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

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            <s xml:id="echoid-s4052" xml:space="preserve">
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            ptarum, ſed eſſe MAXIMAS quoque earum, quæ ad partes verticum in-
              <lb/>
            ſcribuntur; </s>
            <s xml:id="echoid-s4053" xml:space="preserve">id ſequenti Theoremate, in angulo, & </s>
            <s xml:id="echoid-s4054" xml:space="preserve">qualibet coni-ſectione,
              <lb/>
            vel circulo conſequetur, ſimulque dabitur Methodus ipſis inſcribendi ſimiles
              <lb/>
            Ellipſes, quæ ſucceſsiuè ſe mutuò, & </s>
            <s xml:id="echoid-s4055" xml:space="preserve">anguli, vel ſectionum latera contin-
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            gant.</s>
            <s xml:id="echoid-s4056" xml:space="preserve"/>
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        <div xml:id="echoid-div393" type="section" level="1" n="166">
          <head xml:id="echoid-head171" xml:space="preserve">THEOR. XXXVI. PROP. LXXVI.</head>
          <p>
            <s xml:id="echoid-s4057" xml:space="preserve">Ellipſes inſcriptæ eidem angulo, vel Parabolæ, vel Hyperbolę,
              <lb/>
            aut portioni Ellipticæ, vel circulari, quæ non excedat Ellipſis, vel
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            circuli dimidium, ſe mutuò, & </s>
            <s xml:id="echoid-s4058" xml:space="preserve">anguli latera, vel ſectionem, vel
              <lb/>
            circulum contingentes, & </s>
            <s xml:id="echoid-s4059" xml:space="preserve">quarum diagonales menſalium, quibus
              <lb/>
            inſcribuntur, inter ſe æquidiſtent, ſunt ſimiles, & </s>
            <s xml:id="echoid-s4060" xml:space="preserve">quæ propior eſt
              <lb/>
            vertici, minor eſt remotiori.</s>
            <s xml:id="echoid-s4061" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4062" xml:space="preserve">SIt ABC, vel angulusrectilineus, vt in prima figura, vel Parabolæ, vel
              <lb/>
            Hyperbolæ, aut portio non maior ſemi-circuli, vel ſemi-Ellipſis dimi-
              <lb/>
            dio, vt in ſecunda, cuius vertex B, diameter BD, & </s>
            <s xml:id="echoid-s4063" xml:space="preserve">circa ipſius ſegmentum
              <lb/>
            DE, inter applicatas AC, IF, ducta diagonali AF, ſecan@ diametrum ED
              <lb/>
            in K, & </s>
            <s xml:id="echoid-s4064" xml:space="preserve">applicata per K recta GKH, per extrema G, E, H, D,
              <note symbol="a" position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">Coroll.
                <lb/>
              57. h.</note>
            Ellipſis GEHD, quæ per Scholium 62. </s>
            <s xml:id="echoid-s4065" xml:space="preserve">huius, menſali AIFC, hoc eſt dato
              <lb/>
            angulo, vel ſectioni crit inſcripta. </s>
            <s xml:id="echoid-s4066" xml:space="preserve">Et per I ducta IL parallela diagonali AF,
              <lb/>
            diametrum ſecan@ in O, conſimili conſtructione, ac ſupra, deſcribatur in
              <lb/>
            menſali INLF Ellipſis PMQE. </s>
            <s xml:id="echoid-s4067" xml:space="preserve">Dico primùm has Ellipſes inter ſe ſimiles
              <lb/>
            eſſe.</s>
            <s xml:id="echoid-s4068" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4069" xml:space="preserve">Nam, in prima figura, proportio rectanguli GKH, ad rectangulum AKF,
              <lb/>
            componitur ex ratione GK ad KA, ſiue (per triangulorum ſimilitudinem)
              <lb/>
            PO ad OI, & </s>
            <s xml:id="echoid-s4070" xml:space="preserve">ex ratione HK ad KF, ſiue QO ad OL, ſed etiam proportio re-
              <lb/>
            ctanguli POQ, IOL, ex ijſdem rationibus componitur, quare in triangulo,
              <lb/>
            rectangulum GKH ad AKF, eſt vt rectangulum POQ ad IOL.</s>
            <s xml:id="echoid-s4071" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4072" xml:space="preserve">Iam, in ſecunda figura, eadem ratione, vt in 64. </s>
            <s xml:id="echoid-s4073" xml:space="preserve">huius, oſtendetur huiuſ-
              <lb/>
            modi Ellipſium centra cadere infra K, O, nempe in R, S, per quæ ſi appli-
              <lb/>
            centur RT, SV, ipſæ, diagonales ſecabunt in T, V; </s>
            <s xml:id="echoid-s4074" xml:space="preserve">
              <gap/>
            cum ſit DR ęqua-
              <lb/>
            lis RE, erit AT æqualis TF, ob parallelas; </s>
            <s xml:id="echoid-s4075" xml:space="preserve">item IV æqualis VL; </s>
            <s xml:id="echoid-s4076" xml:space="preserve">ſuntque
              <lb/>
            AF, IL æquidiſtanter ductæ in ſectione, vel circulo, quare iuncta TV erit
              <lb/>
            earundem æquidiſtantium diameter, quæ producta ad aliud punctum, præ-
              <lb/>
            ter B, ſectioni occurret, vt in Z, eritque ſectionis, vel circuli diameter: </s>
            <s xml:id="echoid-s4077" xml:space="preserve">
              <note symbol="b" position="left" xlink:label="note-0144-02" xlink:href="note-0144-02a" xml:space="preserve">28. ſec.
                <lb/>
              conic.</note>
            ergo ex verticibus B, Z, agantur BX, ZX ordinatim applicatis GH, AF æ-
              <lb/>
            quidiſtantes, hæ ſectionem contingent, & </s>
            <s xml:id="echoid-s4078" xml:space="preserve">ſimul conuenient in X; </s>
            <s xml:id="echoid-s4079" xml:space="preserve">
              <note symbol="c" position="left" xlink:label="note-0144-03" xlink:href="note-0144-03a" xml:space="preserve">17. pri-
                <lb/>
              mi conic.</note>
              <note symbol="d" position="left" xlink:label="note-0144-04" xlink:href="note-0144-04a" xml:space="preserve">59. h.</note>
            rectangulum GKH ad rectangulum AKF, vt quadratum BX ad quadratum
              <lb/>
            ZX; </s>
            <s xml:id="echoid-s4080" xml:space="preserve">item erit rectangulum POQ ad IOL, vt idem quadratum BX ad
              <note symbol="e" position="left" xlink:label="note-0144-05" xlink:href="note-0144-05a" xml:space="preserve">17. tertij
                <lb/>
              conic.</note>
            ZX, quapropter rectangulum GKH ad AKF, erit vt rectangulum POQ ad
              <lb/>
            IOL, quod etiam ſuperius in prima figura demonſtratum fuit. </s>
            <s xml:id="echoid-s4081" xml:space="preserve">Itaque, cum
              <lb/>
            ſit in vtraque, rectangulum GKH ad AKF, vt rectangulum POQ ad IOL, &</s>
            <s xml:id="echoid-s4082" xml:space="preserve"/>
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