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

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          <p>
            <s xml:id="echoid-s586" xml:space="preserve">
              <pb o="13" file="0033" n="33" rhead=""/>
            L, & </s>
            <s xml:id="echoid-s587" xml:space="preserve">cadat primùm applicata F G infra contingentem D E, ſitque G M
              <lb/>
            ipſi E A, & </s>
            <s xml:id="echoid-s588" xml:space="preserve">G N ipſi E B parallela.</s>
            <s xml:id="echoid-s589" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s590" xml:space="preserve">Iam cum ſit G F parallela ad E
              <lb/>
              <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a" number="9">
                <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0033-01"/>
              </figure>
            D, G M ad E A, & </s>
            <s xml:id="echoid-s591" xml:space="preserve">G N ad E B,
              <lb/>
            erit triangulum A D E ſimile triã-
              <lb/>
            gulo M F G, & </s>
            <s xml:id="echoid-s592" xml:space="preserve">triangulum E D B
              <lb/>
            ſimile triangulo G F N, quare vt
              <lb/>
            A D ad D E, ita M F ad F G, & </s>
            <s xml:id="echoid-s593" xml:space="preserve">
              <lb/>
            vt E D ad D B, ita G F ad F N;
              <lb/>
            </s>
            <s xml:id="echoid-s594" xml:space="preserve">ſuntque A D, D E, D B continuę
              <lb/>
            proportionales, vnde M F, F G,
              <lb/>
            F N, erunt quoque proportiona-
              <lb/>
            les, ſiue rectangulum M F N ęqua-
              <lb/>
            bitur quadrato F G.</s>
            <s xml:id="echoid-s595" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s596" xml:space="preserve">Præterea cum B E circulum
              <lb/>
            contingat, & </s>
            <s xml:id="echoid-s597" xml:space="preserve">E B ſecet, erit an-
              <lb/>
            gulus D E B æqualis angulo B A
              <lb/>
            E, ſed (cum triangula B E C, B A
              <lb/>
            E in ſemicirculo ſint ſimilia) eſt
              <lb/>
            quoque angulus B E C, æqualis
              <lb/>
            lis angulo B A E, ergo angulus
              <lb/>
            D E B, ſiue alternus E I G æqua-
              <lb/>
            lis erit angulo B E C, ergo linea
              <lb/>
            G I ipſi G E æqualis. </s>
            <s xml:id="echoid-s598" xml:space="preserve">Item angu-
              <lb/>
            lus O E A æquatur angulo A B E in alterna portione, ſiue angulo A E C,
              <lb/>
            eſtque angulus O E A alterno G L E æqualis, vnde anguli A E C, G L E
              <lb/>
            æquales erunt, quare linea G L æqualis eidem G E; </s>
            <s xml:id="echoid-s599" xml:space="preserve">erunt ergo L G, G I
              <lb/>
            inter ſe æquales, ſed eſt G F maior I F, habebit ergo L G ad G F mino-
              <lb/>
            rem rationem quàm G I ad I F, & </s>
            <s xml:id="echoid-s600" xml:space="preserve">componendo L F, ad F G, ſiue A F
              <lb/>
            ad F M minorem rationem quàm G F ad F I, vel quàm N F ad F B, qua-
              <lb/>
            re rectangnlum ſub extremis A F, F B, minus erit rectangulo ſub
              <note symbol="a" position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">16 ſept.
                <lb/>
              Pappi.</note>
            dijs M F, F N, ſiue minus quadrato F G. </s>
            <s xml:id="echoid-s601" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s602" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s603" xml:space="preserve">Idem penitùs oſtendetur, quando applicata _F G_ productæ diametro
              <lb/>
            occurrat vltra D; </s>
            <s xml:id="echoid-s604" xml:space="preserve">nam adhibitis angulis ad verticem E, alterniſque pa-
              <lb/>
            rallelarum, item demonſtrabirur _I G_ ipſi _G L_ æqualem eſſe, & </s>
            <s xml:id="echoid-s605" xml:space="preserve">ex _G_ facta
              <lb/>
            fimili conſtructione, demonſtratio, & </s>
            <s xml:id="echoid-s606" xml:space="preserve">concluſio omninò erit eadem, ac
              <lb/>
            ſupra.</s>
            <s xml:id="echoid-s607" xml:space="preserve"/>
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        <div xml:id="echoid-div40" type="section" level="1" n="28">
          <head xml:id="echoid-head33" xml:space="preserve">PROBL. II. PROP. IV.</head>
          <p>
            <s xml:id="echoid-s608" xml:space="preserve">Datæ Hyperbolæ, vel Ellipſi, per punctum in ea datum
              <lb/>
              <note position="right" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">Prop. 34.
                <lb/>
              primi co-
                <lb/>
              nic.</note>
            contingentem lineam ducere.</s>
            <s xml:id="echoid-s609" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s610" xml:space="preserve">SIt Ellipſis, vel Hyperbolæ A B K, cuius tranſuerſum latus ſit B C, & </s>
            <s xml:id="echoid-s611" xml:space="preserve">
              <lb/>
            datum in ſectione punctum ſit A, extra verticem B: </s>
            <s xml:id="echoid-s612" xml:space="preserve">oportet ex A
              <lb/>
            datæ ſectioni contingentem lineam ducere.</s>
            <s xml:id="echoid-s613" xml:space="preserve"/>
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