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

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            <s xml:id="echoid-s733" xml:space="preserve">Siverò CB fuerit maior BA, erit quoque ED maior DA, & </s>
            <s xml:id="echoid-s734" xml:space="preserve">tunc ex edu-
              <lb/>
            cta IDH ſupra ſubiectum planum dematur DH, quæ minor ſit ipſa DA, & </s>
            <s xml:id="echoid-s735" xml:space="preserve">
              <lb/>
            iungatur AH, & </s>
            <s xml:id="echoid-s736" xml:space="preserve">fiat vt HD ad DF, ita DF ad DI; </s>
            <s xml:id="echoid-s737" xml:space="preserve">erit rectangulum HDI æ-
              <lb/>
            quale quadrato DF, ſiue rectangulo EDB, ſed rectangulum EDB maius eſt
              <lb/>
            rectangulo ADB, cum ſit ED maior DA, quare rectangulum HDI maius
              <lb/>
            erit rectangulo ADB. </s>
            <s xml:id="echoid-s738" xml:space="preserve">Iam ex I ducatur IR parallela ad AH, ſecans produ-
              <lb/>
            ctam AD in R; </s>
            <s xml:id="echoid-s739" xml:space="preserve">erit HD ad DA, vt ID ad DR; </s>
            <s xml:id="echoid-s740" xml:space="preserve">ſed HD facta eſt minor DA,
              <lb/>
            ergo & </s>
            <s xml:id="echoid-s741" xml:space="preserve">ID erit minor DR, vnde rectangulum ſub maioribus AD, DR, maius
              <lb/>
            erit rectangulo ſub minoribus HD, DI; </s>
            <s xml:id="echoid-s742" xml:space="preserve">ſed rectangulum HDI demonſtra-
              <lb/>
            tum eſt maius rectangulo ADB, ergo rectangulum ADR eò amplius maius
              <lb/>
            erit rectangulo ADB: </s>
            <s xml:id="echoid-s743" xml:space="preserve">vnde recta BR maior erit recta DB, hoc eſt punctum
              <lb/>
            B cadet inter D, & </s>
            <s xml:id="echoid-s744" xml:space="preserve">R, ſiue inter parallelas AH, IR; </s>
            <s xml:id="echoid-s745" xml:space="preserve">quare iuncta I B, & </s>
            <s xml:id="echoid-s746" xml:space="preserve">pro-
              <lb/>
            ducta conueniet cum producta AH ad partes B, H, veluti in L.</s>
            <s xml:id="echoid-s747" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s748" xml:space="preserve">His itaque conſtructis, & </s>
            <s xml:id="echoid-s749" xml:space="preserve">demonſtratis; </s>
            <s xml:id="echoid-s750" xml:space="preserve">cum factum ſit vt ID ad DF, vel
              <lb/>
            ad DG, ita DG ad DH, ſi circa diametrum IH in plano ſecante deſcribatur
              <lb/>
            circulus ipſe tranſibit per puncta F, G: </s>
            <s xml:id="echoid-s751" xml:space="preserve">ſi ergo intelligatur deſcriptus conus,
              <lb/>
            cuius vertex L, baſis circulus IFHG; </s>
            <s xml:id="echoid-s752" xml:space="preserve">& </s>
            <s xml:id="echoid-s753" xml:space="preserve">in infinitum productus infra baſim,
              <lb/>
            communis ſectio eius conicæ ſuperficiei cum ſubiecto plano ſit linea AMF
              <lb/>
            BGNA. </s>
            <s xml:id="echoid-s754" xml:space="preserve">Dico hanc eſſe Ellipſim quæſitam.</s>
            <s xml:id="echoid-s755" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s756" xml:space="preserve">Eſt enim conus ILH ſectus plano per axem, triangulum facient LIH, & </s>
            <s xml:id="echoid-s757" xml:space="preserve">
              <lb/>
            ſecatur altero plano FBGA, (nempe ſubiecto plano) quod baſi non æquidi-
              <lb/>
            ſtat (cum ſe mutuò ſecent ſecundum rectam FG) & </s>
            <s xml:id="echoid-s758" xml:space="preserve">communis ſectio baſis
              <lb/>
            coni I H, & </s>
            <s xml:id="echoid-s759" xml:space="preserve">ſecantis plani BA eſt recta linea FG, quæ ad IH baſim trianguli
              <lb/>
            per axem eſt ducta perpendicularis, erit, per primam huius, ſectio AMFBGN
              <lb/>
            Ellipſis, cuius vertex B, diameter BA, cui ordinatim ductæ, qualis eſt FG,
              <lb/>
            ad datum angulum P applicantur ex conſtructione. </s>
            <s xml:id="echoid-s760" xml:space="preserve">Cumque factum ſit vt
              <lb/>
            ED ad DF, ita DF ad DB, erit rectangulum EDB ęquale quadrato DF, ſiue
              <lb/>
            rectangulo IDH, vnde rectangulum ADB, ad rectangulum EDB, erit vt
              <lb/>
            idem rectangulum ADB, ad rectangulum IDH; </s>
            <s xml:id="echoid-s761" xml:space="preserve">ſed rectangulum ADB ad
              <lb/>
            EDB, eſt vt AD ad DE, vel vt AB ad BC, ergo rectangulum ADB, ad re-
              <lb/>
            ctangulum IDH, erit vt AB ad BC: </s>
            <s xml:id="echoid-s762" xml:space="preserve">vnde AB eſt latus tranſuerſum, BC ve-
              <lb/>
            rò rectum deſcriptæ Ellipſis BFAG, vt ex prima huius. </s>
            <s xml:id="echoid-s763" xml:space="preserve">Quod erat facien-
              <lb/>
            dum.</s>
            <s xml:id="echoid-s764" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div54" type="section" level="1" n="33">
          <head xml:id="echoid-head38" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s765" xml:space="preserve">CVm ad MAXIMARV M, MINIMARV Mque coni-ſe-
              <lb/>
            ctionum inſcriptibilium, ac circumſcriptibilium inuentionem
              <lb/>
            nobis ſit opus admir andam illam affectionem propagare circa
              <lb/>
            lineas ſemper magis, ac magis inter ſe accedentes, nunquam
              <lb/>
            verò ſimul coeuntes, ab ipſo Apollonio præcipuè animaduerſam inter curuam
              <lb/>
            Hyperbolæ, rectamque lineam, quàm ipſe Aſymptoton appellauit, neceſsè
              <lb/>
            quidem videretur, ad hoc vt integram huius argumenti doctrinam hic ſi-
              <lb/>
            mul habeatur, addere nunc, primam, ſecundam, decimam tertiam, ac de-
              <lb/>
            cimam quartam ſecundi conicorum ad prædictam Aſymptoton ſpectantes; </s>
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