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

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        <div xml:id="echoid-div1073" type="section" level="1" n="643">
          <pb o="458" file="0478" n="478" rhead="GEOMETRIÆ"/>
          <p>
            <s xml:id="echoid-s11713" xml:space="preserve">Huius demoſtratio non erit alia à demoſtratione 13. </s>
            <s xml:id="echoid-s11714" xml:space="preserve">huius,
              <lb/>
            propterea ibi recolatur.</s>
            <s xml:id="echoid-s11715" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1074" type="section" level="1" n="644">
          <head xml:id="echoid-head674" xml:space="preserve">THEOREMA XIX. PROPOS. XIX.</head>
          <p>
            <s xml:id="echoid-s11716" xml:space="preserve">PRimi circuli ſpatium helicum ad ſpatium helicum ſe-
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            cundi circuli erit, vt tertia pars quadrati radij primi
              <lb/>
            circuli ad rectangulum ſub rad io primi, & </s>
            <s xml:id="echoid-s11717" xml:space="preserve">ſecundi circuli,
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            vna cum tertia parte quadrati exceſſus radij ſecundi circu-
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            li ſuper radium primi. </s>
            <s xml:id="echoid-s11718" xml:space="preserve">Spatium verò ſecundi circuli ad
              <lb/>
            ſpatium tertij erit, vt rectangulum ſub radio eiuſdem, & </s>
            <s xml:id="echoid-s11719" xml:space="preserve">
              <lb/>
            ſub radio circuli vnitate minoris, ideſt primi, vna cum ter-
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            tia parte quadrati differentiæ horum radiorũ, ad rectangu-
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            lum ſub radio eiuſdem, & </s>
            <s xml:id="echoid-s11720" xml:space="preserve">ſub radio circuli vnitate maio-
              <lb/>
            ris, ideſt tertij, vna cum tertia parte quadrati differentiæ
              <lb/>
            iſtorum radiorum, & </s>
            <s xml:id="echoid-s11721" xml:space="preserve">ſic deinceps in reliquis.</s>
            <s xml:id="echoid-s11722" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11723" xml:space="preserve">Exponantur ſuper eodem centro, A, circuli, primus, HRVO,
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            ſecundus, kLNM, tertius autem, CDFB, cum ſpatijs ſub ſpirali-
              <lb/>
            bus eiuſdem numeri cum circulis, primo, AGHA, ſecundo, HPK
              <lb/>
            MH, tertio autem, MZSBM. </s>
            <s xml:id="echoid-s11724" xml:space="preserve">Dico ſpatium primum ad ſecundũ
              <lb/>
            eſſe vt {1/3}. </s>
            <s xml:id="echoid-s11725" xml:space="preserve">quadrati HA, ad rectangulum ſub, HA, AM, vna cum {1/3}.
              <lb/>
            </s>
            <s xml:id="echoid-s11726" xml:space="preserve">quadrati, HM, ſecundum verò ad tertium eſſe vt rectangulum ſub,
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            HA, AM, cum {1/3}. </s>
            <s xml:id="echoid-s11727" xml:space="preserve">quadrati, HM, ad rectangulum ſub, MA, AB,
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            vna cum {1/5}. </s>
            <s xml:id="echoid-s11728" xml:space="preserve">quadrati, MB. </s>
            <s xml:id="echoid-s11729" xml:space="preserve">Nam ſpatium, AGH, ad ſpatium, HP
              <lb/>
            kM, habet rationem compoſita ex ratione ſpatij, AGH, ad cir-
              <lb/>
            culum, OVRH, ideſt ex ratione {1/3}. </s>
            <s xml:id="echoid-s11730" xml:space="preserve">quadrati, HA, ad quadratum,
              <lb/>
              <note position="left" xlink:label="note-0478-01" xlink:href="note-0478-01a" xml:space="preserve">9. huius.</note>
            HA, & </s>
            <s xml:id="echoid-s11731" xml:space="preserve">ex ratione circuli, OVRH, ad circulum, MkLN, ideſt ex
              <lb/>
            ratione quadrati, HA, ad quadratum, AM, & </s>
            <s xml:id="echoid-s11732" xml:space="preserve">ex ratione circuli, C
              <lb/>
              <note position="left" xlink:label="note-0478-02" xlink:href="note-0478-02a" xml:space="preserve">Coroll. 2.
                <lb/>
              11. l. 3.
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              15. huius.
                <lb/>
              Defin. 12.
                <lb/>
              l. 1.</note>
            DFB, ad ſpatium, HPMH, ideſt ex ratione quadrati, MA, ad re-
              <lb/>
            ctangulum, MAH, vna cum {1/3}. </s>
            <s xml:id="echoid-s11733" xml:space="preserve">quadrati, MH, quæ rationes com-
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            ponunt rationem {1/3}. </s>
            <s xml:id="echoid-s11734" xml:space="preserve">quadrati, AH, ad rectangulum, MAH, cum
              <lb/>
            {1/3}. </s>
            <s xml:id="echoid-s11735" xml:space="preserve">quadrati, HM. </s>
            <s xml:id="echoid-s11736" xml:space="preserve">Item ſpatium, HPMH, ad ſpatium, MZSBM,
              <lb/>
            habet rationem compoſitam ex ratione ſpatij, HPMH, ad circu-
              <lb/>
            lum, kLNM, ideſt ex ratione rectanguli, HAM, cum {1/3}. </s>
            <s xml:id="echoid-s11737" xml:space="preserve">quadrati,
              <lb/>
              <note position="left" xlink:label="note-0478-03" xlink:href="note-0478-03a" xml:space="preserve">15. huiu.</note>
            HM, ad quadratum, AM, & </s>
            <s xml:id="echoid-s11738" xml:space="preserve">ex ratione circuli, kLNM, ad circu-
              <lb/>
            lum, CDFB, ideſt quadrati, MA, ad quadratum, AB, & </s>
            <s xml:id="echoid-s11739" xml:space="preserve">tandem
              <lb/>
              <note position="left" xlink:label="note-0478-04" xlink:href="note-0478-04a" xml:space="preserve">Coroll. 2.
                <lb/>
              11. l. 3.
                <lb/>
              15. huius.</note>
            ex ratione circuli, CDFB, ad ſpatium, MZSBM, ideſt ex ratione
              <lb/>
            quadrati, BA, ad rectangulum, BAM, cum {1/3}. </s>
            <s xml:id="echoid-s11740" xml:space="preserve">quadrati, MB, </s>
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