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

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          <p>
            <s xml:id="echoid-s11515" xml:space="preserve">
              <pb o="449" file="0469" n="469" rhead="LIBER VI."/>
            ſpatium, OP℟, erit vt omnia quadrata, MG, regula, MI, ad om-
              <lb/>
            nia quadrata trianguli, TLC, eſt autem idem ſector, AOQ, ad ſpa-
              <lb/>
            tium, OP℟Q, vt omnia quadrata, MG, ad omnia quadratatrian-
              <lb/>
            guli, MIC, regula eadem, ergoſector, AOQ, ad reliquum ſpa-
              <lb/>
            tium, dempto ſpatio, OP℟, à ſpatio, OP℟Q, erit vt omnia qua-
              <lb/>
            drata, MG, regula, MI, ad omnia quadrata trapezij, MILT, ſed
              <lb/>
              <note position="right" xlink:label="note-0469-01" xlink:href="note-0469-01a" xml:space="preserve">28. 1. 2.</note>
            hæcſunt, vt quadratum, GT, adrectangulum, GTL, cum {1/3}. </s>
            <s xml:id="echoid-s11516" xml:space="preserve">qua-
              <lb/>
            drati, LG, ergo, conuertendo, ſpatium, ℟OQ, ad ſectorem, AOQ,
              <lb/>
            erit vtrectangulum, AO℟, cum tertia parte quadrati, A℟, ad qua-
              <lb/>
            dratum, AO, quod erat oſtendendum.</s>
            <s xml:id="echoid-s11517" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1060" type="section" level="1" n="637">
          <head xml:id="echoid-head667" xml:space="preserve">THEOREMA XIII. PROPOS. XIII.</head>
          <p>
            <s xml:id="echoid-s11518" xml:space="preserve">IN eadem antecedentis ſigura centro, O, diſtantia, O℟,
              <lb/>
            deſcripta circumferentia, ℟Y, oſtendemus trilineum,
              <lb/>
            A℟Q, ad trilineum, ℟QY, eſſe vt, ℟O, cum {2/3}. </s>
            <s xml:id="echoid-s11519" xml:space="preserve">℟A, ad, ℟
              <lb/>
            O, cum tertia parte ipſius, ℟A.</s>
            <s xml:id="echoid-s11520" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11521" xml:space="preserve">Quia enim ex antecedenteſector, AOQ, ad ſpatium, Q℟O, eſt
              <lb/>
            vt quadratum, AO, ad rectangulum, AO℟, cum {1/3}. </s>
            <s xml:id="echoid-s11522" xml:space="preserve">quadrati, A ℟,
              <lb/>
            per conuerſionem rationis, idem ſector ad trilineum, A℟Q, erit vt
              <lb/>
            quadratum, AO, ad rectangulum, O℟A, cum {2/3}. </s>
            <s xml:id="echoid-s11523" xml:space="preserve">quadrati, ℟A, nã
              <lb/>
            dempto rectangulo, AO℟, à quadrato, AO, remanet rectangulũ,
              <lb/>
            OA℟,. </s>
            <s xml:id="echoid-s11524" xml:space="preserve">i. </s>
            <s xml:id="echoid-s11525" xml:space="preserve">rectangulũ, O℟A, cum quadrato, ℟A, à quo ablato {1/3}. </s>
            <s xml:id="echoid-s11526" xml:space="preserve">re-
              <lb/>
            manet rectangulum, O℟A, cum {2/3}. </s>
            <s xml:id="echoid-s11527" xml:space="preserve">quadrati, ℟A, ideſt cum re-
              <lb/>
              <note position="right" xlink:label="note-0469-02" xlink:href="note-0469-02a" xml:space="preserve">1. Secundi
                <lb/>
              Elem.</note>
            ctanguloſub {2/3}. </s>
            <s xml:id="echoid-s11528" xml:space="preserve">℟ A, & </s>
            <s xml:id="echoid-s11529" xml:space="preserve">ſub, ℟A, quod cum rectangulo, O℟A, cõ-
              <lb/>
            ficit rectangulum ſub compoſita ex, O℟, & </s>
            <s xml:id="echoid-s11530" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s11531" xml:space="preserve">℟A, & </s>
            <s xml:id="echoid-s11532" xml:space="preserve">ſub, ℟A,
              <lb/>
            conuertendo igitur trilineum, A℟Q, ad ſectorem, AOQ, erit vt
              <lb/>
              <note position="right" xlink:label="note-0469-03" xlink:href="note-0469-03a" xml:space="preserve">Coroll. 2.
                <lb/>
              3. huius.</note>
            rectangulum ſub compoſita ex, O℟, & </s>
            <s xml:id="echoid-s11533" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s11534" xml:space="preserve">℟A, & </s>
            <s xml:id="echoid-s11535" xml:space="preserve">ſub, ℟A, ad qua-
              <lb/>
            dratum, OA, inſuper ſector, AOQ, ad ſectorem, ℟OY, eſt vt qua-
              <lb/>
            dratum, AO, ad quadratum, O℟, & </s>
            <s xml:id="echoid-s11536" xml:space="preserve">quia idem ſector, AOQ, ad
              <lb/>
            ſpatium, Q℟O, eſt vt quadratum, AO, ad rectangulum, AO℟, cũ
              <lb/>
            {1/3}. </s>
            <s xml:id="echoid-s11537" xml:space="preserve">quadrati, ℟A, ideò ſector, AOQ, ad reliquum dempto à ſpatio,
              <lb/>
            ℟OQ, ſectore, ℟OY, ideſt ad trilineum, Q℟Y, erit vt quadratu,
              <lb/>
              <note position="right" xlink:label="note-0469-04" xlink:href="note-0469-04a" xml:space="preserve">3. Secundi
                <lb/>
              Elem.</note>
            AO, ad reliquum rectanguli, AO℟, cum {1/3}. </s>
            <s xml:id="echoid-s11538" xml:space="preserve">quadrati, ℟A, ab eo
              <lb/>
            dempto quadrato, ℟O, ideſt ad rectangulum, O℟A, cum {1/3}. </s>
            <s xml:id="echoid-s11539" xml:space="preserve">qua-
              <lb/>
            drati, ℟A, erat autem trilineum, A℟Q, ad rectorem, AOQ, vt re-
              <lb/>
            ctangulum ſub compoſita ex, O℟, & </s>
            <s xml:id="echoid-s11540" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s11541" xml:space="preserve">℟A, ad quadratum, AO,
              <lb/>
            ergo ex æquali trilineum, A℟Q, ad trilineum, ℟QY, erit vt rectã-
              <lb/>
            gulum ſub compoſita ex, O℟, & </s>
            <s xml:id="echoid-s11542" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s11543" xml:space="preserve">℟A, & </s>
            <s xml:id="echoid-s11544" xml:space="preserve">ſub, ℟A, ad rectangu-
              <lb/>
            lum, O℟A, cum 3. </s>
            <s xml:id="echoid-s11545" xml:space="preserve">parte quadrati, ℟A, ideſt ad rectang. </s>
            <s xml:id="echoid-s11546" xml:space="preserve">ſub </s>
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