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

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        <div xml:id="echoid-div377" type="section" level="1" n="159">
          <head xml:id="echoid-head164" xml:space="preserve">LEMMA IX. PROP. LXXII.</head>
          <p>
            <s xml:id="echoid-s3915" xml:space="preserve">Dato angulo rectilineo ABC, cuius diameter BDE, & </s>
            <s xml:id="echoid-s3916" xml:space="preserve">applica-
              <lb/>
            ta ADC: </s>
            <s xml:id="echoid-s3917" xml:space="preserve">oportet ex C, infra ADC, ſecantem ducere CEF, ita vt
              <lb/>
            ſi ex E, & </s>
            <s xml:id="echoid-s3918" xml:space="preserve">F applicentur EH, FG ipſi ADC parallelæ, quadratum
              <lb/>
            HE ad rectangulum DEG, datam habeaa
              <unsure/>
            t rationem O ad P.</s>
            <s xml:id="echoid-s3919" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3920" xml:space="preserve">SVmatur Q media proportionalis inter O, & </s>
            <s xml:id="echoid-s3921" xml:space="preserve">P; </s>
            <s xml:id="echoid-s3922" xml:space="preserve">& </s>
            <s xml:id="echoid-s3923" xml:space="preserve">ex D, niſi DA ſit dia-
              <lb/>
            metro BD perpendicularis, erigatur DL, & </s>
            <s xml:id="echoid-s3924" xml:space="preserve">fiat vt O ad Q, ita DA ad
              <lb/>
            DL, iunctaque BLM, ſumatur LM æqualis LD, & </s>
            <s xml:id="echoid-s3925" xml:space="preserve">per M demittatur ME per-
              <lb/>
            pendicularis ipſi BDE: </s>
            <s xml:id="echoid-s3926" xml:space="preserve">dico per punctum E quæſitam ſecantem tranſire.</s>
            <s xml:id="echoid-s3927" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3928" xml:space="preserve">Nam iuncta MD, ductaque MN ipſi BM
              <lb/>
              <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a" number="107">
                <image file="0140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0140-01"/>
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            perpendiculari cum ſint anguli L M N,
              <lb/>
            LDN recti, erũt anguli NMD, NDM duo-
              <lb/>
            bus rectis minores; </s>
            <s xml:id="echoid-s3929" xml:space="preserve">quare MN ipſi DE oc-
              <lb/>
            curret in N; </s>
            <s xml:id="echoid-s3930" xml:space="preserve">cumque angulus LMD, ęqua-
              <lb/>
            lis ſit angulo LDM, erunt reſidui ex rectis
              <lb/>
            DMN, MDN æquales, hoc eſt ND æqua-
              <lb/>
            lis NM, facto igitur centro N, interuallo
              <lb/>
            ND, deſcribatur circulus DMG, qui vtrã-
              <lb/>
            que LD, LM, continget in D, M, cum an-
              <lb/>
            guli ad D, M ſint recti: </s>
            <s xml:id="echoid-s3931" xml:space="preserve">ducatur tandem
              <lb/>
            EH parallela ad DA.</s>
            <s xml:id="echoid-s3932" xml:space="preserve"/>
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            <s xml:id="echoid-s3933" xml:space="preserve">Iam cum BM circulum DMG contingat
              <lb/>
            in M, ſitque ME diametro DG perpendi-
              <lb/>
            cularis erit GB ad BD, vt GE ad ED, &</s>
            <s xml:id="echoid-s3934" xml:space="preserve">
              <note symbol="a" position="left" xlink:label="note-0140-01" xlink:href="note-0140-01a" xml:space="preserve">36. pri-
                <lb/>
              mi conic.</note>
            permutando BG ad GE, vt BD ad DE, ſed
              <lb/>
            eſt BG maior GE, ergo & </s>
            <s xml:id="echoid-s3935" xml:space="preserve">BD maior DE,
              <lb/>
            eſtque AD æqualis DC, & </s>
            <s xml:id="echoid-s3936" xml:space="preserve">anguli ad verticem D æquales, quare iunctæ
              <lb/>
            BA, CE, productę, conuenient ſimul ad partes A, E, vt in F eritque FB
              <note symbol="b" position="left" xlink:label="note-0140-02" xlink:href="note-0140-02a" xml:space="preserve">71. h.</note>
            BA, vt FH ad HA, & </s>
            <s xml:id="echoid-s3937" xml:space="preserve">permutando BF ad FH, vt BE ad AH, vel vt BD ad
              <lb/>
            DE, vel vt BG ad GE, & </s>
            <s xml:id="echoid-s3938" xml:space="preserve">diuidendo BH ad HF, vt BE ad EG, quare iuncta
              <lb/>
            FG ipſis EH, DA æquidiſtabit. </s>
            <s xml:id="echoid-s3939" xml:space="preserve">Et quoniam eſt HE ad EB, vt AD ad DB,
              <lb/>
            & </s>
            <s xml:id="echoid-s3940" xml:space="preserve">BE ad EM, vt BD ad DL (ob triangulorum ſimilitudinem) erit ex æquo
              <lb/>
            HE ad EM, vt AD ad DL, & </s>
            <s xml:id="echoid-s3941" xml:space="preserve">quadratum HE ad quadratum EM, hoc eſt ad
              <lb/>
            rectangulum DEG, vt quadratum AD ad DL, vel vt quadratum O ad qua-
              <lb/>
            dratum Q, vel vt linea O ad P. </s>
            <s xml:id="echoid-s3942" xml:space="preserve">Quod erat faciendum.</s>
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