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

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          <p>
            <s xml:id="echoid-s2132" xml:space="preserve">
              <pb o="61" file="0085" n="85" rhead=""/>
            genti parallela, erit hæc vna ordinatim ad diametrum ſemi - applicatarum,
              <lb/>
            datumq; </s>
            <s xml:id="echoid-s2133" xml:space="preserve">interuallum ſuperabit: </s>
            <s xml:id="echoid-s2134" xml:space="preserve">vnde patet propoſitum. </s>
            <s xml:id="echoid-s2135" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s2136" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2137" xml:space="preserve"/>
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        </div>
        <div xml:id="echoid-div197" type="section" level="1" n="93">
          <head xml:id="echoid-head98" xml:space="preserve">THEOR. IV. PROP. XXXIII.</head>
          <p>
            <s xml:id="echoid-s2138" xml:space="preserve">Parabolæ inæqualium laterum per eundem verticem ſimul adſcri-
              <lb/>
            ptæ, ſunt inter ſe nunquam alibi coeuntes, & </s>
            <s xml:id="echoid-s2139" xml:space="preserve">inſcripta eſt ea, cuius re-
              <lb/>
            ctum latus minus eſt, ſuntque, in infinitum productæ, iuxta intercepta
              <lb/>
            applicatarum ſegmenta ſemper magis recedentes, & </s>
            <s xml:id="echoid-s2140" xml:space="preserve">ad interuallum
              <lb/>
            perueniunt maius quolibet dato interuallo.</s>
            <s xml:id="echoid-s2141" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2142" xml:space="preserve">SInt duæ Parabolæ ABC, DBE, per eundem verticem B ſimul adſcriptę, qua-
              <lb/>
            rum communis diameter B H, & </s>
            <s xml:id="echoid-s2143" xml:space="preserve">rectum ſectionis ABC ſit linea B F, D B E
              <lb/>
            verò ſit minor B G. </s>
            <s xml:id="echoid-s2144" xml:space="preserve">Dico primùm has nunquam alibi ſimul conuenire, & </s>
            <s xml:id="echoid-s2145" xml:space="preserve">DBE
              <lb/>
            inſcriptam eſſe. </s>
            <s xml:id="echoid-s2146" xml:space="preserve">Hoc enim iam patet ex 2. </s>
            <s xml:id="echoid-s2147" xml:space="preserve">Coroll. </s>
            <s xml:id="echoid-s2148" xml:space="preserve">19. </s>
            <s xml:id="echoid-s2149" xml:space="preserve">huius.</s>
            <s xml:id="echoid-s2150" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2151" xml:space="preserve">Ampliùs, dico has in infinitum productas ſemper
              <lb/>
              <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a" number="55">
                <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0085-01"/>
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            eſſe inter ſe magis recedentes.</s>
            <s xml:id="echoid-s2152" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2153" xml:space="preserve">Ductis enim regulis F O, G P, & </s>
            <s xml:id="echoid-s2154" xml:space="preserve">applicatis vbi-
              <lb/>
              <figure xlink:label="fig-0085-02" xlink:href="fig-0085-02a" number="56">
                <image file="0085-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0085-02"/>
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            cunque duabus ADH, IL M; </s>
            <s xml:id="echoid-s2155" xml:space="preserve">quæ productę ſecent
              <lb/>
            regulas in P, O, N, R; </s>
            <s xml:id="echoid-s2156" xml:space="preserve">manifeſtum iam eſt ex 1. </s>
            <s xml:id="echoid-s2157" xml:space="preserve">h.
              <lb/>
            </s>
            <s xml:id="echoid-s2158" xml:space="preserve">has regulas inter ſe æquidiſtare: </s>
            <s xml:id="echoid-s2159" xml:space="preserve">& </s>
            <s xml:id="echoid-s2160" xml:space="preserve">cum ſit vt qua-
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            dratum I M ad M L, ita recta R M ad M N, vel vt
              <note symbol="a" position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">6. Co-
                <lb/>
              rol. 19. h.</note>
            H ad H P, vel vt quadratum A H ad H D, erit etiam
              <lb/>
            recta IM ad ML, vt AH ad H D, & </s>
            <s xml:id="echoid-s2161" xml:space="preserve">per conuerſionẽ
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            rationis, & </s>
            <s xml:id="echoid-s2162" xml:space="preserve">permutando I M ad A H, vt I L ad A D,
              <lb/>
              <note symbol="b" position="right" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">32. h.</note>
            ſed eſt IM maior AH, quare, & </s>
            <s xml:id="echoid-s2163" xml:space="preserve">IL erit maior AD.</s>
            <s xml:id="echoid-s2164" xml:space="preserve"> Quod ſecundò, &</s>
            <s xml:id="echoid-s2165" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2166" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2167" xml:space="preserve">Demũ dico, has aliquando peruenire ad interuallũ maius quolibet dato NO.</s>
            <s xml:id="echoid-s2168" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2169" xml:space="preserve">Fiat vt AD ad DH, vel vt IL ad LM, quod idem eſt, (modò enim oſtẽdimus
              <lb/>
            omnes huiuſmodi applicatas proportionaliter diuidi à Parabola B D L) ita datũ
              <lb/>
            interuallum N O ad aliud O P, & </s>
            <s xml:id="echoid-s2170" xml:space="preserve">ducta ex vertice contingente B Q R ſumatur
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            B R æqualis P N, & </s>
            <s xml:id="echoid-s2171" xml:space="preserve">B Q æqualis PO, & </s>
            <s xml:id="echoid-s2172" xml:space="preserve">per R agatur R I diametro B M æquidi-
              <lb/>
              <note symbol="c" position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">26. pr.
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              conic.</note>
            ſtans, quæ Parabolæ ABC occurrat in I, & </s>
            <s xml:id="echoid-s2173" xml:space="preserve">per I applicetur IL M. </s>
            <s xml:id="echoid-s2174" xml:space="preserve">Erit ergo I M æqualis R B, ſiue æqualis NP, eſtque vt IL ad LM, ita NO ad OP, ex con-
              <lb/>
            ſtructione, quare IL ipſi N O ęqualis erit, ſed applicatę infra I L inter Parabolas
              <lb/>
            excedunt ipſam IL, vti nuper oſtendimus: </s>
            <s xml:id="echoid-s2175" xml:space="preserve">quare huiuſmodi Parabolæ ad inter-
              <lb/>
            uallum perueniunt maius dato iuteruallo N O. </s>
            <s xml:id="echoid-s2176" xml:space="preserve">Quod vltimò erat, &</s>
            <s xml:id="echoid-s2177" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2178" xml:space="preserve"/>
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        <div xml:id="echoid-div201" type="section" level="1" n="94">
          <head xml:id="echoid-head99" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s2179" xml:space="preserve">_HV_C Franciſcum Barocium ſubaudiet fortaſſe aliquis admurmurantem,
              <lb/>
            nos, qui de aſymptoticis lineis mutuam acceſsionem, vel receſsionem
              <lb/>
            perpendendam ſuſcepimus, æquidiſtantium linearum ſegmentis, inter
              <lb/>
            conuergentes, ac diuergentes aſymptotos interceptis vſos fuiſſe, veluti
              <lb/>
            in præcedenti, vbi iuxta lineas, ſiue portiones A D, I L ex ordinatim
              <lb/>
            app licatis ad communem diametrum, datarum ſectionum diſtantias commetimur;
              <lb/>
            </s>
            <s xml:id="echoid-s2180" xml:space="preserve">dum tamen ipſæ à breuiſsimis, ſeu MINIMIS lineis ſint determinandæ, atque </s>
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