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

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          <p>
            <s xml:id="echoid-s2264" xml:space="preserve">
              <pb o="65" file="0089" n="89" rhead=""/>
            maior erit ipſa CE. </s>
            <s xml:id="echoid-s2265" xml:space="preserve">Quò ergo in-
              <lb/>
              <figure xlink:label="fig-0089-01" xlink:href="fig-0089-01a" number="59">
                <image file="0089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0089-01"/>
              </figure>
            terceptæ PO magis remouentur à
              <lb/>
            vertice B, eò ſunt maiores: </s>
            <s xml:id="echoid-s2266" xml:space="preserve">quare
              <lb/>
            huiuſmodi ſectiones inter ſe ſunt
              <lb/>
            ſemper recedentes. </s>
            <s xml:id="echoid-s2267" xml:space="preserve">Quod ſecun-
              <lb/>
            dò, &</s>
            <s xml:id="echoid-s2268" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2269" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2270" xml:space="preserve">Præterea ſit BY ſectionem con-
              <lb/>
            tingens in B, & </s>
            <s xml:id="echoid-s2271" xml:space="preserve">bifariam ſectis trãſ-
              <lb/>
            uerſis lateribus, nempe GB in V,
              <lb/>
            & </s>
            <s xml:id="echoid-s2272" xml:space="preserve">HB in X: </s>
            <s xml:id="echoid-s2273" xml:space="preserve">cum ſit tranſuerſum
              <lb/>
            GB maius BH, erit dimidium BV
              <lb/>
            maius dimidio BX: </s>
            <s xml:id="echoid-s2274" xml:space="preserve">iam ex centro
              <lb/>
            V ducatur VY aſymptotos inſcri-
              <lb/>
            ptæ Hyperbolæ DBE, & </s>
            <s xml:id="echoid-s2275" xml:space="preserve">ex cen-
              <lb/>
            tro X agatur XZ aſymptotos cir-
              <lb/>
            cumſcriptæ ABC, quæ aſymptoti
              <lb/>
            contingentem BI ſecent in Y, Z, & </s>
            <s xml:id="echoid-s2276" xml:space="preserve">
              <lb/>
            per X agatur X & </s>
            <s xml:id="echoid-s2277" xml:space="preserve">parallela ad BY contingentem ſecans in &</s>
            <s xml:id="echoid-s2278" xml:space="preserve">.</s>
          </p>
          <p>
            <s xml:id="echoid-s2279" xml:space="preserve">Itaque quadratum BY ad BZ eſt, vt rectangulum GBF ad rectangulum
              <lb/>
            HBF (vtrumque enim quadratorum eſt quarta pars ſuæ figuræ) vel vt
              <note symbol="a" position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">8. huius.</note>
            GB ad BH, vel ſumptis ſubduplis, vt VB ad BX, vel ob parallelas, vt YB ad
              <lb/>
            B&</s>
            <s xml:id="echoid-s2280" xml:space="preserve">, quare BZ eſt media proportionalis inter BY, & </s>
            <s xml:id="echoid-s2281" xml:space="preserve">B&</s>
            <s xml:id="echoid-s2282" xml:space="preserve">: cum ergo inter pa-
              <lb/>
            rallelas VY, Z& </s>
            <s xml:id="echoid-s2283" xml:space="preserve">recta ZX ſecet alteram parallelarum X& </s>
            <s xml:id="echoid-s2284" xml:space="preserve">in X, ipſa produ-
              <lb/>
            cta ad partes Z ſecabit quoque alteram parallelam VY infra BY: </s>
            <s xml:id="echoid-s2285" xml:space="preserve">vnde ha-
              <lb/>
            rum ſectionum aſymptoti infra contingentem ex vertice inter ſe conueniunt.</s>
            <s xml:id="echoid-s2286" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2287" xml:space="preserve">Amplius cum aſymptotos VY inſcriptæ occurrat diametro BG vltra cen-
              <lb/>
            trum X circumſcriptæ Hyperbolę ABC in puncto V, ipſaque aſymptotos
              <lb/>
            VY conueniat cum XZ aſymptoto circumſcriptæ ABC, vt modò oſtendi-
              <lb/>
            mus, ſi producatur, ſecabit quoque Hyperbolen ABC. </s>
            <s xml:id="echoid-s2288" xml:space="preserve">Quare
              <note symbol="b" position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">35. h.</note>
            inſcriptæ ſecat Hyperbolen circumſcriptam.</s>
            <s xml:id="echoid-s2289" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2290" xml:space="preserve">Tandem cum harum ſectionum aſymptoti infra contingentem BY ſe mu-
              <lb/>
            tuò ſecent, & </s>
            <s xml:id="echoid-s2291" xml:space="preserve">XZ aſymptotos circũſcriptæ BCP, cadat totas extra ipsã BCP,
              <lb/>
            harum aſymptoton occurſus erit extra eandem BCP, vt in 2: </s>
            <s xml:id="echoid-s2292" xml:space="preserve">& </s>
            <s xml:id="echoid-s2293" xml:space="preserve">cum VY 2
              <lb/>
            aſymptotos inſcriptæ, ſecet Hyperbolen BCP circumſcriptam, eſto earum
              <lb/>
            communis ſectio in 3, & </s>
            <s xml:id="echoid-s2294" xml:space="preserve">recta 2 3, producatur ad inferiores partes 4, atque
              <lb/>
            ex 3 ducatur recta 3 5 parallela ad X 2 aſymptoton circumſcriptæ BC 3, quę
              <lb/>
            recta 3 5 nunquam conueniet cum ſectione 3 7 ad inferiores partes,
              <note symbol="c" position="right" xlink:label="note-0089-03" xlink:href="note-0089-03a" xml:space="preserve">34. h.</note>
            etiam recta 3 4 nunquã conuenit cum ſectione BE 6 ad eaſdem partes (nam
              <lb/>
            eſt eius aſymptotos) & </s>
            <s xml:id="echoid-s2295" xml:space="preserve">duæ rectæ 3 5, 3 4 ſunt ſemper ſimul recedentes,
              <lb/>
            & </s>
            <s xml:id="echoid-s2296" xml:space="preserve">ad interuallum perueniunt maius quolibet dato interuallo; </s>
            <s xml:id="echoid-s2297" xml:space="preserve">quare eò ma-
              <lb/>
            gis interuallum ſectionum BC7, BE6, datum quodcunque interuallum ex-
              <lb/>
            cedet. </s>
            <s xml:id="echoid-s2298" xml:space="preserve">Quod erat vltimò demonſtrandum.</s>
            <s xml:id="echoid-s2299" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div214" type="section" level="1" n="98">
          <head xml:id="echoid-head103" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s2300" xml:space="preserve">EX hac manifeſtum fit, quod Hyperbolarum per eundem verticem ſimul
              <lb/>
            adſcriptarum, & </s>
            <s xml:id="echoid-s2301" xml:space="preserve">idem rectum latus habentium aſymptoti infra </s>
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