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

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        <div xml:id="echoid-div69" type="section" level="1" n="42">
          <pb o="25" file="0045" n="45" rhead=""/>
        </div>
        <div xml:id="echoid-div70" type="section" level="1" n="43">
          <head xml:id="echoid-head48" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s925" xml:space="preserve">EX hucuſque demonſtratis liceat animaduertere quamcumque
              <lb/>
            aſymptoton quodam-modo eſſe primam ex centro ducibilium,
              <lb/>
            ſed Hyperbolæ non occurrentium; </s>
            <s xml:id="echoid-s926" xml:space="preserve">itemque eſſe primam ſibi ipſi
              <lb/>
            æquidiſtantium, ſed Hyperbolen non ſecantium.</s>
            <s xml:id="echoid-s927" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s928" xml:space="preserve">QVæcunque enim educta ex C diuidens angulum DCE ſecat Hyperbo-
              <lb/>
            len, quæcunque verò ex C ducta extra CD, Hyperbolæ quidem non
              <lb/>
            occurrit, cum neque ipſa CD interior, cum ſectione conueniat.
              <lb/>
            </s>
            <s xml:id="echoid-s929" xml:space="preserve">Quare angulus DCE dici poterit MINIMVS ex centro C Hyperbolen com-
              <lb/>
            prehendentium, rectis lineis nunquam ei occurrentibus.</s>
            <s xml:id="echoid-s930" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s931" xml:space="preserve">Item quælibet SX aſymptoto CD æquidiſtanter ducta intra angulũ DCE,
              <lb/>
            Hyperbolen ſecat, quælibet verò extra angulum ducta eidem CD parallela,
              <lb/>
            nunquam conuenit cum CD, & </s>
            <s xml:id="echoid-s932" xml:space="preserve">eò minus cum ſectione: </s>
            <s xml:id="echoid-s933" xml:space="preserve">ex quo aſymptoton
              <unsure/>
              <lb/>
            Hyperbolæ appellari quodammodo poſſet vltima tangentium Hyperbolen,
              <lb/>
            ad infinitum tamen interuallum. </s>
            <s xml:id="echoid-s934" xml:space="preserve">Nam, quæcumque contingens Hyperbo-
              <lb/>
            len ad finitam diſtantiam, ſecat ſemper diametrum CB infra C, & </s>
            <s xml:id="echoid-s935" xml:space="preserve">quò pun-
              <lb/>
            ctum contactus remotius fuerit à vertice eò magis occurſus contingẽtis cum
              <lb/>
            diametro, centro C fiet propior; </s>
            <s xml:id="echoid-s936" xml:space="preserve">donec, cum punctum contactus per infini-
              <lb/>
            tum interuallum abierit à centro, prædictus occurſus cum ipſo centro con-
              <lb/>
            ueniat.</s>
            <s xml:id="echoid-s937" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s938" xml:space="preserve">Sed ne ſuſcipiendam materiam interpellare nobis ſit opus, cum in ipſius
              <lb/>
            progreſſu Parabolæ quadratura indigeamus, inter alias, quas habemus,
              <lb/>
            apponemus hic̀ tantùm eam, quæ, licet expeditior non ſit, nonnulla tamen
              <lb/>
            Lemmata, ac Theoremata præmittit, quorum prima ad aliquas de MA-
              <lb/>
            XIMIS, & </s>
            <s xml:id="echoid-s939" xml:space="preserve">MINIMIS propoſitiones omnino ſunt neceſſaria.</s>
            <s xml:id="echoid-s940" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div71" type="section" level="1" n="44">
          <head xml:id="echoid-head49" xml:space="preserve">LEMMA III. PROP. XII.</head>
          <p>
            <s xml:id="echoid-s941" xml:space="preserve">Si fuerit vt recta AD ad DC, ita quadratum AB ad BC. </s>
            <s xml:id="echoid-s942" xml:space="preserve">Dico
              <lb/>
            tres AD, DB, DC eſſe in continua eademque ratione geometrica.</s>
            <s xml:id="echoid-s943" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s944" xml:space="preserve">NAm ſumpta BE tertia proportionali poſt AB, BC;
              <lb/>
            </s>
            <s xml:id="echoid-s945" xml:space="preserve">
              <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a" number="20">
                <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0045-01"/>
              </figure>
            cum ſit in prima figura, AB ad BC, vt BC ad BE,
              <lb/>
            erit componendo AB cum BC ad BC, vt BC cum BE ad
              <lb/>
            BE, & </s>
            <s xml:id="echoid-s946" xml:space="preserve">permutando, AB cum BC, ſiue AC, ad BC cum
              <lb/>
            BE, ſiue ad CE, vt BC ad BE, vel vt AB ad BC, ex con-
              <lb/>
            ſtructione: </s>
            <s xml:id="echoid-s947" xml:space="preserve">quod memento.</s>
            <s xml:id="echoid-s948" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s949" xml:space="preserve">Et cum ſit, ex ſuppoſitione, linea AD ad DC vt qua-
              <lb/>
            dratum AB ad BC, & </s>
            <s xml:id="echoid-s950" xml:space="preserve">quadratum AB ad BC, vt linea AB
              <lb/>
            ad BE, ex conſtructione, erit AD ad DC, vt AB ad BE,
              <lb/>
            & </s>
            <s xml:id="echoid-s951" xml:space="preserve">per conuerſionem rationis, & </s>
            <s xml:id="echoid-s952" xml:space="preserve">permutando, & </s>
            <s xml:id="echoid-s953" xml:space="preserve">iterum
              <lb/>
            per conuerſionem rationis AD ad DB, vt AC ad </s>
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