb/>
            bitur ipſam HA, & </s>
            <s xml:id="echoid-s4182" xml:space="preserve">Parabolen, & </s>
            <s xml:id="echoid-s4183" xml:space="preserve">Hyperbolen contingere, vnde ſectiones
              <lb/>
            ſe mutuò contingent, & </s>
            <s xml:id="echoid-s4184" xml:space="preserve">Hyperbole ABC erit Parabolæ circumſcripta:</s>
            <s xml:id="echoid-s4185" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0147-08" xlink:href="note-0147-08a" xml:space="preserve">61. h.</note>
            eritque _MINIMA_; </s>
            <s xml:id="echoid-s4186" xml:space="preserve">nam quælibet alia Hyperbole per B adſcripta cum eo-
              <lb/>
            dem tranſuerſo, ſed cum recto maiori, maior eſt ipſa ABC, quæ verò cum
              <lb/>
            minori, eſt quidem minor, ſed cum ipſi ABC ſit inſcripta, & </s>
            <s xml:id="echoid-s4187" xml:space="preserve">ad partes </s>
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