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

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          <pb o="193" file="0213" n="213" rhead="LIBER II."/>
          <p>
            <s xml:id="echoid-s4772" xml:space="preserve">Hæc manifeſta eſt, nam habebunt baſes ipſis altitudinibus recipro-
              <lb/>
            cas, quod etiam vniuerſalius oſtenditur Vndecimo Elementorum
              <lb/>
            Propoſ. </s>
            <s xml:id="echoid-s4773" xml:space="preserve">36.</s>
            <s xml:id="echoid-s4774" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div489" type="section" level="1" n="295">
          <head xml:id="echoid-head311" xml:space="preserve">THEOREMA XLII. PROPOS. XLII.</head>
          <p>
            <s xml:id="echoid-s4775" xml:space="preserve">DAta recta linea terminata, vtcumque in puncto diuiſa,
              <lb/>
            poſſibile eſt ipſam ad alteram eiuſdem partium ita
              <lb/>
            producere, vt cubus compoſitæ ex propoſita linea, & </s>
            <s xml:id="echoid-s4776" xml:space="preserve">adiun-
              <lb/>
            cta, ſit æqualis cubo propoſitæ lineæ, ſimul cum cubo com-
              <lb/>
            poſitæ ex adiecta, & </s>
            <s xml:id="echoid-s4777" xml:space="preserve">illi conterminante portione ſectæ lineę.</s>
            <s xml:id="echoid-s4778" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4779" xml:space="preserve">Sit data recta linea, AC, terminata, diuiſaq; </s>
            <s xml:id="echoid-s4780" xml:space="preserve">vtcumque in pun-
              <lb/>
            cto, B, oftendendum eſt poſſibile eſſe ipſam ita producere ad alteram
              <lb/>
            illius partium, vt ad, C, vt cubus compoſitę ex, AC, & </s>
            <s xml:id="echoid-s4781" xml:space="preserve">adiecta, ſit
              <lb/>
            æqualis cubo, AC, cum cubo compoſitæ ex eadem adiecta, & </s>
            <s xml:id="echoid-s4782" xml:space="preserve">ex,
              <lb/>
            BC, portione, AC, adiectæ conterminante. </s>
            <s xml:id="echoid-s4783" xml:space="preserve">Producatur ergo, C
              <lb/>
            A, ad partes, A, vt in, N, ita quod, NB, ſit tripla, BA, ſiat dein-
              <lb/>
            de, vt, NB, ad, BC, ita quadratum, BC, ad quadratum rectæ li-
              <lb/>
            neę, M, ſeorſim poſitæ: </s>
            <s xml:id="echoid-s4784" xml:space="preserve">Vlterius exponatur recta, EF, æqualis com-
              <lb/>
            poſitæ ex, AC, CB, cui applicetur rectangulum æquale quadrato,
              <lb/>
              <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a" number="128">
                <image file="0213-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0213-01"/>
              </figure>
            M, excedens figura quadrata,
              <lb/>
              <note position="right" xlink:label="note-0213-01" xlink:href="note-0213-01a" xml:space="preserve">29. Sex.
                <lb/>
              lem,</note>
            cuius latus ſit, FH, producatur
              <lb/>
            autem, AC, verſus, C, vt in,
              <lb/>
            D, ita nempè, vt, CD, ſit
              <lb/>
            æqualis, FH. </s>
            <s xml:id="echoid-s4785" xml:space="preserve">Dico cubum to-
              <lb/>
            tius, AD, æquari duobus cubis,
              <lb/>
            AC, BD. </s>
            <s xml:id="echoid-s4786" xml:space="preserve">Cum.</s>
            <s xml:id="echoid-s4787" xml:space="preserve">n. </s>
            <s xml:id="echoid-s4788" xml:space="preserve">ſit, vt, N
              <lb/>
            B, ad, BC, ita quadratum, BC,
              <lb/>
            ad quadratum, M, ideò parallelepipedum ſub altitudine, AB, (qu
              <lb/>
            eſt, {1/3}, prædictę altitudinis, NB,) & </s>
            <s xml:id="echoid-s4789" xml:space="preserve">quadrato, M, æquabitur ter-
              <lb/>
            tiæ parti cubi, BC. </s>
            <s xml:id="echoid-s4790" xml:space="preserve">Quoniam verò quadratum, M, æquatur rectan-
              <lb/>
              <note position="right" xlink:label="note-0213-02" xlink:href="note-0213-02a" xml:space="preserve">E. Cor. 4
                <lb/>
              Gen. 34.
                <lb/>
              huius.</note>
            gulo, EHF, ideſt rectangulo ſub compoſita ex, AD, BC, & </s>
            <s xml:id="echoid-s4791" xml:space="preserve">ſub,
              <lb/>
            CD, ideò parallelepipedum ſub altitudine, AB, & </s>
            <s xml:id="echoid-s4792" xml:space="preserve">baſi rectangulo
              <lb/>
            ſub compoſita ex, AD, BC, & </s>
            <s xml:id="echoid-s4793" xml:space="preserve">ſub, DC, æquabitur tertiæ parti
              <lb/>
            cubi, BC, addatur commune parallelepipedum ſub, BC, & </s>
            <s xml:id="echoid-s4794" xml:space="preserve">baſi re-
              <lb/>
            ctangulo, BDC, erit ex vna parte hoc parallelepipedum cum, {1/3}, cu-
              <lb/>
            bi, BC, ex alia verò hæcſumma; </s>
            <s xml:id="echoid-s4795" xml:space="preserve">ſcilicet parallelepipedum ſub, AB,
              <lb/>
            & </s>
            <s xml:id="echoid-s4796" xml:space="preserve">ſub rectangulo ſub compoſita ex, AD, BC, & </s>
            <s xml:id="echoid-s4797" xml:space="preserve">ſub, DC, vna
              <lb/>
            cum parallelepipedo ſub, BC, & </s>
            <s xml:id="echoid-s4798" xml:space="preserve">rectangulo, BDC, quæ quidem
              <lb/>
            ſumma efficit parallelepipedum ſub, AC, & </s>
            <s xml:id="echoid-s4799" xml:space="preserve">rectangulo, ADC, </s>
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