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

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          <p>
            <s xml:id="echoid-s953" xml:space="preserve">
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            velvt AB ad BC, vt ſuperius oſtendimus: </s>
            <s xml:id="echoid-s954" xml:space="preserve">& </s>
            <s xml:id="echoid-s955" xml:space="preserve">permutan-
              <lb/>
              <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a" number="21">
                <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0046-01"/>
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            do, & </s>
            <s xml:id="echoid-s956" xml:space="preserve">per conuerſionem rationis, AD ad DB vt DB ad
              <lb/>
            DC. </s>
            <s xml:id="echoid-s957" xml:space="preserve">Quod in prima figura oſtendendum erat.</s>
            <s xml:id="echoid-s958" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s959" xml:space="preserve">In ſecunda verò: </s>
            <s xml:id="echoid-s960" xml:space="preserve">cum ſit AB ad BC, vt BC ad BE, erit
              <lb/>
            diuidendo, & </s>
            <s xml:id="echoid-s961" xml:space="preserve">permutando, AC ad CE vt BC ad BE, vel
              <lb/>
            vt AB ad BC, ex conſtructione: </s>
            <s xml:id="echoid-s962" xml:space="preserve">quod ſerua.</s>
            <s xml:id="echoid-s963" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s964" xml:space="preserve">Et cum ſit AD ad DC vt quadratũ AB ad BC, & </s>
            <s xml:id="echoid-s965" xml:space="preserve">qua-
              <lb/>
            dratum AB ad BC vt linea AB ad BE, ex conſtructione,
              <lb/>
            erit AD ad DC vt AB ad BE, & </s>
            <s xml:id="echoid-s966" xml:space="preserve">per conuerſionem ratio-
              <lb/>
            nis, permutando, conuertendo, diuidendo, & </s>
            <s xml:id="echoid-s967" xml:space="preserve">iterum
              <lb/>
            conuertendo AD ad DB, vt AC ad CE, vel vt AB ad
              <lb/>
            BC, vt modò oſtenſum fuit, & </s>
            <s xml:id="echoid-s968" xml:space="preserve">permutando, conuerten-
              <lb/>
            d
              <unsure/>
            o, per conuerſionem rationis, & </s>
            <s xml:id="echoid-s969" xml:space="preserve">diuidendo AD ad DB vt BD ad DC. </s>
            <s xml:id="echoid-s970" xml:space="preserve">Quod
              <lb/>
            erat in ſecunda demonſtrandum.</s>
            <s xml:id="echoid-s971" xml:space="preserve"/>
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        <div xml:id="echoid-div74" type="section" level="1" n="45">
          <head xml:id="echoid-head50" xml:space="preserve">ALITER idem breuiùs.</head>
          <p>
            <s xml:id="echoid-s972" xml:space="preserve">Ijſdem poſitis: </s>
            <s xml:id="echoid-s973" xml:space="preserve">dico iterùm vt in præcedenti.</s>
            <s xml:id="echoid-s974" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s975" xml:space="preserve">DEſcribatur ſuper AD ſemicirculus AED, & </s>
            <s xml:id="echoid-s976" xml:space="preserve">per Cerigatur CE diame-
              <lb/>
            tro AD perpendicularis, iunganturque DE, AE, & </s>
            <s xml:id="echoid-s977" xml:space="preserve">BE, quæ produ-
              <lb/>
            cta in ſecuuda figura, occurrat cum AF ipſi CE parallela in puncto F.</s>
            <s xml:id="echoid-s978" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s979" xml:space="preserve">Iam in vtraque figura, cum
              <lb/>
              <figure xlink:label="fig-0046-02" xlink:href="fig-0046-02a" number="22">
                <image file="0046-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0046-02"/>
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            ſit per hypoteſim quadratum
              <lb/>
            AB ad BC, vt recta AD ad DC,
              <lb/>
            vel vt quadratum AD ad DE,
              <lb/>
            vel vt quadratum AE ad EC,
              <lb/>
            ob triangulorum ſimilitudiné,
              <lb/>
            erit recta AB ad BC, vt recta
              <lb/>
            AE ad EC: </s>
            <s xml:id="echoid-s980" xml:space="preserve">quare in prima fi-
              <lb/>
            gura erit angulus AEB, æqua-
              <lb/>
            lis angulo BEC, ſed angulus
              <lb/>
            BAE æquatur angulo D E C,
              <lb/>
            quare duo ſimul AEB, BAE,
              <lb/>
            ſiue vnicus DBE, æqualis erit
              <lb/>
            duobus ſimul BEC, DEC, ſiue vnico DEB, ergo BD eſt æqualis ipſi DE. </s>
            <s xml:id="echoid-s981" xml:space="preserve">In
              <lb/>
            ſecunda verò, cum ſit AB ad BC, vel FA ad EC, vt AE ad EC erunt AF,
              <lb/>
            A E interſe æquales, vnde angulus AEF, æqualis angulo AFE ſiue paralle-
              <lb/>
            larum externo CEB, ſed eſt AEF æqualis duobus ſimul ABE, EAB, quare
              <lb/>
            & </s>
            <s xml:id="echoid-s982" xml:space="preserve">CEB ijſdem angulis ABE, EAB æqualis crit, eſtque pars CED æqualis
              <lb/>
            vnico angulo EAD, ergo reliquus angulus DEB reliquo DBE æqualis erit,
              <lb/>
            hoc eſt recta DB æqualis DE. </s>
            <s xml:id="echoid-s983" xml:space="preserve">Itaque in vtraque figura cum DB ſit æqualis
              <lb/>
            DE, ſitque DE media proportionalis inter AD, DC, erit quoq; </s>
            <s xml:id="echoid-s984" xml:space="preserve">DB media
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            inter eaſdem AD, DC. </s>
            <s xml:id="echoid-s985" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s986" xml:space="preserve"/>
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