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

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            ctiones contingentes, quæ productæ, communi diametro G B E
              <note symbol="a" position="right" xlink:label="note-0241-01" xlink:href="note-0241-01a" xml:space="preserve">24. 25.
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              primi co-
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              nic.</note>
            in L, M. </s>
            <s xml:id="echoid-s6689" xml:space="preserve">Dico primùm G A ad A D eſſe vt G B ad B E, & </s>
            <s xml:id="echoid-s6690" xml:space="preserve">contingentes
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            A L, D L inter ſe æquidiſtare.</s>
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          <p>
            <s xml:id="echoid-s6692" xml:space="preserve">Applicentur ex A, D ad diametrum communem G B M rectę A I, D H.
              <lb/>
            </s>
            <s xml:id="echoid-s6693" xml:space="preserve">Erit iam in ſectione D E F, rectangulum G H M ad quadratum H D,
              <note symbol="b" position="right" xlink:label="note-0241-02" xlink:href="note-0241-02a" xml:space="preserve">37. ibid.</note>
            tranſuerſum ad rectum, vel, ob ſectionum ſimilitudinem, vt tranſuerſum ſe-
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            ctionis A B C ad eius rectum, vel vt rectangulum G I L ad quadratum I A,
              <lb/>
            & </s>
            <s xml:id="echoid-s6694" xml:space="preserve">quadratum D H ad H G, eſt vt quadratum A I ad I G, ergo ex æquo
              <lb/>
            rectangulum G H M ad quadratum G H, erit vt rectangulum G I L ad qua-
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            dratum I G, & </s>
            <s xml:id="echoid-s6695" xml:space="preserve">conuertendo quadratum G H ad rectangulum G H M, vt
              <lb/>
            quadratum I G ad rectangulum G I L, & </s>
            <s xml:id="echoid-s6696" xml:space="preserve">per conuerſionem rationis in pri-
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            ma figura, & </s>
            <s xml:id="echoid-s6697" xml:space="preserve">componendo in ſecunda, quadratum G H ad rectangulum.
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            </s>
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              <figure xlink:label="fig-0241-01" xlink:href="fig-0241-01a" number="199">
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            H G M, vt quadratum I G ad rectangulum I G L, & </s>
            <s xml:id="echoid-s6699" xml:space="preserve">permutando quadra-
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            tum H G ad G I, vel quadratum D G ad G A, erit vt rectangulum H G M
              <lb/>
            ad rectangulum I G L, vel permutatis æqualibus, vt quadratum E G
              <note symbol="c" position="right" xlink:label="note-0241-03" xlink:href="note-0241-03a" xml:space="preserve">ibidem.</note>
            quadratum G B, ſeulinea D G ad G A, vt linea E G ad G B, & </s>
            <s xml:id="echoid-s6700" xml:space="preserve">diuiden-
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            do, & </s>
            <s xml:id="echoid-s6701" xml:space="preserve">conuertendo G A ad A D, vt G B ad B E. </s>
            <s xml:id="echoid-s6702" xml:space="preserve">Quod primò erat, &</s>
            <s xml:id="echoid-s6703" xml:space="preserve">c.</s>
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            <s xml:id="echoid-s6705" xml:space="preserve">Præterea, cum ſuperiùs demonſtratum ſit eſſe rectangulum G H M ad
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            quadratum H D, vt rectangulum G I L ad quadratum I A, erit permutan-
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            do rectangulum G H M ad G I L, vt quadratum H D ad I A; </s>
            <s xml:id="echoid-s6706" xml:space="preserve">ſed propor-
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            tio quadrati H D ad I A componitur ex du@bus rationibus H D ad I A,
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            vel ex duobus rationibus H G ad G I, & </s>
            <s xml:id="echoid-s6707" xml:space="preserve">proportio rectanguli G H M ad
              <lb/>
            G I L componitur ex duobus rationibus, nempe ex G H ad G I, & </s>
            <s xml:id="echoid-s6708" xml:space="preserve">ex H M
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            ad I L; </s>
            <s xml:id="echoid-s6709" xml:space="preserve">ergo proportio G H ad G I, hoc eſt H D ad I A, æqualis eſt pro-
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            portioni H M ad I I.</s>
            <s xml:id="echoid-s6710" xml:space="preserve">, & </s>
            <s xml:id="echoid-s6711" xml:space="preserve">permutando D H ad H M, erit vt A I ad I L, &</s>
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