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

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          <p>
            <s xml:id="echoid-s4273" xml:space="preserve">
              <pb o="127" file="0151" n="151" rhead=""/>
            quarta, & </s>
            <s xml:id="echoid-s4274" xml:space="preserve">in ipſis concipiatur quædam AC ad diametrum BF ordinatim du-
              <lb/>
            cta; </s>
            <s xml:id="echoid-s4275" xml:space="preserve">oportet per eius terminos A, C, dato angulo, velſectioni, _MAXIMAM_
              <lb/>
            Ellipſim inſcribere, cuius tranſuerſa diameter æqualis ſit datæ lineæ DE,
              <lb/>
            quæ tamen, pro Ellipſi ABCO, quartæ figuræ, minor ſit eius tranſuerſa dia-
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            metro BO.</s>
            <s xml:id="echoid-s4276" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4277" xml:space="preserve">Ducatur ex A,
              <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a" number="118">
                <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0151-01"/>
              </figure>
              <note symbol="a" position="right" xlink:label="note-0151-01" xlink:href="note-0151-01a" xml:space="preserve">2.4. h.</note>
            ABC contingens AK, quæ dia-
              <lb/>
            metro occurret in K, & </s>
            <s xml:id="echoid-s4278" xml:space="preserve">KF,
              <note symbol="b" position="right" xlink:label="note-0151-02" xlink:href="note-0151-02a" xml:space="preserve">24. 25.
                <lb/>
              pr. conic.</note>
            angulo etiam rectilineo, bifariam
              <lb/>
            ſecetur in puncto G, quod in Pa-
              <lb/>
            rabola cadet in ipſō B, cum (ob
              <lb/>
            tangentem AK) ſit KB
              <note symbol="c" position="right" xlink:label="note-0151-03" xlink:href="note-0151-03a" xml:space="preserve">35. pri-
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              mi conic.</note>
            BF, & </s>
            <s xml:id="echoid-s4279" xml:space="preserve">in Hyperbola cadet infra
              <lb/>
            B, cum ſit FB maior BK (ſumpta
              <lb/>
            enim eius tranſuerſa diametro
              <lb/>
              <note symbol="d" position="right" xlink:label="note-0151-04" xlink:href="note-0151-04a" xml:space="preserve">36. pri-
                <lb/>
              mi conic.</note>
            BO, eſt OF ad FB, vt OK ad KB, & </s>
            <s xml:id="echoid-s4280" xml:space="preserve">permutando OF ad OK,
              <lb/>
            vt FB ad BK, ſed eſt OF maior
              <lb/>
            OK, quare, & </s>
            <s xml:id="echoid-s4281" xml:space="preserve">FB erit maior BK)
              <lb/>
            in Ellipſi verò cadet ſupra B, cũ
              <lb/>
            ſit KB maior BF (nam eſt OK ad
              <lb/>
              <note symbol="e" position="right" xlink:label="note-0151-05" xlink:href="note-0151-05a" xml:space="preserve">ibidem.</note>
            KB, vt OF ad FB, & </s>
            <s xml:id="echoid-s4282" xml:space="preserve">KF bifa- riam ſecta eſt in G, ac ideo G ca-
              <lb/>
            det ſupra B.) </s>
            <s xml:id="echoid-s4283" xml:space="preserve">Præterea ad datam rectam DE applicetur parallelogrammum
              <lb/>
            æquale quadrato GF, excedens figura quadrata, idque ſit rectangulum
              <lb/>
            DHE; </s>
            <s xml:id="echoid-s4284" xml:space="preserve">ſumptaque HI media proportionali inter DH, HE, erit rectangulum
              <lb/>
            DHE, ſiue quadratum GF, æquale quadrato HI, ergo rectæ GF, HI æqua-
              <lb/>
            les inter ſe. </s>
            <s xml:id="echoid-s4285" xml:space="preserve">Inſuper ſumatur GL æqualis HE, & </s>
            <s xml:id="echoid-s4286" xml:space="preserve">erit reliqua LF æqualis re-
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            liquæ EI; </s>
            <s xml:id="echoid-s4287" xml:space="preserve">& </s>
            <s xml:id="echoid-s4288" xml:space="preserve">punctum L cadet omnino infra B, ſiue intra angulum, vel ſe-
              <lb/>
            ctionem, cum in angulo, & </s>
            <s xml:id="echoid-s4289" xml:space="preserve">Hyperbola cadat infra G, quod eſt intra angu-
              <lb/>
            lum, vel ſectionem, & </s>
            <s xml:id="echoid-s4290" xml:space="preserve">in Parabola cadat infra G, quod eſt in ipſa ſectione;
              <lb/>
            </s>
            <s xml:id="echoid-s4291" xml:space="preserve">in Ellipſi verò, prædictum punctum L cadet infra B; </s>
            <s xml:id="echoid-s4292" xml:space="preserve">quoniam cum ſit OK
              <lb/>
            ad KB, vt OF ad FB, & </s>
            <s xml:id="echoid-s4293" xml:space="preserve">KF bifariam ſecta in G, per conſtructionem, erit re-
              <lb/>
            ctangulum OGB æquale quadrato GF, (hic notatione dignum
              <note symbol="f" position="right" xlink:label="note-0151-06" xlink:href="note-0151-06a" xml:space="preserve">79. h.</note>
            hanc ipſam affectionem verificari etiam in Hyperbola, nempe rectangulo
              <lb/>
            OGB æquari quadrato GF, vel GK) ſiue quadrato HI, ſiue rectangulo
              <lb/>
            DHE; </s>
            <s xml:id="echoid-s4294" xml:space="preserve">ſed eſt OB maior DE, quare GB erit minor HE, ſiue minor
              <note symbol="g" position="right" xlink:label="note-0151-07" xlink:href="note-0151-07a" xml:space="preserve">80. h.</note>
            hoc eſt punctum L erit quoque intra Ellipſim A B C O. </s>
            <s xml:id="echoid-s4295" xml:space="preserve">Sumatur præte-
              <lb/>
            rea in quacumque figura FN æqualis ID, erit ergo LN æqualis datæ ED
              <lb/>
            (cum ſit quoque LF æqualis EI) & </s>
            <s xml:id="echoid-s4296" xml:space="preserve">punctum N in quarta figura cadet omni-
              <lb/>
            no intra Eilipſim ABCO: </s>
            <s xml:id="echoid-s4297" xml:space="preserve">quoniam cum ſit rectangulum DHE, ſiue NGL,
              <lb/>
            æquale quadrato HI, ſiue GE, & </s>
            <s xml:id="echoid-s4298" xml:space="preserve">ſit etiam rectangulum OGB æquale eidem
              <lb/>
            quadrato GF, vt ſuperiùs demonſtrauimus, erunt rectangulo
              <unsure/>
            OGB, NGL
              <lb/>
            inter ſe æquale
              <unsure/>
            @, & </s>
            <s xml:id="echoid-s4299" xml:space="preserve">ideo, vt OG ad GN, ita LG ad GB, ſed eſt LG maior
              <lb/>
            GB, vt paulò ante oſtendimus, quapropter, & </s>
            <s xml:id="echoid-s4300" xml:space="preserve">OG erit maior GN, ſiue pun-
              <lb/>
            ctum N cadet intra Ellipſim ABCO.</s>
            <s xml:id="echoid-s4301" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4302" xml:space="preserve">Tandem cum trãſuerſo LN, quod æquatur datę lineæ ED, circa </s>
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