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

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            <s xml:id="echoid-s6470" xml:space="preserve">
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            C B, erunt A C, B D æquales, & </s>
            <s xml:id="echoid-s6471" xml:space="preserve">parallelæ, idcirco angulus G B D æ-
              <lb/>
            qualis angulo B A C, ergo reliquus angulus D B F, æqualis erit reliquo
              <lb/>
            C A E; </s>
            <s xml:id="echoid-s6472" xml:space="preserve">eadem ratione oſtendetur angulum B D F æquari angulo A C E,
              <lb/>
            & </s>
            <s xml:id="echoid-s6473" xml:space="preserve">B D demonſtrata eſt æqualis ipſi A C, ergo in triangulis B F D, A E
              <lb/>
            C, cum æqualia latera B D, A C æqualibus angulis adiaceant, erit ter-
              <lb/>
            tius angulus B F D, tertio A E C æqualis, & </s>
            <s xml:id="echoid-s6474" xml:space="preserve">reliqua latera B F, A E,
              <lb/>
            itemque D F, C E, inter ſe æqualia, ſed ſunt quoque parallela, ob hy-
              <lb/>
            poteſim, ergo E F angulorum vertices iungens, erit æqualis, & </s>
            <s xml:id="echoid-s6475" xml:space="preserve">paralle-
              <lb/>
            la ad A B, vel ad C D; </s>
            <s xml:id="echoid-s6476" xml:space="preserve">cadetque inter datas A B, C D, cum punctum
              <lb/>
            E, ex quo ducitur ſit inter eas, ſicque angulus B F D cadet intra datas
              <lb/>
            æquidiſtantes A B, C D. </s>
            <s xml:id="echoid-s6477" xml:space="preserve">Quod primò oſtendere opus erat.</s>
            <s xml:id="echoid-s6478" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6479" xml:space="preserve">Sint verò, in reli-
              <lb/>
            quis tribus figuris,
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            datæ rectæ A B, A C
              <lb/>
              <figure xlink:label="fig-0231-01" xlink:href="fig-0231-01a" number="193">
                <image file="0231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0231-01"/>
              </figure>
            terminatæ angulum
              <lb/>
            B A C conſtituentes,
              <lb/>
            quę proportionaliter
              <lb/>
            ſecentur, vel produ-
              <lb/>
            cantur in D, E; </s>
            <s xml:id="echoid-s6480" xml:space="preserve">& </s>
            <s xml:id="echoid-s6481" xml:space="preserve">
              <lb/>
            ex punctis D, E du-
              <lb/>
            ctæ ſint D F, E F ſibi
              <lb/>
            ipſis occurrentes in-
              <lb/>
            tra datum angulum
              <lb/>
            in F, ex reliquis ve-
              <lb/>
            rò punctis B C, alię
              <lb/>
            ipſis æquidiſtantes B
              <lb/>
            G, C H. </s>
            <s xml:id="echoid-s6482" xml:space="preserve">Dico item
              <lb/>
            has intra angulum B
              <lb/>
            A C inter ſe conue-
              <lb/>
            nire, vt in G, ac tres
              <lb/>
            angulorum occurſus
              <lb/>
            A, F, G in eadem re-
              <lb/>
            cta linea reperiri.</s>
            <s xml:id="echoid-s6483" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6484" xml:space="preserve">Nam iuncta A F, & </s>
            <s xml:id="echoid-s6485" xml:space="preserve">producta cum B G ipſi D F ęquidiſtet, & </s>
            <s xml:id="echoid-s6486" xml:space="preserve">A F cum
              <lb/>
            D F conueniat, conueniet quoque producta cum B G, ſit ergo G punctum
              <lb/>
            occurſus. </s>
            <s xml:id="echoid-s6487" xml:space="preserve">Item cum C H æquidiſtet ipſi E F, & </s>
            <s xml:id="echoid-s6488" xml:space="preserve">A F ſecet E F, producta
              <lb/>
            ſecabit quoque C H: </s>
            <s xml:id="echoid-s6489" xml:space="preserve">ſecet in H. </s>
            <s xml:id="echoid-s6490" xml:space="preserve">Oſtendam puncta G, H, quæ iam in
              <lb/>
            recta A F reperiri demonſtratum eſt, eſſe vnum idemque punctum rectæ
              <lb/>
            A F: </s>
            <s xml:id="echoid-s6491" xml:space="preserve">eſt enim in triangulo A G B vt G A ad A F, ita B A ad A D, vel,
              <lb/>
            ob hypoteſim, vt C A ad A E, vel vt H A ad A F, ergo G A, & </s>
            <s xml:id="echoid-s6492" xml:space="preserve">H A
              <lb/>
            ſunt æquales, hoc eſt puncta G, & </s>
            <s xml:id="echoid-s6493" xml:space="preserve">H non ſunt duo, ſed vnum tantùm,
              <lb/>
            & </s>
            <s xml:id="echoid-s6494" xml:space="preserve">in eadem recta linea in qua ſunt puncta A, F. </s>
            <s xml:id="echoid-s6495" xml:space="preserve">Ergo B G, C H inter
              <lb/>
            ſe conueniunt intra datum angulum, ac trium angulorum vertices ſunt
              <lb/>
            in directum poſiti. </s>
            <s xml:id="echoid-s6496" xml:space="preserve">Quod vltimò oſtendere propoſitum fuit.</s>
            <s xml:id="echoid-s6497" xml:space="preserve"/>
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