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

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        <div xml:id="echoid-div677" type="section" level="1" n="271">
          <head xml:id="echoid-head280" xml:space="preserve">THEOR. XXIII. PROP. XXXX.</head>
          <p>
            <s xml:id="echoid-s6509" xml:space="preserve">Si in Parabola, ex binis ipſius diametris duo æqualia ſegmen-
              <lb/>
            ta ſint abſciſſa: </s>
            <s xml:id="echoid-s6510" xml:space="preserve">in Hyperbola verò, Ellipſi, vel circulo duæ ſe-
              <lb/>
            mi-diametri proportionaliter intra ſectionem ſectę fuerint, & </s>
            <s xml:id="echoid-s6511" xml:space="preserve">ex
              <lb/>
              <note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">Schema-
                <lb/>
              tiſmus 3.</note>
            terminis æqualium diametrorum in Parabola, vel ex punctis di-
              <lb/>
            uiſionum, in reliquis ſectionibus, ordinatim applicentur lineæ ad
              <lb/>
            fuas diametros, & </s>
            <s xml:id="echoid-s6512" xml:space="preserve">producantur, donec ad vtranque partem ſe-
              <lb/>
            ctioni occurrant: </s>
            <s xml:id="echoid-s6513" xml:space="preserve">coni- ſectionum portiones; </s>
            <s xml:id="echoid-s6514" xml:space="preserve">at in Ellipſi, vel
              <lb/>
            circulo, minores portiones ijſdem applicatis, tanquam baſibus
              <lb/>
            inſiſtentes, inter ſe æquales erunt.</s>
            <s xml:id="echoid-s6515" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6516" xml:space="preserve">ESto A B C Parabole, in prima, ſecunda, & </s>
            <s xml:id="echoid-s6517" xml:space="preserve">tertia figura, vel Hyper-
              <lb/>
            bole in quarta, quinta, & </s>
            <s xml:id="echoid-s6518" xml:space="preserve">ſexta, aut Ellipſis in ſeptima, octaua, & </s>
            <s xml:id="echoid-s6519" xml:space="preserve">
              <lb/>
            nona, aut circulus, in reliquis, quarum ſectionum binæ diametri in Pa-
              <lb/>
            rabola ſint D B, D E, à quibus dempta ſint æqualia ſegmenta B F, E G,
              <lb/>
            & </s>
            <s xml:id="echoid-s6520" xml:space="preserve">in reliquis binæ ſemi-diametri D B, D E (quæ primò in Ellipſi, vel
              <lb/>
            circulo omnino conſtituant angulum B D E) ita intra ſectiones ſectæ ſint
              <lb/>
            in F, G, vt D B ad B F, ſit vt D E ad E G, & </s>
            <s xml:id="echoid-s6521" xml:space="preserve">per puncta F, G, in ſin-
              <lb/>
            gulis figuris ſint ad diametros D B, D E ordinatim ductæ A F C, H G I,
              <lb/>
            quæ ad vtranque partem ſectioni occurrent in punctis A, C; </s>
            <s xml:id="echoid-s6522" xml:space="preserve">H, I, &</s>
            <s xml:id="echoid-s6523" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-0235-02" xlink:href="note-0235-02a" xml:space="preserve">19. primi
                <lb/>
              conic.</note>
            bifariam in F, G ſecabuntur, cum D B, D G, ſint ipſarum diametri. </s>
            <s xml:id="echoid-s6524" xml:space="preserve">Di-
              <lb/>
            co portiones A B C, H E I ſuper ijſdem applicatis, tanquam baſibus inſi-
              <lb/>
            ſtentes, inter ſe æquales eſſe.</s>
            <s xml:id="echoid-s6525" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6526" xml:space="preserve">Nam, ductis ex B, E rectis B N, E N ſectionem contingentibus in B,
              <lb/>
            E; </s>
            <s xml:id="echoid-s6527" xml:space="preserve">ipſæ occurrent ſimul in N inter diametros D B, D E, & </s>
            <s xml:id="echoid-s6528" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0235-03" xlink:href="note-0235-03a" xml:space="preserve">58. primi
                <lb/>
              huius.</note>
            H I, A C æquidiſtabunt. </s>
            <s xml:id="echoid-s6529" xml:space="preserve">Iungantur præterea E B, G F.</s>
            <s xml:id="echoid-s6530" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6531" xml:space="preserve">Iam in Parabolis, cum ſint E G, B F inter ſe æquales, & </s>
            <s xml:id="echoid-s6532" xml:space="preserve">parallelę, iun-
              <lb/>
            ctæ quoq; </s>
            <s xml:id="echoid-s6533" xml:space="preserve">E B, G F inter ſe æquidiſtabunt, & </s>
            <s xml:id="echoid-s6534" xml:space="preserve">cum ex illarum terminis E,
              <lb/>
            B, ductæ ſint rectę E N, B N angulum E N B inter eas conſtituentes, atq;
              <lb/>
            </s>
            <s xml:id="echoid-s6535" xml:space="preserve">ex reliquis terminis G, F, ſint G I, F A, ipſis E N, B N æqurdiſtantes; </s>
            <s xml:id="echoid-s6536" xml:space="preserve">
              <lb/>
            ipſæ G I, F A inter eaſdem E G, B F ſimul conuenient, vt in M, & </s>
            <s xml:id="echoid-s6537" xml:space="preserve">iuncta
              <lb/>
            N M ijſdem E G, B F æquidiſtabit, ſiue erit altera Parabolæ diameter.</s>
            <s xml:id="echoid-s6538" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-0235-04" xlink:href="note-0235-04a" xml:space="preserve">38. h.</note>
            Cum ergo ſit E G parallela ad N M, & </s>
            <s xml:id="echoid-s6539" xml:space="preserve">E N ad G M, erit E N
              <note symbol="d" position="right" xlink:label="note-0235-05" xlink:href="note-0235-05a" xml:space="preserve">46. pri-
                <lb/>
              mi conic.</note>
            G M; </s>
            <s xml:id="echoid-s6540" xml:space="preserve">eademque ratione B N æqualis F M, quare vt E N ad N B, ita G
              <lb/>
            M ad M F.</s>
            <s xml:id="echoid-s6541" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6542" xml:space="preserve">In reliquis verò figuris, cum rectæ D B, D E angulum E D B efficien-
              <lb/>
            tes, proportionaliter ſectæ, aut productæ ſint in G, F, ſintque ex earum
              <lb/>
            homologis terminis E, B ductæ E N, B N angulum inter ipſas conſti-
              <lb/>
            tuentes E N B, & </s>
            <s xml:id="echoid-s6543" xml:space="preserve">ex reliquis diuiſionum punctis G, F, ſint G I, F A ijſdem
              <lb/>
            E N, B N parallelę, hæ intra datum angulum E D B ſimul conuenient, vt
              <lb/>
            in M; </s>
            <s xml:id="echoid-s6544" xml:space="preserve">& </s>
            <s xml:id="echoid-s6545" xml:space="preserve">recta iungens puncta D, M, per occurſum M omnino tranſibit,
              <lb/>
            ſiue erit alia ſectionis diameter. </s>
            <s xml:id="echoid-s6546" xml:space="preserve">Cumque ob parallelas G M, E N ſit
              <note symbol="e" position="right" xlink:label="note-0235-06" xlink:href="note-0235-06a" xml:space="preserve">38. h.</note>
            M ad E N, vt M D ad D N, & </s>
            <s xml:id="echoid-s6547" xml:space="preserve">ob parallelas M F, N B ſit M F ad N B,
              <lb/>
              <note position="right" xlink:label="note-0235-07" xlink:href="note-0235-07a" xml:space="preserve">_f_ 47. primi
                <lb/>
              conic.</note>
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