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

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        <div xml:id="echoid-div227" type="section" level="1" n="104">
          <p>
            <s xml:id="echoid-s2435" xml:space="preserve">
              <pb o="71" file="0095" n="95" rhead=""/>
            quiori, & </s>
            <s xml:id="echoid-s2436" xml:space="preserve">hoc ſemper, &</s>
            <s xml:id="echoid-s2437" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2438" xml:space="preserve">Quapropter huiuſmodi Hyperbolæ ſunt ſemper
              <lb/>
            ſimul recedentes. </s>
            <s xml:id="echoid-s2439" xml:space="preserve">Quod ſecundò. </s>
            <s xml:id="echoid-s2440" xml:space="preserve">&</s>
            <s xml:id="echoid-s2441" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2442" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2443" xml:space="preserve">Præterea ſit TX aſymptotos inſcriptæ DBE, & </s>
            <s xml:id="echoid-s2444" xml:space="preserve">VZ aſymptotos circum-
              <lb/>
            ſcriptæ, quæ contingentem GB productam ſecent in X, Z; </s>
            <s xml:id="echoid-s2445" xml:space="preserve">& </s>
            <s xml:id="echoid-s2446" xml:space="preserve">cum huiuſmo-
              <lb/>
            di Hyperbole ſint ſimiles, ſintque earum aſymptoti VZ, TX ad partes ęqua-
              <lb/>
            lium inclinationum ductæ, erit angulus ZVB æqualis angulo XTB,
              <note symbol="a" position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">Coroll.
                <lb/>
              40. h.</note>
            TX æquidiſtat VZ, ſed eſt VZ. </s>
            <s xml:id="echoid-s2447" xml:space="preserve">Aſymptotos circumſcriptæ, vnde TX pro-
              <lb/>
            ducta ſecabit circumſcriptam Hyperbolen ABC; </s>
            <s xml:id="echoid-s2448" xml:space="preserve">ſecet ergo eam in 2, &</s>
            <s xml:id="echoid-s2449" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0095-02" xlink:href="note-0095-02a" xml:space="preserve">11. h.</note>
            per 2 applicetur 3 2 4 5 alteram aſymptoton, inſcriptam ſectionem, ac
              <lb/>
            diametrum ſecans in 3,4,5 dico huiuſmodi Hyperbolas, licet ſemper inter
              <lb/>
            ſe magis recedant, nnnquam tamen ad interuallum peruenire æquale inter-
              <lb/>
            uallo 3 2, quod inter æquidiſtantes aſymptotos intercedit, & </s>
            <s xml:id="echoid-s2450" xml:space="preserve">iuxta ordina-
              <lb/>
            tim ductas metitur.</s>
            <s xml:id="echoid-s2451" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2452" xml:space="preserve">Nam cum in ſimilibus Hyperbolis ABC, DBE, ex æqualibus, immo ex
              <lb/>
            eodem diametri ſegmento B 5, ducta ſit quædam applicata 5 4 2 3 ſimi-
              <lb/>
            limium Hyperbolarum aſymptotos ſecans in 2, 3; </s>
            <s xml:id="echoid-s2453" xml:space="preserve">erit intercepta
              <note symbol="c" position="right" xlink:label="note-0095-03" xlink:href="note-0095-03a" xml:space="preserve">40. h.</note>
            applicatæ portio 3 2 in Hyperbola maiorum laterum, maior intercepta
              <lb/>
            portione 2 4, in Hyperbola minorum. </s>
            <s xml:id="echoid-s2454" xml:space="preserve">Ampliùs applicata infra 3 2 4 5,
              <lb/>
            qualibet alia 6 7 8 9; </s>
            <s xml:id="echoid-s2455" xml:space="preserve">erit ob eandem rationem, & </s>
            <s xml:id="echoid-s2456" xml:space="preserve">portio 6 7 maior por-
              <lb/>
            tione 8 9, quare addita communi 7 8; </s>
            <s xml:id="echoid-s2457" xml:space="preserve">erit 6 8 ſiue 3 2 maior 7 9, & </s>
            <s xml:id="echoid-s2458" xml:space="preserve">
              <lb/>
            hoc ſemper, vbicunque ſit intercepta 8 9 infra 2 4 licet ipſae; </s>
            <s xml:id="echoid-s2459" xml:space="preserve">interceptæ
              <lb/>
            continuè augeantur. </s>
            <s xml:id="echoid-s2460" xml:space="preserve">Vnde ſimiles Hyperbolæ per eundem verticem ſimul
              <lb/>
            adſcriptę, quamuis ſint ſemper magis recedentes ad interuallum tamen non
              <lb/>
            perueniunt æquale cuidam dato interuallo. </s>
            <s xml:id="echoid-s2461" xml:space="preserve">Quod erat vltimò, &</s>
            <s xml:id="echoid-s2462" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2463" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div232" type="section" level="1" n="105">
          <head xml:id="echoid-head110" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s2464" xml:space="preserve">HInc eſt, quod ſimiles Hyperbolæ per eundem verticem ſimul adſcriptę
              <lb/>
            habent aſymptotos parallelas, & </s>
            <s xml:id="echoid-s2465" xml:space="preserve">aſymptotos inſcriptæ ſecat Hyper-
              <lb/>
            bolen circumſcriptam: </s>
            <s xml:id="echoid-s2466" xml:space="preserve">nam vltimò loco oſtẽdimus TX ęquidiſtare ipſi VZ,
              <lb/>
            & </s>
            <s xml:id="echoid-s2467" xml:space="preserve">ſecare inſcriptam in 2.</s>
            <s xml:id="echoid-s2468" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div233" type="section" level="1" n="106">
          <head xml:id="echoid-head111" xml:space="preserve">THEOR. XXII. PROP. XXXXII.</head>
          <p>
            <s xml:id="echoid-s2469" xml:space="preserve">Parabolæ congruentes, per diuerſos vertices ſimul adſcriptæ,
              <lb/>
            ſunt inter ſe nunquam coeuntes, & </s>
            <s xml:id="echoid-s2470" xml:space="preserve">in infinitum productæ ad ſe pro-
              <lb/>
            pius accedunt, & </s>
            <s xml:id="echoid-s2471" xml:space="preserve">ad interuallum perueniunt minus quolibet dato
              <lb/>
            interuallo.</s>
            <s xml:id="echoid-s2472" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2473" xml:space="preserve">SInt duæ congruentes Parabolæ ABC, DEF per diuerſos vertices B, E,
              <lb/>
            ſimul adſcriptæ, quarum recta latera ſint BG, EH (quæ inter ſe æqua-
              <lb/>
            lia erunt, cum ſectiones ponantur congruentes.) </s>
            <s xml:id="echoid-s2474" xml:space="preserve">Dico primùm has in
              <note symbol="a" position="right" xlink:label="note-0095-04" xlink:href="note-0095-04a" xml:space="preserve">1. Co-
                <lb/>
              roll. 19. h.</note>
            finitum productas nunquam inter ſe conuenire.</s>
            <s xml:id="echoid-s2475" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2476" xml:space="preserve">Nam producta contingente HE ſectioni ABC occurrent in A, & </s>
            <s xml:id="echoid-s2477" xml:space="preserve">C, hæc
              <lb/>
            erit quoque ordinatim ducta in ſectione ABC (cum ſint ſectiones ſimul ad-
              <lb/>
            ſcriptæ) & </s>
            <s xml:id="echoid-s2478" xml:space="preserve">Parabole DEF tota cadet infra contingentem AEC, </s>
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