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

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        <div xml:id="echoid-div372" type="section" level="1" n="157">
          <p>
            <s xml:id="echoid-s3882" xml:space="preserve">
              <pb o="115" file="0139" n="139" rhead=""/>
            angulo per D adſcripta, cum eodem tranſuerſo
              <lb/>
              <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a" number="105">
                <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0139-01"/>
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            latere DE, ſed cum recto, quod minus ſit recto
              <lb/>
            adſcriptæ DAEC, eſt ipſa minor,
              <note symbol="a" position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">2. Co-
                <lb/>
              roll. 19. h.</note>
            verò cum recto maiori, eſt quidem maior ea-
              <lb/>
            dem, ſed omnino ſecat anguli latera BA, BC,
              <lb/>
            vt ſatis conſtat. </s>
            <s xml:id="echoid-s3883" xml:space="preserve">Ampliùs, Ellipſis, quæ per
              <lb/>
            D ſupra applicatam FG eidem angulo contin-
              <lb/>
            genter inſcribitur, cum tranſuerſo latere ęqua-
              <lb/>
            li ipſo DE, vel dato R (ſi tamen interceptum
              <lb/>
            diametri ſegmentum DB maius fuerit DE) eſt
              <lb/>
            omnino minor prædicta ADCE: </s>
            <s xml:id="echoid-s3884" xml:space="preserve">quare
              <note symbol="b" position="right" xlink:label="note-0139-02" xlink:href="note-0139-02a" xml:space="preserve">64. h.</note>
            eſt _MAXIMA_ inſcripta quæſita. </s>
            <s xml:id="echoid-s3885" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s3886" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3887" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3888" xml:space="preserve">Notandum eſt autem, quod ſi ABC fuerit
              <lb/>
            quælibet coni- ſectio, vel circulus, eadem penitùs conſtructione, ac demon-
              <lb/>
            ſtratione inſcribetur ei _MAXIMA_ Ellipſis ADCE, cum dato tranſuerſo R,
              <lb/>
            quod tamen in Ellipſi, vel circulo, non excedat maius diametri ſegmentum.</s>
            <s xml:id="echoid-s3889" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3890" xml:space="preserve">SI verò data ſit Ellipſis ADCE, & </s>
            <s xml:id="echoid-s3891" xml:space="preserve">per punctum B extra ipſam datum cir-
              <lb/>
            cumſcribendus ſit ei _MINIMV S_ angulus rectilineus. </s>
            <s xml:id="echoid-s3892" xml:space="preserve"> Ducantur ex B
              <note symbol="c" position="right" xlink:label="note-0139-03" xlink:href="note-0139-03a" xml:space="preserve">49. ſec.
                <lb/>
              conic.</note>
            lipſim contingentes BA, BC; </s>
            <s xml:id="echoid-s3893" xml:space="preserve">nam angulus ABC erit _MINIMV S_ circum-
              <lb/>
            ſcriptus quæſitus: </s>
            <s xml:id="echoid-s3894" xml:space="preserve">quoniam ducta AC, ac bifariam ſecta in M, iunctaque
              <lb/>
            BME, ipſa erit Ellipſis diameter, ſimulque dati anguli ABC, cum
              <note symbol="d" position="right" xlink:label="note-0139-04" xlink:href="note-0139-04a" xml:space="preserve">29. ſec.
                <lb/>
              conic.</note>
            ipſi AC æquidiſtanter ductæ ab eadem BM bifariam ſecentur: </s>
            <s xml:id="echoid-s3895" xml:space="preserve">vnde angulus
              <lb/>
            ABC erit datæ Ellipſi ADCE circumſcriptus: </s>
            <s xml:id="echoid-s3896" xml:space="preserve">eritque _MINIMV S_; </s>
            <s xml:id="echoid-s3897" xml:space="preserve">quoniam
              <lb/>
            quæcunque recta, quæ ex B intra angulum ABC ducitur, cum altera con-
              <lb/>
            tingentium minorem angulum conſtituens, neceſſariò ſecat datam Ellipſim
              <lb/>
            ADC: </s>
            <s xml:id="echoid-s3898" xml:space="preserve">quare angulus ABC eſt _MIMIMV S_ circũſcriptus quæſitus. </s>
            <s xml:id="echoid-s3899" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s3900" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3901" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div375" type="section" level="1" n="158">
          <head xml:id="echoid-head163" xml:space="preserve">LEMMA VIII. PROP. LXXI.</head>
          <p>
            <s xml:id="echoid-s3902" xml:space="preserve">Si duæ rectæ AB, CD ſe mutuò ſecent in E, ſitque AE æqualis
              <lb/>
            EB, ſed CE maior ED, dico iunctas CA, BD, ſi producantur, con-
              <lb/>
            uenire ſimul ad partes A, D, vt in F, & </s>
            <s xml:id="echoid-s3903" xml:space="preserve">ſi per D ducatur DG pa-
              <lb/>
            rallela ad AE, eſſe FC ad CA, vt FG ad GA.</s>
            <s xml:id="echoid-s3904" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3905" xml:space="preserve">SVmpta enim EH æquali ipſi EC, erit EH ma-
              <lb/>
              <figure xlink:label="fig-0139-02" xlink:href="fig-0139-02a" number="106">
                <image file="0139-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0139-02"/>
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            ior ED, & </s>
            <s xml:id="echoid-s3906" xml:space="preserve">iuncta BH, in triangulis BEH, AE
              <lb/>
            C erunt latera circùm æquales angulos ad E, æ-
              <lb/>
            qualia: </s>
            <s xml:id="echoid-s3907" xml:space="preserve">quare reliqui anguli EBH, EAC ęquales,
              <lb/>
            vnde BH parallela ad CA, hoc eſt anguli BAG,
              <lb/>
            ABH duobus rectis æquales, ideoque duo BAG,
              <lb/>
            ABD minores duobus rectis: </s>
            <s xml:id="echoid-s3908" xml:space="preserve">occurrit ergo BD
              <lb/>
            cum CA producta ad partes D, A; </s>
            <s xml:id="echoid-s3909" xml:space="preserve">ſitque occur-
              <lb/>
            ſus in F, ex quo ducatur FI parallela ad DG, vel
              <lb/>
            ad AE.</s>
            <s xml:id="echoid-s3910" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3911" xml:space="preserve">Cum ſint ergo triangula FDI, BDE ſimilia, erit
              <lb/>
            FI ad EB, vel ad AE, hoc eſt FC ad CA, vt FD
              <lb/>
            ad DB, vel vt FG ad GA. </s>
            <s xml:id="echoid-s3912" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s3913" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3914" xml:space="preserve"/>
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