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

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          <head xml:id="echoid-head305" xml:space="preserve">THEOR. XXXV. PROP. LIV.</head>
          <p>
            <s xml:id="echoid-s7076" xml:space="preserve">Si Conus rectus plano per axem ſecetur, per in quo verticem du-
              <lb/>
            cta ſit quędam linea, quę non in directum ſit poſita cum aliquo late-
              <lb/>
            rum trianguli per axem perque ipſam agatur planum, quod rectum
              <lb/>
            ſit ad idem planum, per axem ductum: </s>
            <s xml:id="echoid-s7077" xml:space="preserve">Huiuſmodi planum in ipſo
              <lb/>
            tantùm vertice coni ſuperficiem continget, quæ tota cadet ad alte-
              <lb/>
            ram partem ducti plani.</s>
            <s xml:id="echoid-s7078" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7079" xml:space="preserve">SIt conus rectus A B C plano per axem B D ſectus efficiente triangulum
              <lb/>
            A B C, in cuius plano, & </s>
            <s xml:id="echoid-s7080" xml:space="preserve">per verticem B ſit quælibet linea E B F, non
              <lb/>
            tamen cum aliquo laterum B A, B C ſit in directũ poſita, per quam tranſeat
              <lb/>
            planum G H I K, quod ad planum per axem A B C ſit rectum. </s>
            <s xml:id="echoid-s7081" xml:space="preserve">Dico tale
              <lb/>
            planum G I in nullo alio puncto, quàm in vertice B conicam ſuperficiem
              <lb/>
            contingere, &</s>
            <s xml:id="echoid-s7082" xml:space="preserve">c.</s>
            <s xml:id="echoid-s7083" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7084" xml:space="preserve">Quoniam ſi recta E B F ęquidiſtat
              <lb/>
              <figure xlink:label="fig-0255-01" xlink:href="fig-0255-01a" number="211">
                <image file="0255-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0255-01"/>
              </figure>
            ipſi A C baſi trianguli per axem, an-
              <lb/>
            guli interiores E B D, A D B duobus
              <lb/>
            rectis æquales erunt, ſed A D B re-
              <lb/>
            ctus eſt, cum ſit axis B D plano baſis
              <lb/>
            A C perpendicularis, quare, & </s>
            <s xml:id="echoid-s7085" xml:space="preserve">an-
              <lb/>
            gulus E B D rectus erit, ſed planum
              <lb/>
            A B C ponitur rectum ad planum G
              <lb/>
            I, & </s>
            <s xml:id="echoid-s7086" xml:space="preserve">in eo ad communem horum ſe-
              <lb/>
            ctionem E B F ducta eſt perpendi-
              <lb/>
            cularis D B, ergo ipſa D B erit
              <note symbol="a" position="right" xlink:label="note-0255-01" xlink:href="note-0255-01a" xml:space="preserve">4. defin.
                <lb/>
              vndec. E-
                <lb/>
              lem.</note>
            cta ad planum G I, eſtque eadem B
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            D recta ad planum baſis A C, quare
              <lb/>
            duo plana G I, A C inter ſe æquidiſtant, atque eſt punctum B in vno
              <note symbol="b" position="right" xlink:label="note-0255-02" xlink:href="note-0255-02a" xml:space="preserve">14. vnd.
                <lb/>
              Elem.</note>
            no G I, & </s>
            <s xml:id="echoid-s7087" xml:space="preserve">circuli peripheria A C in altero A C, ergo recta B A, quæ ma-
              <lb/>
            nente puncto B circa peripheriam C A circumducitur conicam ſuperficiem
              <lb/>
            deſcribens, hoc eſt ipſa conica ſuperficies tota cadet inter plana ęquidiſtan-
              <lb/>
            tia (vbicunque enim ducatur planum per axem, habentur communes æqui-
              <lb/>
            diſtantium planorum fectiones inter ſe parallelę, inter quas cadit communis
              <lb/>
            ſectio ſecantis plani cum ſuperficie) ac ideò planum G I in ipſo tantùm ver-
              <lb/>
            tice B, coni ſuperficiem continget.</s>
            <s xml:id="echoid-s7088" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s7089" xml:space="preserve">Si verò recta F B E conueniet cum C A, vt in E; </s>
            <s xml:id="echoid-s7090" xml:space="preserve">patet, dum triangulum
              <lb/>
            B E D circa axim B D conuerti concipitur, rectam B E coni B E L ſuperfi-
              <lb/>
            ciem deſcribere, cuius triangulum per axem eſt B E L idem cum plano A B
              <lb/>
            C, cui rectum eſt planum G I ductum per latus B E, quare idem planum G
              <lb/>
            I continget conicam B E L in ipſo tantùm latere B E, ſed latus B E
              <note symbol="c" position="right" xlink:label="note-0255-03" xlink:href="note-0255-03a" xml:space="preserve">53. h.</note>
            tingit conicam B C in vnico tantùm vertice B, ergo planum G I conicam
              <lb/>
            A B C in ipſo tantùm vertice B contingit, ac propterea ipſa coni ſuperficies
              <lb/>
            cadit tota infra planum G I. </s>
            <s xml:id="echoid-s7091" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s7092" xml:space="preserve"/>
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