Cavalieri, Buonaventura, Geometria indivisibilibvs continvorvm : noua quadam ratione promota

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        <div xml:id="echoid-div1113" type="section" level="1" n="666">
          <head xml:id="echoid-head696" xml:space="preserve">PROBLEMA V. PROPOS. XXVIII.</head>
          <p>
            <s xml:id="echoid-s12118" xml:space="preserve">DAto cylindro tympanum ſphærale eidem æquale cõ-
              <lb/>
            ſtituere, cuius axis ſemidiametro baſis ſit æqualis.</s>
            <s xml:id="echoid-s12119" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12120" xml:space="preserve">Sit datus cylindrus,
              <lb/>
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                <image file="0493-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0493-01"/>
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            ABC, cuius axis, AD,
              <lb/>
            baſis, BC, oportet illi
              <lb/>
            æquale tympanu ſphę-
              <lb/>
            rale conſtituere, cuius
              <lb/>
            axis ſemidiametro ba-
              <lb/>
            ſis ſit æqualis. </s>
            <s xml:id="echoid-s12121" xml:space="preserve">Vt hoc
              <lb/>
            fiat exponatur recta li-
              <lb/>
            nea terminata quæcũ-
              <lb/>
            que, EI, quæ ſit bifariã
              <lb/>
            ſecta in, G, & </s>
            <s xml:id="echoid-s12122" xml:space="preserve">vt quad.
              <lb/>
            </s>
            <s xml:id="echoid-s12123" xml:space="preserve">circulo cuicunq; </s>
            <s xml:id="echoid-s12124" xml:space="preserve">circũ. </s>
            <s xml:id="echoid-s12125" xml:space="preserve">
              <lb/>
            ſcriptum ad eundem
              <lb/>
            circulum, ita fiat, GE,
              <lb/>
            ad, EF, ſumatur dein-
              <lb/>
            de, FH, quæ ſit exceſ-
              <lb/>
            ſus, quo, FE, ſuperat
              <lb/>
            {2/3}. </s>
            <s xml:id="echoid-s12126" xml:space="preserve">tertiæ proportiona-
              <lb/>
            lis duarum, IE, EG,
              <lb/>
            deinde vt, HI, ad, IE, ita fiat, AD, ad, LO, altitudinem alterius
              <lb/>
            cylindri in baſi, NP, æquali ipſi, BC, exiſtentis, & </s>
            <s xml:id="echoid-s12127" xml:space="preserve">tandem inter
              <lb/>
            LO, &</s>
            <s xml:id="echoid-s12128" xml:space="preserve">, ON, baſis ſemidiametrum, ſumantur duæ mediæ conti-
              <lb/>
            nuò proportionales, QO, OR, & </s>
            <s xml:id="echoid-s12129" xml:space="preserve">in altitudine æquali, OR, nem-
              <lb/>
            pè, T8, baſi, ℟8, æquali eidem, OR, fiat parallelogrammum re-
              <lb/>
            ctangulum, 58, in cuius plano deſcribatur ſemicirculus, SY℟, & </s>
            <s xml:id="echoid-s12130" xml:space="preserve">
              <lb/>
            ipſum cum parallelogrammo reuo uatur circa manentem axim,
              <lb/>
            T8, donec redeat vnde diſceſſit, patet autem, quod ex parallelo-
              <lb/>
            grammo fiet cylindrus, vt, Sφ, & </s>
            <s xml:id="echoid-s12131" xml:space="preserve">ex figura, SY℟8T, tympanum
              <lb/>
            ſphærale, 5Yφ. </s>
            <s xml:id="echoid-s12132" xml:space="preserve">Dico igitur hoc tympanum eſſe æquale dato cy-
              <lb/>
            lindro, ABC, &</s>
            <s xml:id="echoid-s12133" xml:space="preserve">, T8, æqualem ipſi, 8℟, ſemidiametro baſis, ℟φ. </s>
            <s xml:id="echoid-s12134" xml:space="preserve">
              <lb/>
            Sint parallelogrammum, Sφ, & </s>
            <s xml:id="echoid-s12135" xml:space="preserve">figura, SYφ, per axem tranſeun-
              <lb/>
            tia, & </s>
            <s xml:id="echoid-s12136" xml:space="preserve">X&</s>
            <s xml:id="echoid-s12137" xml:space="preserve">, per centra, X, &</s>
            <s xml:id="echoid-s12138" xml:space="preserve">, circulorum ducta, ſecans, T8, in, Z,
              <lb/>
            manifeſtum eſt igitur, quod, XZ, bifariam ſecabitur à circumfe-
              <lb/>
            rentia, SY℟, vt in, Y, cum, 8℟, ſit æqualis, ℟S, & </s>
            <s xml:id="echoid-s12139" xml:space="preserve">ipſi, XZ, S℟,
              <lb/>
            autem ſit dupla, XY, vnde ſi ſecetur, ℟8, bifariam in, 4@erit, ℟4,
              <lb/>
            æqualis ipſi, XY, ſit autem ab ea dempta, ℟@, ad qua ?</s>
            <s xml:id="echoid-s12140" xml:space="preserve">? 4℟, ſit </s>
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