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

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            <s xml:id="echoid-s7684" xml:space="preserve">
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            ipſis æqualiter inclinata; </s>
            <s xml:id="echoid-s7685" xml:space="preserve">quare applicatæ in ſectione I G H ad diametrum
              <lb/>
            B G æquidiſtabunt applicatis in ſectione A B C ad eandem diametrum,
              <lb/>
            quarum vna eſt A C per verticé G ducta, cum in G ſit bifariam ſecta; </s>
            <s xml:id="echoid-s7686" xml:space="preserve">ergo
              <lb/>
            ipſa A C continget in G ſectionem I G H.</s>
            <s xml:id="echoid-s7687" xml:space="preserve"/>
          </p>
          <note symbol="a" position="right" xml:space="preserve">ibidem.</note>
          <p style="it">
            <s xml:id="echoid-s7688" xml:space="preserve">Sed hoc idem breuiùs, tùm in angulo, tùm in qualibet coni-ſectione,
              <lb/>
            omiſſo precedenti Lemmate.</s>
            <s xml:id="echoid-s7689" xml:space="preserve"/>
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          <figure number="226">
            <image file="0275-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0275-01"/>
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          <p>
            <s xml:id="echoid-s7690" xml:space="preserve">COncedatur ſectionem I G H occurrere rectæ A C in alio puncto quàm
              <lb/>
            G, quod ſit K. </s>
            <s xml:id="echoid-s7691" xml:space="preserve">Dico tamen punctum K idem eſſe ac G.</s>
            <s xml:id="echoid-s7692" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7693" xml:space="preserve">Quoniam erit A K æqualis G C, ſed eſt quoque A G æqualis
              <note symbol="b" position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">8. ſec.
                <lb/>
              conic. &
                <lb/>
              ex 1. Co-
                <lb/>
              roll 46. h.</note>
            G C, ergo A K, & </s>
            <s xml:id="echoid-s7694" xml:space="preserve">A G ſunt æquales, ſed hæ habent communes terminos
              <lb/>
            ad A, ergo, & </s>
            <s xml:id="echoid-s7695" xml:space="preserve">punctum K congruet cum G. </s>
            <s xml:id="echoid-s7696" xml:space="preserve">Quare ipſa baſis A C con-
              <lb/>
            tingit omnino ſectionem I G H in G.</s>
            <s xml:id="echoid-s7697" xml:space="preserve"/>
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            <s xml:id="echoid-s7698" xml:space="preserve">Ampliùs, in prima figura, iungatur E H, quæ eſt diameter
              <note symbol="c" position="right" xlink:label="note-0275-03" xlink:href="note-0275-03a" xml:space="preserve">8. pr. h.</note>
            Hyperbolæ I G H, & </s>
            <s xml:id="echoid-s7699" xml:space="preserve">in ſecunda ex H ducatur vnius ſectionis diameter H
              <lb/>
            E, quæ erit quoque diameter alterius (cum ponantur concentricæ, &</s>
            <s xml:id="echoid-s7700" xml:space="preserve">c.) </s>
            <s xml:id="echoid-s7701" xml:space="preserve">Si
              <lb/>
            ergo hæc diameter E H producatur, ipſa ſecabit interiorem ſectionem I G
              <lb/>
            H in aliquo puncto, vt in L, ex quo ducatur in ſectione A B F recta M L N
              <lb/>
            ipſi D F æquidiſtans.</s>
            <s xml:id="echoid-s7702" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7703" xml:space="preserve">Et quoniam, in ſingulis, figuris D F eſt bifariam ſecta in H, erit quoque
              <lb/>
            M N bifariam ſecta in L (cum M N ex conſtructione æquidiſter ordinatim
              <lb/>
            ductæ D F in eadem ſectione A B F) ſed ſectio I G tranſit per L, quare
              <lb/>
            ſectio ipſa I G continget omnino rectam M N in L (quod ijſdem rationi-
              <lb/>
            bus, ac ſupra de A C oſtenſum fuit, demonſtrabitur) ergo portio M E N
              <lb/>
            æquabitur portioni A B C, ſed portio quoque D E F æquatur eidem
              <note symbol="d" position="right" xlink:label="note-0275-04" xlink:href="note-0275-04a" xml:space="preserve">45. h.</note>
            tioni A B C, ex hypotheſi, quare portiones M E N, D E F inter ſe æqua-
              <lb/>
            les erunt, ſuntque de eodem angulo, vel de eadem coni- ſectione, vel cir-
              <lb/>
            culo, & </s>
            <s xml:id="echoid-s7704" xml:space="preserve">circa communem diametrum E H L, & </s>
            <s xml:id="echoid-s7705" xml:space="preserve">ipſarum baſes ſimul æqui-
              <lb/>
            diſtant, qua propter, & </s>
            <s xml:id="echoid-s7706" xml:space="preserve">baſes quoque ſimul in totum congruent, nempe M
              <lb/>
            N cum D F, ac ideò punctum L cum puncto H. </s>
            <s xml:id="echoid-s7707" xml:space="preserve">Recta igitur D F, quæ
              <lb/>
            eadem eſt cum M N, contingit ſectionem I G in H. </s>
            <s xml:id="echoid-s7708" xml:space="preserve">Quod tandem erat
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            demonſtrandum.</s>
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