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

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        <div xml:id="echoid-div263" type="section" level="1" n="118">
          <head xml:id="echoid-head123" xml:space="preserve">ALITER.</head>
          <p>
            <s xml:id="echoid-s2765" xml:space="preserve">DVcantur ex communi centro G aſymptoti GX, GZ ſectionis ABC, quę
              <lb/>
            alterius ſimilis, & </s>
            <s xml:id="echoid-s2766" xml:space="preserve">concentricæ ſectionis DEF erunt quoque
              <note symbol="a" position="right" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve">Coroll.
                <lb/>
              40. huius.</note>
            ptoti, & </s>
            <s xml:id="echoid-s2767" xml:space="preserve">ipſi GX, productæ contingentes IB, ME, occurrant in X, Y, & </s>
            <s xml:id="echoid-s2768" xml:space="preserve">per
              <lb/>
            G ſit G 2, regulis HI, LM parallela, recta latera ſecans in 2, & </s>
            <s xml:id="echoid-s2769" xml:space="preserve">3; </s>
            <s xml:id="echoid-s2770" xml:space="preserve">cum ſit GE
              <lb/>
            æqualis GL, & </s>
            <s xml:id="echoid-s2771" xml:space="preserve">GB æqualis GH, erit E 3 æqualis 3 M, & </s>
            <s xml:id="echoid-s2772" xml:space="preserve">B 2 æqualis 2 I,
              <lb/>
            ſiue 3 4, quare E 4 eſt aggregatum E 3 cum B 2. </s>
            <s xml:id="echoid-s2773" xml:space="preserve">Iam cum rectangulum
              <lb/>
            GE 3 ſit quarta pars rectanguli LEM, & </s>
            <s xml:id="echoid-s2774" xml:space="preserve">quadratum EY eiuſdem rectangu-
              <lb/>
            li ſubquadruplum, ergo quadratum EY ęquatur rectangulo GE 3: </s>
            <s xml:id="echoid-s2775" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0107-02" xlink:href="note-0107-02a" xml:space="preserve">8. huius.</note>
            que ratione eſt quadratum BX æquale rectangulo GB 2, ſed rectangulum
              <lb/>
            GE 3 excedit rectangulum GB 2 rectangulo BE 4, ſiue quadrato KE,
              <note symbol="c" position="right" xlink:label="note-0107-03" xlink:href="note-0107-03a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            re quadratum EY ſuperat quadratum BX quadrato EK: </s>
            <s xml:id="echoid-s2776" xml:space="preserve">ſed productis
              <note symbol="*" position="right" xlink:label="note-0107-04" xlink:href="note-0107-04a" xml:space="preserve">46. h.</note>
            plicatis AN, QS vſque ad communes aſymptotos, ipſas, ac ſectiones ſecan-
              <lb/>
            tibus in 5 ADFC 6, & </s>
            <s xml:id="echoid-s2777" xml:space="preserve">in 7 QR 8 9, eſt quadratum EY æquale
              <note symbol="d" position="right" xlink:label="note-0107-05" xlink:href="note-0107-05a" xml:space="preserve">ibidem.</note>
            lo 5 D 6, & </s>
            <s xml:id="echoid-s2778" xml:space="preserve">quadratum BX ęquale rectangulo 5 A 6; </s>
            <s xml:id="echoid-s2779" xml:space="preserve">vnde quadratorum
              <lb/>
            exceſſus æquatur exceſſui rectãgulorum, ſed exceſſus quadratorum eſt qua-
              <lb/>
            dratum EK, & </s>
            <s xml:id="echoid-s2780" xml:space="preserve">exceſſus rectangulorum 5 D 6, 5 A 6 eſt
              <note symbol="e" position="right" xlink:label="note-0107-06" xlink:href="note-0107-06a" xml:space="preserve">179. ſe-
                <lb/>
              pt. Pappi.</note>
            ADC; </s>
            <s xml:id="echoid-s2781" xml:space="preserve">vnde quadratum EK æquatur rectangulo ADC; </s>
            <s xml:id="echoid-s2782" xml:space="preserve">eademque ratione
              <lb/>
            oſtendetur idem quadratum EK æquale rectangulo QR 8, quare rectangu-
              <lb/>
            la ADC, QR 8 inter ſe ſunt æqualia, ideoque R 8 ad DC, vt DA ad QR,
              <lb/>
            ſed eſt R 8 maior DC (cum ſit RS maior DN, & </s>
            <s xml:id="echoid-s2783" xml:space="preserve">S 8 maior NC) ergo AD
              <lb/>
            erit maior QR, & </s>
            <s xml:id="echoid-s2784" xml:space="preserve">hoc ſemper, &</s>
            <s xml:id="echoid-s2785" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2786" xml:space="preserve"> Quod iterum erat ſecundò
              <note symbol="*" position="right" xlink:label="note-0107-07" xlink:href="note-0107-07a" xml:space="preserve">32. h.</note>
            dum.</s>
            <s xml:id="echoid-s2787" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2788" xml:space="preserve">Dico tandem has ſimiles concentricas Hyperbolas in infinitum produ-
              <lb/>
            ctas ad interuallum peruenire minus quolibet dato interuallo R
              <unsure/>
            . </s>
            <s xml:id="echoid-s2789" xml:space="preserve">Nam facta
              <lb/>
            eadem penitus conſtructione, ac in vltima parte 42. </s>
            <s xml:id="echoid-s2790" xml:space="preserve">huius, hoc quod expo-
              <lb/>
            nitur, non abſimili eiuſdem argumento demonſtrabitur. </s>
            <s xml:id="echoid-s2791" xml:space="preserve">Quod vltimò, &</s>
            <s xml:id="echoid-s2792" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2793" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div265" type="section" level="1" n="119">
          <head xml:id="echoid-head124" xml:space="preserve">COROLL. I.</head>
          <p>
            <s xml:id="echoid-s2794" xml:space="preserve">EX hac elicitur ſimilium, & </s>
            <s xml:id="echoid-s2795" xml:space="preserve">concentricarum Hyperbolarum, per diuer-
              <lb/>
            ſos vertices ſimul adſcriptarum, Aſymptotos communes eſſe.</s>
            <s xml:id="echoid-s2796" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div266" type="section" level="1" n="120">
          <head xml:id="echoid-head125" xml:space="preserve">COROLL. II.</head>
          <p>
            <s xml:id="echoid-s2797" xml:space="preserve">COnſtat etiam ex penultima parte huius, in prædictis Hyperbolis rectan-
              <lb/>
            gula ſegmentorum applicatarum vtranque Hyperbolen ſecantium,
              <lb/>
            qualia ſunt rectangula ADC, QR8, omnia inter ſe æqualia eſſe.</s>
            <s xml:id="echoid-s2798" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2799" xml:space="preserve">Quod in prima parte præcedentium 44. </s>
            <s xml:id="echoid-s2800" xml:space="preserve">45. </s>
            <s xml:id="echoid-s2801" xml:space="preserve">47. </s>
            <s xml:id="echoid-s2802" xml:space="preserve">earumque primis Co-
              <lb/>
            rollarijs oſtendimus, vniuerſaliùs ſequenti Theoremate demonſtrabitur.</s>
            <s xml:id="echoid-s2803" xml:space="preserve"/>
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