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

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        <div xml:id="echoid-div906" type="section" level="1" n="541">
          <pb o="376" file="0396" n="396" rhead="GEOMETRIÆ"/>
          <p>
            <s xml:id="echoid-s9671" xml:space="preserve">Ducantur per puncta, X, R, XN, RM, rectæ lineæ parallelæ
              <lb/>
            axi, vel diametro hyperbolæ, OV, occurrentes, TP, productæ, in,
              <lb/>
            N, M: </s>
            <s xml:id="echoid-s9672" xml:space="preserve">Omnia ergo quadr. </s>
            <s xml:id="echoid-s9673" xml:space="preserve">trapezij, GPRV, ad omnia quadrata
              <lb/>
            quadlilinei, GVXY, habent rationem compoſitam ex ea, quam
              <lb/>
            habent omnia quadrata trapezij, PGVR, ad omnia quadrata,
              <lb/>
            GR, .</s>
            <s xml:id="echoid-s9674" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9675" xml:space="preserve">ex ea, quam habet rectangulum ſub, PG, VR, cum {1/3}. </s>
            <s xml:id="echoid-s9676" xml:space="preserve">qua-
              <lb/>
            drati, PM, ad quadratum, VR, & </s>
            <s xml:id="echoid-s9677" xml:space="preserve">ex ea, quam habent omnia qua-
              <lb/>
            drata, GR, ad omnia quadrata, GX, ideſt ex ea, quam habet qua-
              <lb/>
            dratum, RV, ad quadratum, VX; </s>
            <s xml:id="echoid-s9678" xml:space="preserve">quæ duæ rationes componunt
              <lb/>
              <figure xlink:label="fig-0396-01" xlink:href="fig-0396-01a" number="271">
                <image file="0396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0396-01"/>
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            rationem, quam habet rectangulum ſub,
              <lb/>
            GP, VR, cum {1/3}. </s>
            <s xml:id="echoid-s9679" xml:space="preserve">quadrati, PM, ad qua
              <lb/>
              <note position="left" xlink:label="note-0396-01" xlink:href="note-0396-01a" xml:space="preserve">28.2.</note>
            dratum, VX; </s>
            <s xml:id="echoid-s9680" xml:space="preserve">& </s>
            <s xml:id="echoid-s9681" xml:space="preserve">tandem ex ea, quam ha-
              <lb/>
            bent omnia quadrata, GX, ad omnia
              <lb/>
            quadrata, GYXV, .</s>
            <s xml:id="echoid-s9682" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9683" xml:space="preserve">ex ea, quam habet
              <lb/>
              <note position="left" xlink:label="note-0396-02" xlink:href="note-0396-02a" xml:space="preserve">9 2.</note>
            rectangulum, BVO, ad rectangulum ſub,
              <lb/>
            BV, GO, vna cum rectangulo ſub com-
              <lb/>
            poſita ex {1/2}. </s>
            <s xml:id="echoid-s9684" xml:space="preserve">BO, & </s>
            <s xml:id="echoid-s9685" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9686" xml:space="preserve">GV, & </s>
            <s xml:id="echoid-s9687" xml:space="preserve">ſub, GV,
              <lb/>
              <note position="left" xlink:label="note-0396-03" xlink:href="note-0396-03a" xml:space="preserve">6.2.</note>
            ergo omnia quadrata trapezij, PGVR,
              <lb/>
            ad omnia quadrata quadrilinei, YGVX,
              <lb/>
            vel eorum quadrupla .</s>
            <s xml:id="echoid-s9688" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9689" xml:space="preserve">omnia quadrata
              <lb/>
            trapezij, THRP, ad omnia quadrata fru-
              <lb/>
            ſti, ISXY, habebunt rationem compoſi
              <lb/>
              <note position="left" xlink:label="note-0396-04" xlink:href="note-0396-04a" xml:space="preserve">3. huius.</note>
            tam ex ea; </s>
            <s xml:id="echoid-s9690" xml:space="preserve">quam habet rectangulum ſub,
              <lb/>
            GP, VR, cum {1/3}. </s>
            <s xml:id="echoid-s9691" xml:space="preserve">quadrati, PM, ad quadratum, VX, & </s>
            <s xml:id="echoid-s9692" xml:space="preserve">ex ea, quã
              <lb/>
            habet rectangulum, BVO, ad rectangulum ſub, BV, GO, vna cum
              <lb/>
            rectangulo ſub compoſita ex {1/2}. </s>
            <s xml:id="echoid-s9693" xml:space="preserve">BO, & </s>
            <s xml:id="echoid-s9694" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9695" xml:space="preserve">GV, & </s>
            <s xml:id="echoid-s9696" xml:space="preserve">ſub, GV, quod
              <lb/>
            oſtendere opus erat.</s>
            <s xml:id="echoid-s9697" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div908" type="section" level="1" n="542">
          <head xml:id="echoid-head566" xml:space="preserve">THEOREMA VIII. PROPOS. IX.</head>
          <p>
            <s xml:id="echoid-s9698" xml:space="preserve">VIſa adhuc anteced. </s>
            <s xml:id="echoid-s9699" xml:space="preserve">figura, exponemus aliter rationẽ
              <lb/>
            ibi adinuentam tantummodo compoſitam ex dua-
              <lb/>
            bus, ad vnam ſolum eandem reducentes, probando .</s>
            <s xml:id="echoid-s9700" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s9701" xml:space="preserve">om-
              <lb/>
            nia quadrata trianguli, HCR, regula eadem, HR, retenta
              <lb/>
            ad omnia quadrata hyperbolæ, SOX, eſſe vt cubus, CV,
              <lb/>
            eſt ad parallelepipedum ter ſub, CO, & </s>
            <s xml:id="echoid-s9702" xml:space="preserve">quadrato, OV,
              <lb/>
            cum cubo, OV.</s>
            <s xml:id="echoid-s9703" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9704" xml:space="preserve">Nam vt in ſupradicta Propoſit. </s>
            <s xml:id="echoid-s9705" xml:space="preserve">oſtenſum eſt, omnia quadrata
              <lb/>
              <note position="left" xlink:label="note-0396-05" xlink:href="note-0396-05a" xml:space="preserve">6. huius</note>
            trianguli, CHR, ad omnia quadrata hyperbolæ, SOX, conuertẽ.
              <lb/>
            </s>
            <s xml:id="echoid-s9706" xml:space="preserve">do, habent rationem compoſitam ex ea, quam habet </s>
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