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

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            <s xml:id="echoid-s8242" xml:space="preserve">
              <pb o="110" file="0296" n="296" rhead=""/>
            longo erunt ipſę C D, E F, in prolato verò erunt axes minores)
              <note symbol="a" position="left" xlink:label="note-0296-01" xlink:href="note-0296-01a" xml:space="preserve">ibidem.</note>
            tiaque ſolidas portiones C A D, E A F, quarum recti Canones erunt ipſæ
              <lb/>
            æquales portiones planæ C A D, E A F. </s>
            <s xml:id="echoid-s8243" xml:space="preserve">Dico tales portiones ſolidas inter
              <lb/>
            ſe æquales eſſe.</s>
            <s xml:id="echoid-s8244" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8245" xml:space="preserve">Nam bifariam ſectis E F in G, & </s>
            <s xml:id="echoid-s8246" xml:space="preserve">C D in H, patet puncta G, H eſſe cen-
              <lb/>
            tra circulorum, ſiue Ellipſium E O F, C P D; </s>
            <s xml:id="echoid-s8247" xml:space="preserve">& </s>
            <s xml:id="echoid-s8248" xml:space="preserve">ſi per punctum G deſcri-
              <lb/>
              <note symbol="b" position="left" xlink:label="note-0296-02" xlink:href="note-0296-02a" xml:space="preserve">4. ſec.
                <lb/>
              Conic &
                <lb/>
              5. 6. 7. pri-
                <lb/>
              mi huius.</note>
            batur in vtraque ſigura eiuſdem nominis Coni-ſectio G H, que ſimilis & </s>
            <s xml:id="echoid-s8249" xml:space="preserve">concentrica ſectioni E A D, & </s>
            <s xml:id="echoid-s8250" xml:space="preserve">qualis in Monito poſt 68. </s>
            <s xml:id="echoid-s8251" xml:space="preserve">h. </s>
            <s xml:id="echoid-s8252" xml:space="preserve">definiuimus,
              <lb/>
            patet inquam ipſam ſectionem G H omnino tranſire per H, ſimulque E F,
              <lb/>
            & </s>
            <s xml:id="echoid-s8253" xml:space="preserve">C D in punctis medijs G, H contingere.</s>
            <s xml:id="echoid-s8254" xml:space="preserve"/>
          </p>
          <note symbol="c" position="left" xml:space="preserve">68. h.</note>
          <p>
            <s xml:id="echoid-s8255" xml:space="preserve">Iam ductis per G, H, rectis I G L, M H N ad axem A B perpendicula-
              <lb/>
            ribus, concipiantur per ipſas duci plana ad planum per axem E A D erecta,
              <lb/>
              <note symbol="d" position="left" xlink:label="note-0296-04" xlink:href="note-0296-04a" xml:space="preserve">4. primi
                <lb/>
              Conic. &
                <lb/>
              12. Arch.
                <lb/>
              de Co-
                <lb/>
              noid. &c.</note>
            quæ efficient in exteriori ſolido circulos circa diametros I L, M N, &</s>
            <s xml:id="echoid-s8256" xml:space="preserve"> communes eorum ſectiones cum planis per E F, C D ductis, erunt rectæ G O, H P, quæ ad planum E A D rectæ erunt (ſunt enim communes ſe- ctiones duorum planorum ad idem planum erectorum) hoc eſt, tùm O G
              <lb/>
              <note symbol="e" position="left" xlink:label="note-0296-05" xlink:href="note-0296-05a" xml:space="preserve">3. vnd.
                <lb/>
              Elem.</note>
            cum vtriſque E F, I L, tùm P H cum vtriſque C D, M N rectos eſſiciet
              <lb/>
              <note symbol="f" position="left" xlink:label="note-0296-06" xlink:href="note-0296-06a" xml:space="preserve">19. ibid.</note>
            angulos; </s>
            <s xml:id="echoid-s8257" xml:space="preserve">vnde in circulis tranſeuntibus per I L, M N, rectangulum I G L
              <lb/>
            æquabitur quadrato G O, & </s>
            <s xml:id="echoid-s8258" xml:space="preserve">re-
              <lb/>
              <figure xlink:label="fig-0296-01" xlink:href="fig-0296-01a" number="241">
                <image file="0296-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0296-01"/>
              </figure>
            ctangulum M H N quadrato H
              <lb/>
            P, atque ipſæ G O, H P erunt
              <lb/>
            circulorum, aut Ellipſium E O
              <lb/>
            F, C P D minores ſemi-axes,
              <lb/>
            in Cono tamen, vel Conoide
              <lb/>
            Parabolico, aut Hyperbolico,
              <lb/>
            vel in Sphæroide oblongo; </s>
            <s xml:id="echoid-s8259" xml:space="preserve">nam
              <lb/>
            in prolato, erunt maiores ſemi-
              <lb/>
            axes: </s>
            <s xml:id="echoid-s8260" xml:space="preserve">ſed rectangula I G L, M
              <lb/>
            H N ſunt æqualia,
              <note symbol="g" position="left" xlink:label="note-0296-07" xlink:href="note-0296-07a" xml:space="preserve">3. Co-
                <lb/>
              roll. 46. h.</note>
            enim æquatur quadrato ſemi-tangentis per verticem interioris ſectionis, &</s>
            <s xml:id="echoid-s8261" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s8262" xml:space="preserve">ergo, & </s>
            <s xml:id="echoid-s8263" xml:space="preserve">quadrato G O, H P æqualia erunt, ſiue ſemi-axis G O æqualis ſe-
              <lb/>
            mi- axi H P; </s>
            <s xml:id="echoid-s8264" xml:space="preserve">ſed circulus, aut Ellipſis E O F ad C P D, eſt vt
              <note symbol="h" position="left" xlink:label="note-0296-08" xlink:href="note-0296-08a" xml:space="preserve">7. Arch.
                <lb/>
              de Co-
                <lb/>
              noid. &c.</note>
            lum ſub E F, G O, ad rectangulum ſub C D, H P, & </s>
            <s xml:id="echoid-s8265" xml:space="preserve">rectangulum ſub E F,
              <lb/>
            G O ad rectangulum ſub C D, H P eſt vt E F ad C D (cum eorum latitu-
              <lb/>
            dines G O, H P ſint æquales) ergo circulus, vel Ellipſis E O F ad C P D
              <lb/>
            erit in ſolido Parabolico, vel Hyperbolico, aut Sphæroide oblongo, vt ma-
              <lb/>
            ior axis E F ad maiorem axim C D, vel in Sphæroide prolato, vt minor
              <lb/>
            axis E F ad minorem C D: </s>
            <s xml:id="echoid-s8266" xml:space="preserve">ſed E F ad C D eſt vt altitudo Canonis C
              <note symbol="i" position="left" xlink:label="note-0296-09" xlink:href="note-0296-09a" xml:space="preserve">65. h.</note>
            D, ad altitudinem Canonis E A F, cum ipſi ſint æquales portiones eiuſ-
              <lb/>
            dem coni-ſectionis, & </s>
            <s xml:id="echoid-s8267" xml:space="preserve">horum Canonum altitudines ſunt eædem, ac
              <note symbol="l" position="left" xlink:label="note-0296-10" xlink:href="note-0296-10a" xml:space="preserve">3. Schol.
                <lb/>
              69. h.</note>
            tudines ſolidarum portionum C A D, E A F, quare circulus, vel Ellipſis E
              <lb/>
            O F ad C P D, erit reciprocè vt altitudo ſolidæ portionis C A D, ad alti-
              <lb/>
            tucinem ſolidæ E A F: </s>
            <s xml:id="echoid-s8268" xml:space="preserve">at huiuſmodi portiones ſunt ſolida
              <note symbol="m" position="left" xlink:label="note-0296-11" xlink:href="note-0296-11a" xml:space="preserve">Coroll.
                <lb/>
              70. h.</note>
            proportionalia, & </s>
            <s xml:id="echoid-s8269" xml:space="preserve">ipſorum baſes altitudinibus reciprocantur, ergo ipſæ ſo-
              <lb/>
            lidæ portiones inter ſe ſunt æquales. </s>
            <s xml:id="echoid-s8270" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s8271" xml:space="preserve"/>
          </p>
          <note symbol="n" position="left" xml:space="preserve">74. h.</note>
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