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

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          <pb o="85" file="0109" n="109" rhead=""/>
          <p>
            <s xml:id="echoid-s2843" xml:space="preserve">Si primùm; </s>
            <s xml:id="echoid-s2844" xml:space="preserve">cum ſit EH æqualis HT, eſſet etiam EH æqualis ST, vnde
              <lb/>
            eius fegmentum EB mins eſſet diſtantia ST. </s>
            <s xml:id="echoid-s2845" xml:space="preserve">Si ſecundùm; </s>
            <s xml:id="echoid-s2846" xml:space="preserve">cum ſit HT æ-
              <lb/>
            qualis HE omnino ST maior eſſet eadem HE, & </s>
            <s xml:id="echoid-s2847" xml:space="preserve">eò maior ipſius ſegmento
              <lb/>
            BE. </s>
            <s xml:id="echoid-s2848" xml:space="preserve">Si tertiùm; </s>
            <s xml:id="echoid-s2849" xml:space="preserve">vt in hac ipſa figura, in qua centrum H interioris cadit inter
              <lb/>
            S, & </s>
            <s xml:id="echoid-s2850" xml:space="preserve">G; </s>
            <s xml:id="echoid-s2851" xml:space="preserve">cum ſit HE æqualis HT, & </s>
            <s xml:id="echoid-s2852" xml:space="preserve">ablata HB maior ablata HS (nam eſt to-
              <lb/>
            ta SB ſecta bifariam in G) erit reliqua BE maior reliqua ST. </s>
            <s xml:id="echoid-s2853" xml:space="preserve">Quapropter in
              <lb/>
            hoc caſu, in quo centrum H interioris cadit vltra centrum G exterioris, vbi-
              <lb/>
            cunq; </s>
            <s xml:id="echoid-s2854" xml:space="preserve">ſit eius incidentia, demonſtratum eſt ſemper diſtantiam verticum B, E,
              <lb/>
            minorem eſſe ipſa ST diſtantia inter ſuperiora extrema tranſuerſorum late-
              <lb/>
            rum ET, BS. </s>
            <s xml:id="echoid-s2855" xml:space="preserve">Quod memento.</s>
            <s xml:id="echoid-s2856" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2857" xml:space="preserve">Ampliùs ſint harum ſectionum recta latera BV, EX, & </s>
            <s xml:id="echoid-s2858" xml:space="preserve">regulæ TV, TX.
              <lb/>
            </s>
            <s xml:id="echoid-s2859" xml:space="preserve">Patet ob ſectionum ſimilitudinem, vt SB ad BV, ita eſſe TE ad EX, ſed an-
              <lb/>
            guli ad B, E, ſunt æquales (cum ſectiones ſint ſimul adſcriptæ, &</s>
            <s xml:id="echoid-s2860" xml:space="preserve">c.) </s>
            <s xml:id="echoid-s2861" xml:space="preserve">quare
              <lb/>
            in triangulis SBV, TEX, anguli ad S, T, æquales erunt, ac ideò regulæ SV,
              <lb/>
            TX inter ſe æquidiſtabunt. </s>
            <s xml:id="echoid-s2862" xml:space="preserve">Cumque ſit ST maior BE, ſi dematur SK ipſi BE
              <lb/>
            æqualis, ducaturque SY parallela ad EX, & </s>
            <s xml:id="echoid-s2863" xml:space="preserve">abſcindatur EL æqualis SY, ac
              <lb/>
            iungantur KY, BL: </s>
            <s xml:id="echoid-s2864" xml:space="preserve">erunt in triangulis KSY, BEL, in quibus latera circum
              <lb/>
            æquales angulos S, E, ſunt æqualia, vtrunque vtrique, anguli quoq; </s>
            <s xml:id="echoid-s2865" xml:space="preserve">SKY,
              <lb/>
            EBL æquales; </s>
            <s xml:id="echoid-s2866" xml:space="preserve">ſuntque alterni, quare KY, & </s>
            <s xml:id="echoid-s2867" xml:space="preserve">BL inter ſe ęquidiſtant, ſed KY
              <lb/>
            ſecat TX, vnde & </s>
            <s xml:id="echoid-s2868" xml:space="preserve">BL producta ſecabit TX, vt in N: </s>
            <s xml:id="echoid-s2869" xml:space="preserve">Iam per N ordinatim
              <lb/>
            ductis æquidiſtans applicetur NQDP, regulam SV, ſecans in Z, communem
              <lb/>
            diametrum in Q, exteriorem ſectionem CBA in P, & </s>
            <s xml:id="echoid-s2870" xml:space="preserve">interiorem in D: </s>
            <s xml:id="echoid-s2871" xml:space="preserve">Cum
              <lb/>
            in triangulo BQN ſit EL ipſi QN parallela, erit BQ ad QN, vt BE ad EL,
              <lb/>
            & </s>
            <s xml:id="echoid-s2872" xml:space="preserve">permutando QB ad BE, vt QN ad EL, ſiue ad SY, vel ZN, & </s>
            <s xml:id="echoid-s2873" xml:space="preserve">per con-
              <lb/>
            uerſionem rationis BQ ad QE, vt NQ ad QZ, vnde rectangulum BQZ
              <note symbol="a" position="right" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            quadratum applicatæ PQ æquale eſt rectangulo EQN, ſiue quadrato appli-
              <lb/>
            catæ DQ ex quo puncta P, D in vnum conueniunt, hoc eſt interior Hyper-
              <lb/>
            bole FED exteriori ABC occurrit in D; </s>
            <s xml:id="echoid-s2874" xml:space="preserve">eademque ratione oſtendetur ipſas
              <lb/>
            ſimul occurrere in F, altero extremo eiuſdem applicatæ DQF, quare in ipſis
              <lb/>
            occurſibus ſe mutuò ſecant: </s>
            <s xml:id="echoid-s2875" xml:space="preserve">quoniam ſi exempli gratia, huiuſmodi ſectiones
              <lb/>
            non ſe ſecarent, ſed contigerent in D, contingerent ſe quoque in F, vt fa-
              <lb/>
            cillimum eſt demonſtrare, ſed Hyperbole ED ſecat omnino rectam GI extra
              <lb/>
            ſectionem BA, vti ſuperius oſtendimus, quare hæc inter ſectio alio in loco
              <lb/>
            cadet quàm in D, pariterque ad alteram partem ſectio EF ſecabit BC in alio
              <lb/>
            puncto, præter in F: </s>
            <s xml:id="echoid-s2876" xml:space="preserve">Quapropter coni-ſectio coni-ſectionem contingeret in
              <lb/>
            duobus punctis D, F, & </s>
            <s xml:id="echoid-s2877" xml:space="preserve">in alijs duobus punctis ſibi ipſis occurrerent, quod
              <lb/>
            eſt impoſſibile: </s>
            <s xml:id="echoid-s2878" xml:space="preserve">vnde in ipſis occurſibus D, F ſe mutuò ſecant; </s>
            <s xml:id="echoid-s2879" xml:space="preserve">quod
              <note symbol="b" position="right" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">37. 4.
                <lb/>
              conic.</note>
            abundanti oſtendere propoſuimus.</s>
            <s xml:id="echoid-s2880" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2881" xml:space="preserve">Si verò centrum H interioris idem fuerit cum G centro exterioris, etiam
              <lb/>
            aſymptotos GI eadem erit cum aſymptoto HM, cum angulus IGB æqualis,
              <lb/>
            vel idem ſit cum angulo MHE; </s>
            <s xml:id="echoid-s2882" xml:space="preserve">Ergo ſimilium concentricarum
              <note symbol="c" position="right" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">Coroll.
                <lb/>
              40. huius.</note>
            larum aſymptoti communes ſunt. </s>
            <s xml:id="echoid-s2883" xml:space="preserve">Quod quartò erat, &</s>
            <s xml:id="echoid-s2884" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2885" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2886" xml:space="preserve">Quod autem ſint ſimul nunquam coeuntes ſatis patet ex prima parte 47.
              <lb/>
            </s>
            <s xml:id="echoid-s2887" xml:space="preserve">huius, vel quàm breuiſſimè ex propoſ. </s>
            <s xml:id="echoid-s2888" xml:space="preserve">208. </s>
            <s xml:id="echoid-s2889" xml:space="preserve">ſeptimi Pappi. </s>
            <s xml:id="echoid-s2890" xml:space="preserve">Quod quintò, &</s>
            <s xml:id="echoid-s2891" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2892" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2893" xml:space="preserve">Si autem centrum H interioris DEF cadat infra G centrum exterioris
              <lb/>
            ABC, vt in ſecunda figura, per verticem E contingenter applicata CEA;
              <lb/>
            </s>
            <s xml:id="echoid-s2894" xml:space="preserve">cum HM ſit intra angulum IGO ab aſymptotis factum, ac ipſi GI </s>
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