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

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            <s xml:id="echoid-s6159" xml:space="preserve">
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            vt quadratum D C ad C F, & </s>
            <s xml:id="echoid-s6160" xml:space="preserve">per con-
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            uerſionem rationis, quadratum A B ad
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              <figure xlink:label="fig-0220-01" xlink:href="fig-0220-01a" number="181">
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            rectangulum G B E, vt quadratum C D
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            ad rectangulum H D F, & </s>
            <s xml:id="echoid-s6161" xml:space="preserve">conuertendo,
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            rectangulum G B E ad quadratum A B,
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            vt rectangulum H D F ad quadratum C
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            D, & </s>
            <s xml:id="echoid-s6162" xml:space="preserve">quadratum A B ad B I, eſt vt qua-
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            dratum C D ad D L, ob triangulorum
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            I A B, L C D ſimilitudinem; </s>
            <s xml:id="echoid-s6163" xml:space="preserve">quare ex
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            æquo rectangulum G B E ad quadratum
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            B I, erit vt rectangulum H D F ad quadra-
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            tum D L. </s>
            <s xml:id="echoid-s6164" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s6165" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6166" xml:space="preserve"/>
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        <div xml:id="echoid-div640" type="section" level="1" n="255">
          <head xml:id="echoid-head263" xml:space="preserve">LEMMA VIII. PROP. XXIX.</head>
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            <s xml:id="echoid-s6167" xml:space="preserve">Si quatuor magnitudinum eiuſdem generis, prima A ad ſe-
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            cundam B maiorem habuerit rationem, quàm tertia C ad quar-
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            tam D E, ſitque prima minor tertia, erit ſecunda minor quar-
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            ta.</s>
            <s xml:id="echoid-s6168" xml:space="preserve"/>
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            <s xml:id="echoid-s6169" xml:space="preserve">FIat, vt A ad B, ita C ad D F, & </s>
            <s xml:id="echoid-s6170" xml:space="preserve">cum
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            A ad B habeat maiorem rationem,
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              <figure xlink:label="fig-0220-02" xlink:href="fig-0220-02a" number="182">
                <image file="0220-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0220-02"/>
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            quàm C ad D E, habebit quoque C ad D
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            F maiorem quàm ad D E, vnde D F erit
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            minor D E, & </s>
            <s xml:id="echoid-s6171" xml:space="preserve">eſt A ad B, vt C ad D F,
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            erit permutando A ad C, vt B ad D F,
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            eſtque A minor C, ergo B erit minor D
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            F, & </s>
            <s xml:id="echoid-s6172" xml:space="preserve">D F oſtenſa eſt minor D E, quare B
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            eò ampliùs erit minor D E. </s>
            <s xml:id="echoid-s6173" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s6174" xml:space="preserve">c.</s>
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        <div xml:id="echoid-div642" type="section" level="1" n="256">
          <head xml:id="echoid-head264" xml:space="preserve">THEOR. XIX. PROP. XXX.</head>
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            <s xml:id="echoid-s6176" xml:space="preserve">Rectorum laterum in Hyperbola, cuius axis tranſuerſus non
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            ſit minor eius recto latere, MINIMVM eſt rectum axis.</s>
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          <p>
            <s xml:id="echoid-s6178" xml:space="preserve">ESto Hyperbole A B C, cuius centrum D, axis tranſnerſus E B, qui
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            primò ſit minor recto B F. </s>
            <s xml:id="echoid-s6179" xml:space="preserve">Dico rectum B F eſſe rectorum laterum
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            _MINIMVM._</s>
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          <p>
            <s xml:id="echoid-s6181" xml:space="preserve">Sit quæcunque alia tranſuerſa diameter G D A, in ſectione producta
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            ad I, cuius rectum ſit A K ex A contingenter applicatum, & </s>
            <s xml:id="echoid-s6182" xml:space="preserve">axi occur-
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            rens in H; </s>
            <s xml:id="echoid-s6183" xml:space="preserve">& </s>
            <s xml:id="echoid-s6184" xml:space="preserve">ſit B I æquidiſtans A H, quæ ad diametrum G A I erit or-
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            dinatim ducta, atque ex I ſit I L ipſi D I perpendicularis, ex A verò A
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            M axi applicata, cui ex vertice B ſit parallela, vel contingens B O, ſe-
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            cans A H in P, iunganturque A B, O H.</s>
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            <s xml:id="echoid-s6186" xml:space="preserve">Iam cum rectangulum D M H ad quadratum M A, ſit vt E B ad B
              <note symbol="a" position="left" xlink:label="note-0220-01" xlink:href="note-0220-01a" xml:space="preserve">25. pri-
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              miconic.</note>
            ſitque E B maior B F, erit rectangulum D M H maius quadrato M </s>
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