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

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            vt eadem M D ad D N, erit G M ad E N, vt M F ad N B, & </s>
            <s xml:id="echoid-s6548" xml:space="preserve">permu-
              <lb/>
            tando G M ad M F, vt E N ad N B.</s>
            <s xml:id="echoid-s6549" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6550" xml:space="preserve">Cum ergo, in figuris prima, ſecunda, quarta, quinta, ſeptima, octaua,
              <lb/>
            decima, ac decimaprima ſit G M ad M F, vt E N ad N B, erit quoq; </s>
            <s xml:id="echoid-s6551" xml:space="preserve">qua-
              <lb/>
            dratum G M ad M F, vt quadratum E N ad N B, vel vt rectangulum
              <note symbol="a" position="left" xlink:label="note-0236-01" xlink:href="note-0236-01a" xml:space="preserve">17. tertij
                <lb/>
              conic.</note>
            M I ad rectangulum C M A, & </s>
            <s xml:id="echoid-s6552" xml:space="preserve">permutando quadratum G M ad rectan-
              <lb/>
            gulum H M I, vt quadratum F M ad rectangulum C M A, & </s>
            <s xml:id="echoid-s6553" xml:space="preserve">couertendo
              <lb/>
            in prima, quarta, ſeptima, & </s>
            <s xml:id="echoid-s6554" xml:space="preserve">decima figura (in quibus applicatæ H I, C A
              <lb/>
            ſecant ſe mutuò intra ſectionem in puncto M) rectangulum H M I ad qua-
              <lb/>
            dratum G M, vtrectangulum C M A ad quadratum F M, & </s>
            <s xml:id="echoid-s6555" xml:space="preserve">componendo
              <lb/>
            rectangulum H M I cum quadrato G M, ſiue vnicum quadratum H G, (nam
              <lb/>
            eſt A C bifariam ſecta in G, & </s>
            <s xml:id="echoid-s6556" xml:space="preserve">non bifariam in M) ad quadratum G M, vt
              <lb/>
            rectangulum C M A cum quadrato F M, ſiue vt vnicum quadratum C F
              <lb/>
            (cum A C quoque ſecta ſit bifariam in F, & </s>
            <s xml:id="echoid-s6557" xml:space="preserve">non bifariam in M) ad quadra-
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            tum F M. </s>
            <s xml:id="echoid-s6558" xml:space="preserve">In figuris verò ſecunda, quinta, octaua, & </s>
            <s xml:id="echoid-s6559" xml:space="preserve">vndecima, in quibus
              <lb/>
            applicatæ H I, C A ſe mutuò ſecant extra ſectionem in puncto M, cum ſit
              <lb/>
            G M quadratum ad rectangulum H M I, vt quadratum F M ad rectangu-
              <lb/>
            lum C M A, erit per conuerſionem rationis quadratum M G ad quadratum
              <lb/>
            G H (eſt enim rectangulum H M I cum quadrato G H æquale quadrato G
              <lb/>
            M, cum ſit H I bifariam ſecta in G, & </s>
            <s xml:id="echoid-s6560" xml:space="preserve">ei adiecta ſit I M) vt quadratum M
              <lb/>
            F ad quadratum F C, ob eandem rationem, (nam C A quoq; </s>
            <s xml:id="echoid-s6561" xml:space="preserve">bifariam ſecta
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            eſt in C, eiq; </s>
            <s xml:id="echoid-s6562" xml:space="preserve">addita eſt in directum A M) & </s>
            <s xml:id="echoid-s6563" xml:space="preserve">conuertendo quadratum H G ad
              <lb/>
            G M quadratum, erit vt quadratum C F ad F M. </s>
            <s xml:id="echoid-s6564" xml:space="preserve">Itaq; </s>
            <s xml:id="echoid-s6565" xml:space="preserve">in ſingulis prædictis
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            figuris, déptis tertia, ſexta, nona, & </s>
            <s xml:id="echoid-s6566" xml:space="preserve">duodecima, cum demonſtratum ſit qua-
              <lb/>
            dratum H G ad G M eſſe vt quadratum C F ad F M, erit quoque linea H G
              <lb/>
            G M, vt linea C F ad F M. </s>
            <s xml:id="echoid-s6567" xml:space="preserve">In figuris deniq; </s>
            <s xml:id="echoid-s6568" xml:space="preserve">tertia, ſexta, nona, & </s>
            <s xml:id="echoid-s6569" xml:space="preserve">duodeci-
              <lb/>
            ma, in quibus applicatę H I, C A conueniunt ſimul cum ipſa ſectione in pun-
              <lb/>
            cto M, patet quoque eſſe H G ad G M, vt C F ad F M, cum ipſæ H I, C
              <lb/>
            A, vel H M, C M bifariam ſecentur in G, F ab earum diametris E G, B F.
              <lb/>
            </s>
            <s xml:id="echoid-s6570" xml:space="preserve">Eſt igitur in qualibet datarum figurarum huius ſchematiſmi, H G ad G M, vt
              <lb/>
            C F ad F M, quare iuncta H C æquidiſtabit iunctæ G F; </s>
            <s xml:id="echoid-s6571" xml:space="preserve">ſed eſt I G æqua-
              <lb/>
            lis H G, & </s>
            <s xml:id="echoid-s6572" xml:space="preserve">A F æqualis C F, ergo etiam I G ad G M erit vt A F ad F M,
              <lb/>
            ideoque iuncta A I æquidiſtabit eidem G F, ſed E B quoque ipſi G F ęqui-
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            diſtat (vt iam ſupra oſtendimus in Parabolis, & </s>
            <s xml:id="echoid-s6573" xml:space="preserve">cum in reliquis ſectionibus
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            ſit D E ad E G, vt D B ad B F ex hypoteſi) ergo quatuor iunctæ rectæ lineæ
              <lb/>
            E B, A I, G F, H C ſunt inter ſe parallelæ; </s>
            <s xml:id="echoid-s6574" xml:space="preserve">ſed N M, quàm ſuperiùs oſten-
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            dimus eſſe ſectionis diametrum, tranſit per N occurſum contingentium E
              <lb/>
            N, B N, ergo recta E B puncta contactuum iungens, ab eadem diametro N
              <lb/>
            M D bifariam ſecabitur, vt in O, ac ideò omnes aliæ in ſectione
              <note symbol="b" position="left" xlink:label="note-0236-02" xlink:href="note-0236-02a" xml:space="preserve">30.ſecũ-
                <lb/>
              di conic.</note>
            ipſi E B ęquidiſtantes, nempe A I, G F, H C, ab eadem D N M bifariam
              <lb/>
            ſecabuntur, vt H C in P.</s>
            <s xml:id="echoid-s6575" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6576" xml:space="preserve">Denique iungantur rectæ H E, C B, & </s>
            <s xml:id="echoid-s6577" xml:space="preserve">fiet quadrilaterum H E B C, cu-
              <lb/>
            ius oppoſita latera H C, E B ſunt parallela, & </s>
            <s xml:id="echoid-s6578" xml:space="preserve">bifariam ſecta à recta P O, in
              <lb/>
            qua ſumptum eſt punctum M, & </s>
            <s xml:id="echoid-s6579" xml:space="preserve">ab ipſo ad terminos alterius ęquidiſtantium
              <lb/>
            nempe ad H, C ductę ſunt rectæ M H, M C, ac in triangulo H M C eſt G F
              <lb/>
            ipſi H C parallela, quare iunctę E G, B F auferent triangula E G H, B F C
              <lb/>
            inter ſe æqualia; </s>
            <s xml:id="echoid-s6580" xml:space="preserve">quapropter baſis H G ad baſim C F erit reciprocè, vt
              <note symbol="c" position="left" xlink:label="note-0236-03" xlink:href="note-0236-03a" xml:space="preserve">39. h.</note>
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