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

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          <p>
            <s xml:id="echoid-s5916" xml:space="preserve">4. </s>
            <s xml:id="echoid-s5917" xml:space="preserve">Si verò, in ſecunda figura, datum punctum G fuerit in maiori ſemi-
              <lb/>
            axe, at diſtet à vertice B per interuallum G B non maius dimidio recti
              <lb/>
            B F, ipſa G D, in qua centrum, erit _MAXIMA_, & </s>
            <s xml:id="echoid-s5918" xml:space="preserve">reliqua G B _MINIMA_.</s>
            <s xml:id="echoid-s5919" xml:space="preserve"/>
          </p>
          <note symbol="a" position="right" xml:space="preserve">9. huius
            <lb/>
          ad nu. 1. 2.</note>
          <p>
            <s xml:id="echoid-s5920" xml:space="preserve">5. </s>
            <s xml:id="echoid-s5921" xml:space="preserve">At ſi in eadem figura datum punctum G item fuerit, in maiori ſemi-
              <lb/>
            axe E B, ſed diſter à vertice B per interuallum maius dimidio recti B F
              <lb/>
            (nam ſemi-axis maior E B, eſt ſemper maior ſemi-recto B F, cum totus
              <lb/>
            axis B D ſit maior toto recto B F) _MAXIMA_ erit GD, in qua centrum:</s>
            <s xml:id="echoid-s5922" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0213-02" xlink:href="note-0213-02a" xml:space="preserve">6. huius.</note>
            _MINIMA_ verò venabitur ſic.</s>
            <s xml:id="echoid-s5923" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5924" xml:space="preserve">Cum ſit B G maior ſemi-recto B F, habebit E B ad B G minorem ra-
              <lb/>
            tionem, quàm E B ad ſemi-rectum B F, vel ſumptis duplis, quàm tranſ-
              <lb/>
            uerſum D B ad rectum B F, ſuntque hæ rationes maioris inæqualitatis:
              <lb/>
            </s>
            <s xml:id="echoid-s5925" xml:space="preserve">Itaque diuidatur B G in H, ita vt E H ad H G ſit vt D B ad B F, & </s>
            <s xml:id="echoid-s5926" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-0213-03" xlink:href="note-0213-03a" xml:space="preserve">16. h.</note>
            H applicetur I H K, & </s>
            <s xml:id="echoid-s5927" xml:space="preserve">iungantur G I, G K: </s>
            <s xml:id="echoid-s5928" xml:space="preserve">nam ipſæ, quæ ſunt ęquales,
              <lb/>
            erunt _MINIMAE_.</s>
            <s xml:id="echoid-s5929" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5930" xml:space="preserve">Quoniam ducta I L contingente, hæc axi occurret in L: </s>
            <s xml:id="echoid-s5931" xml:space="preserve">& </s>
            <s xml:id="echoid-s5932" xml:space="preserve">cum
              <note symbol="d" position="right" xlink:label="note-0213-04" xlink:href="note-0213-04a" xml:space="preserve">25. pri-
                <lb/>
              mi conic.</note>
            E H ad H G, vt tranſuerſum D B ad rectum B F, ſumpta communi altitu-
              <lb/>
            dine H L, erit rectangulum E H L ad G H L, vt tranſuerſum ad rectum,
              <lb/>
            ſed eſt quoque rectangulum E H L ad quadratum H I, vt
              <note symbol="e" position="right" xlink:label="note-0213-05" xlink:href="note-0213-05a" xml:space="preserve">37. ibid.</note>
            ad rectum, ergo rectangulum E H L ad G H L, eſt vt idem E H L ad qua-
              <lb/>
            dratum H I, quare rectangulum G H L æquale eſt quadrato H I: </s>
            <s xml:id="echoid-s5933" xml:space="preserve">eſtque
              <lb/>
            H I ipſi G L perpendicularis, ergo angulus G I L rectus erit, & </s>
            <s xml:id="echoid-s5934" xml:space="preserve">I L ſectio-
              <lb/>
            nem contingit in I, à quo ducta eſt I G perpendicularis, & </s>
            <s xml:id="echoid-s5935" xml:space="preserve">maiori axi
              <lb/>
            occurrens, quapropter G I erit _MINIMA_, eſtque G K æqualis G I. </s>
            <s xml:id="echoid-s5936" xml:space="preserve">
              <note symbol="f" position="right" xlink:label="note-0213-06" xlink:href="note-0213-06a" xml:space="preserve">11. h.</note>
            de in hoc caſu duæ erunt _MINIMAE_, & </s>
            <s xml:id="echoid-s5937" xml:space="preserve">vna tantùm _MAXIMA_.</s>
            <s xml:id="echoid-s5938" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5939" xml:space="preserve">6. </s>
            <s xml:id="echoid-s5940" xml:space="preserve">Si verò datum punctum G fuerit in axe minori, vt in tertia figura, & </s>
            <s xml:id="echoid-s5941" xml:space="preserve">
              <lb/>
            diſtantia G B ſit non minor dimidio recti lateris B E: </s>
            <s xml:id="echoid-s5942" xml:space="preserve">(quæ G B omnino
              <lb/>
            maior erit ſemi-axe B E, vt ad finem 9. </s>
            <s xml:id="echoid-s5943" xml:space="preserve">huius monuimus) tunc ipſa G B
              <lb/>
            erit _MAXIMA_, & </s>
            <s xml:id="echoid-s5944" xml:space="preserve">G D _MINIMA_, vel punctum G cadat infra D; </s>
            <s xml:id="echoid-s5945" xml:space="preserve">vel
              <note symbol="g" position="right" xlink:label="note-0213-07" xlink:href="note-0213-07a" xml:space="preserve">9. huius
                <lb/>
              ad num. 3.</note>
            pra inter D, & </s>
            <s xml:id="echoid-s5946" xml:space="preserve">E. </s>
            <s xml:id="echoid-s5947" xml:space="preserve">Nam ſi caderet in ipſo puncto D (dummodo D B ſit
              <lb/>
            vt ponitur, nempe non minor dimidio recti) ipſa D B eſſet _MAXIMA_,
              <lb/>
            nec daretur _MINIMA_, cum hæc in punctum euaneſcat.</s>
            <s xml:id="echoid-s5948" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5949" xml:space="preserve">7. </s>
            <s xml:id="echoid-s5950" xml:space="preserve">Verùm, ſi datum punctum G ſit in axe minori, ſed diſtet à vertice B
              <lb/>
            per interuallum minus dimidio recti B E, & </s>
            <s xml:id="echoid-s5951" xml:space="preserve">cadat infra centrum E, vel
              <lb/>
            inter E, & </s>
            <s xml:id="echoid-s5952" xml:space="preserve">D; </s>
            <s xml:id="echoid-s5953" xml:space="preserve">vt in quarta figura, aut infra D, vt in quinta. </s>
            <s xml:id="echoid-s5954" xml:space="preserve">Cum ſit
              <lb/>
            G B minor ſemi-recto, & </s>
            <s xml:id="echoid-s5955" xml:space="preserve">E B æqualis ſemi-tranſuerſo B D, habebit G B
              <lb/>
            ad B E minorem rationem, quàm ſemi-rectum ad ſemi-tranſuerſum, vel
              <lb/>
            quàm rectum F B ad tranſuerſum B D. </s>
            <s xml:id="echoid-s5956" xml:space="preserve">Diuidatur ergo B E in H, ita vt
              <lb/>
            G H ad H E, ſit vt rectum F B ad B D tranſuerſum, & </s>
            <s xml:id="echoid-s5957" xml:space="preserve">per H agatur
              <note symbol="h" position="right" xlink:label="note-0213-08" xlink:href="note-0213-08a" xml:space="preserve">16. h.</note>
            dinata H I, & </s>
            <s xml:id="echoid-s5958" xml:space="preserve">I L ſectionem contingens, & </s>
            <s xml:id="echoid-s5959" xml:space="preserve">axi occurrens in L, iunga-
              <lb/>
            turque G I. </s>
            <s xml:id="echoid-s5960" xml:space="preserve">Dico G I eſſe _MAXIMAM_.</s>
            <s xml:id="echoid-s5961" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5962" xml:space="preserve">Cum ſit enim G H ad H E, vt F B ad B D, ſumpta communi altitudi-
              <lb/>
            ne H L erit rectangulum G H L ad E H L, vt F B ad B D, vel vt
              <note symbol="i" position="right" xlink:label="note-0213-09" xlink:href="note-0213-09a" xml:space="preserve">37. primi
                <lb/>
              conic.</note>
            dratum G
              <unsure/>
            I
              <unsure/>
            H ad idem rectangulum E H L, quare rectangulum G H L æ-
              <lb/>
            quale eſt quadrato G H, eſtque H I perpendicularis ad G L; </s>
            <s xml:id="echoid-s5963" xml:space="preserve">ergo angu-
              <lb/>
            lus G I L rectus erit, eſtque I L ſectionem contingens in L, à quo ducta
              <lb/>
            eſt I G perpendicularis, & </s>
            <s xml:id="echoid-s5964" xml:space="preserve">minori axi in G, occurrens, quare ipſa G I
              <note symbol="l" position="right" xlink:label="note-0213-10" xlink:href="note-0213-10a" xml:space="preserve">11. h.</note>
            _MAXIMA_, & </s>
            <s xml:id="echoid-s5965" xml:space="preserve">eſt G K æqualis G I: </s>
            <s xml:id="echoid-s5966" xml:space="preserve">ergo ex G duæ erunt _MAXIMAE. </s>
            <s xml:id="echoid-s5967" xml:space="preserve"/>
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