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

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              <pb o="205" file="0225" n="225" rhead="LIBER III."/>
            ad quadratum, IM, ergo per conuerſionem rationis rectangulum,
              <lb/>
              <note position="right" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve">Ex 40. l. 1.
                <lb/>
              & ex eius
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              Scholio.</note>
              <figure xlink:label="fig-0225-01" xlink:href="fig-0225-01a" number="138">
                <image file="0225-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0225-01"/>
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            HEB, .</s>
            <s xml:id="echoid-s5053" xml:space="preserve">i. </s>
            <s xml:id="echoid-s5054" xml:space="preserve">quadratum, BE, ad qua-
              <lb/>
            dratum, ME, (quod eſt exceſſus re
              <lb/>
            ctanguli, HEB, ſub rectangulum, H
              <lb/>
              <note position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">5. 2. elem.</note>
            MB,) erit vt quadratum, RM, ad ſui
              <lb/>
            reliquum, dempto quadrato, MI, ſed
              <lb/>
            vt quadratum, BE, ad quadratum, E
              <lb/>
            M, ita quadratum, BC, ideſt quadra-
              <lb/>
            tum, MR, ad quadratum, MN, quia
              <lb/>
              <note position="right" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">4. 6. elem.</note>
            triangula, BEC, MEN, ſunt æquian-
              <lb/>
            gula; </s>
            <s xml:id="echoid-s5055" xml:space="preserve">ergo quadratum, BC, ideſt qua-
              <lb/>
            dratum, MR, ad quadratum, MN,
              <lb/>
            erit vt idem quadratum, MR, ad ſui
              <lb/>
            reliquum, dempto quadrato, MI, & </s>
            <s xml:id="echoid-s5056" xml:space="preserve">
              <lb/>
            eorum quadrupla .</s>
            <s xml:id="echoid-s5057" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s5058" xml:space="preserve">quadratum, SN, æquabitur reliquo quadrati,
              <lb/>
            VR, dempto quadrato, TI, quod erat oſtendendum.</s>
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        <div xml:id="echoid-div510" type="section" level="1" n="306">
          <head xml:id="echoid-head323" xml:space="preserve">COROLLARIV M.</head>
          <p style="it">
            <s xml:id="echoid-s5060" xml:space="preserve">_Q_VONIAM autem punctum, M, ſumptum eſt vtcumque hinc
              <lb/>
            patet, quod omnia quadrata trianguli, AEC, (regula, DF,)
              <lb/>
            æquantur reliquo omnium quadratorum parallelogrammi, AF, dem-
              <lb/>
            ptis omnibus quadratis ſemicirculi, vel ſemiellipſis, DBF, & </s>
            <s xml:id="echoid-s5061" xml:space="preserve">duabus
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            vtcunq; </s>
            <s xml:id="echoid-s5062" xml:space="preserve">ductis ipſis, DF, parallelis, vt, XG, VR, patet, quod om-
              <lb/>
            nia quadra@a trapezij, γSN℟, æquabuntur reſiduo omnium quadra-
              <lb/>
            torumpar allelogrammi, XR, demptis omnibus quadratis portionis ſe-
              <lb/>
            micirculi, vel ſemiellipſis inter, ZL, TI, concluſæ: </s>
            <s xml:id="echoid-s5063" xml:space="preserve">Quia verò oſten-
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            ſa eſtratio omnium quadratorum cuiuſuis parallelogrammorum in alti-
              <lb/>
            tudine eadem cum portionibus, baſi autem æquali ſecundæ diametre,
              <lb/>
              <note position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">_24, & 28._
                <lb/>
              _lib. 2._</note>
            ad omnia quadrata trapeziorum, vel triangulorum in ijſdem exiſten-
              <lb/>
            tium, hine manifeſta eſt ratio eorundem ad dictareſidua, & </s>
            <s xml:id="echoid-s5064" xml:space="preserve">conſequen-
              <lb/>
            ter ad omnia quadrata portionum ſemicirculi, vel ſemiellipſis, DBF,
              <lb/>
            dictis parallelis interpoſitarum, vt ex. </s>
            <s xml:id="echoid-s5065" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s5066" xml:space="preserve">nota erit ratio, quam babent
              <lb/>
            omnia quadrata parallelogrammi, XR, ad omnia quadrata portionis,
              <lb/>
            ZTIL, & </s>
            <s xml:id="echoid-s5067" xml:space="preserve">ſic in reliquis. </s>
            <s xml:id="echoid-s5068" xml:space="preserve">Quia verò omnia quadrata trianguli, AE
              <lb/>
              <note position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">_F. Cor. 22_
                <lb/>
              _lib. 2._</note>
            C, ad omnia quadrata trianguli, SEN, ſunt in tripla ratione ipſius,
              <lb/>
            BE, ad, EM, ideò etiam patebit, quod omnia quadrata parallelogram-
              <lb/>
            mi, AF, demptis omnibus quadratis ſemicirculi, vel ſemiellipſis, D
              <lb/>
            BF, ad omnia quadrata parallelogrammi, VF, demptis omnibus qua-
              <lb/>
            dratis fruſti, TDFR, ſint in tripla ratione ipſius, BE, ad, EM, ideſt
              <lb/>
            vt cubus, BE, ad cubum, EM.</s>
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