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

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            omnibus verò per I applicetur ordinatim ad E I recta L I M, quæ rectæ D
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            F æquidiſtabit, & </s>
            <s xml:id="echoid-s8461" xml:space="preserve">per ipſam L I M concipiatur duci planum, quod plano
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            per D F tranſeunti, ſiue baſi portionis ſolidæ D E F æquidiftet, aliam por-
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            tionem ſolidam abſcindens L E M, quæ portioni ſolidæ A B C
              <note symbol="a" position="right" xlink:label="note-0305-01" xlink:href="note-0305-01a" xml:space="preserve">79. h.</note>
            erit; </s>
            <s xml:id="echoid-s8462" xml:space="preserve">ſed ponitur etiam D E F eidem A B C æqualis; </s>
            <s xml:id="echoid-s8463" xml:space="preserve">ergo duæ L E M, D
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            E F inter ſe æquales erunt, ſed vtraque eſt de eodem ſolido, circa commu-
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            nem axim E H I, & </s>
            <s xml:id="echoid-s8464" xml:space="preserve">ſuper baſes parallelas, quare planum baſis ductum per
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            L M, congruet cum plano baſis, quod tranſit per D F, vnde, & </s>
            <s xml:id="echoid-s8465" xml:space="preserve">axis termi-
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            nus I, cum termino axis H. </s>
            <s xml:id="echoid-s8466" xml:space="preserve">Erit ergo axis E I æqualis axi E H. </s>
            <s xml:id="echoid-s8467" xml:space="preserve">Sed in
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            prima, factus fuit E I æqualis B G, & </s>
            <s xml:id="echoid-s8468" xml:space="preserve">in reliquis O E ad E I, vt O B ad
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            B G, quare axis quoque E H, in prima, æquabitur axi B G, in alijs verò
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            erit O E ad E H, vt O B ad B G, & </s>
            <s xml:id="echoid-s8469" xml:space="preserve">conuertendo H E ad E O, vt G B
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            ad B O.</s>
            <s xml:id="echoid-s8470" xml:space="preserve"/>
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            <s xml:id="echoid-s8471" xml:space="preserve">Sint tandem duæ æquales portiones de eodem Cono recto A B C, D B
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            E, quarum recti Canones concipiantur coaptari ſuper eadem ſectione A B
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            E per ſolidi axem ducta, & </s>
            <s xml:id="echoid-s8472" xml:space="preserve">ſint A B C, D B E, quarum baſes A C, D E,
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            & </s>
            <s xml:id="echoid-s8473" xml:space="preserve">diametri B F, B G, (quæ iam ſunt axes ſolidarum portionum.) </s>
            <s xml:id="echoid-s8474" xml:space="preserve">Et
              <note symbol="b" position="right" xlink:label="note-0305-02" xlink:href="note-0305-02a" xml:space="preserve">3. Schol.
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              69. h.</note>
            F cum aſymptotis B A, B C deſcribatur Hyperbole F G; </s>
            <s xml:id="echoid-s8475" xml:space="preserve">quæ omnino
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            continget A C in F, termino axis B F. </s>
            <s xml:id="echoid-s8476" xml:space="preserve">Dico iam extremum G axis
              <note symbol="c" position="right" xlink:label="note-0305-03" xlink:href="note-0305-03a" xml:space="preserve">1. Co-
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              roll. 68. h.</note>
            G, ad eandem quoque ſectionem pertin-
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            gere: </s>
            <s xml:id="echoid-s8477" xml:space="preserve">hoc eſt ſectionem F G ſecare dia-
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              <figure xlink:label="fig-0305-01" xlink:href="fig-0305-01a" number="247">
                <image file="0305-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0305-01"/>
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            metrum B G in puncto G. </s>
            <s xml:id="echoid-s8478" xml:space="preserve">Si poffibile
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            eſt ſectio F G alibi ſecet axim B G, vt in-
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            fra G in puncto H, & </s>
            <s xml:id="echoid-s8479" xml:space="preserve">per H ducatur L
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            H M ipſi D E æquidiſtans: </s>
            <s xml:id="echoid-s8480" xml:space="preserve">erit D G ad
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            G E, vt L H ad H M, eſtque D G ęqua-
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            lis G E, quare L M quoque bifariam ſe-
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            cta erit in H: </s>
            <s xml:id="echoid-s8481" xml:space="preserve">ſed dicitur per H tranſire
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            ſectionem, ergo L M ipfam
              <note symbol="d" position="right" xlink:label="note-0305-04" xlink:href="note-0305-04a" xml:space="preserve">ibidem.</note>
            in H, quapropter portio plana L B M
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            æquabitur portioni A B C, & </s>
            <s xml:id="echoid-s8482" xml:space="preserve">ſi per
              <note symbol="e" position="right" xlink:label="note-0305-05" xlink:href="note-0305-05a" xml:space="preserve">45. h.</note>
            M agatur planum ſecans Conum, & </s>
            <s xml:id="echoid-s8483" xml:space="preserve">ad planum L B M rectum, quod & </s>
            <s xml:id="echoid-s8484" xml:space="preserve">
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            plano datæ portionis ſolidæ D B E per D E ductum æquidiſtabit, cum hoc
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            ad idem planum L B M ponatur rectum eſſe; </s>
            <s xml:id="echoid-s8485" xml:space="preserve">erit ſolida portio L B M ęqua-
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            lis portioni A B C, cum earum recti Canones L B M, A B C
              <note symbol="f" position="right" xlink:label="note-0305-06" xlink:href="note-0305-06a" xml:space="preserve">78. h.</note>
            ſint oſtenſi; </s>
            <s xml:id="echoid-s8486" xml:space="preserve">ſed D B E quoque eidem A B C data eſt æqualis, ergo duæ
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            portiones L B M, D B E ſimul æquales erunt, totum ſuæ parti, quod eſt
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            abſurdum: </s>
            <s xml:id="echoid-s8487" xml:space="preserve">non ergo ſectio F G ſecat axim B G infra H; </s>
            <s xml:id="echoid-s8488" xml:space="preserve">& </s>
            <s xml:id="echoid-s8489" xml:space="preserve">ob eandem ra-
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            tionem neque ſupra; </s>
            <s xml:id="echoid-s8490" xml:space="preserve">ergo ſectio F G omnino tranſibit per G extremum
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            axis B G: </s>
            <s xml:id="echoid-s8491" xml:space="preserve">ſed facta reuolutione anguli, ac ſectionis circa communem axim
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            procreatur Conus, & </s>
            <s xml:id="echoid-s8492" xml:space="preserve">Conoides Hyperbolicum ſimile, ac concentricum:
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            </s>
            <s xml:id="echoid-s8493" xml:space="preserve">ergo F, G, extrema puncta axium æqualium portionum ſolidarum A B C,
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            D B E, ex eodem Cono recto, pertingunt ad idem Conoides Hyperboli-
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            cum ſimile, & </s>
            <s xml:id="echoid-s8494" xml:space="preserve">concentricum inſcriptum. </s>
            <s xml:id="echoid-s8495" xml:space="preserve">Quod vltimò demonſtrandum
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            erat.</s>
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