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

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          <p>
            <s xml:id="echoid-s4044" xml:space="preserve">
              <pb o="168" file="0188" n="188" rhead="GEOMETRIÆ"/>
            lela eſt ipſi, ED. </s>
            <s xml:id="echoid-s4045" xml:space="preserve">Ducaturintra trapezia parallela ipſi, AB, vtcun-
              <lb/>
              <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a" number="108">
                <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0188-01"/>
              </figure>
            que, VC, ſecans, XA, in, S, &</s>
            <s xml:id="echoid-s4046" xml:space="preserve">, O
              <lb/>
            B, in, T, ſunt igitur triangula, AO
              <lb/>
            B, VOT, ſimilia, & </s>
            <s xml:id="echoid-s4047" xml:space="preserve">pariter ſunt ſi-
              <lb/>
            milia triangula, AXB, SXC, ergo,
              <lb/>
            AB, ad, VT, erit vt, BO, ad, OT,
              <lb/>
            .</s>
            <s xml:id="echoid-s4048" xml:space="preserve">i. </s>
            <s xml:id="echoid-s4049" xml:space="preserve">vt, BX, ad, XC, (quia, VC, pa-
              <lb/>
            rallela eſtipſi, AB, & </s>
            <s xml:id="echoid-s4050" xml:space="preserve">conſequenter
              <lb/>
            ipſi, OX,) .</s>
            <s xml:id="echoid-s4051" xml:space="preserve">i. </s>
            <s xml:id="echoid-s4052" xml:space="preserve">vt, AB, ad, SC, er-
              <lb/>
            go, VT, SC, erunt &</s>
            <s xml:id="echoid-s4053" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4054" xml:space="preserve">quales. </s>
            <s xml:id="echoid-s4055" xml:space="preserve">& </s>
            <s xml:id="echoid-s4056" xml:space="preserve">eo-
              <lb/>
            rum quadrata pariter &</s>
            <s xml:id="echoid-s4057" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4058" xml:space="preserve">qualia, ſic au-
              <lb/>
            tem de cæteris ipſi, AB, parallelis
              <lb/>
            idem oſtendetur, ergo omnes lineæ
              <lb/>
            trapezij, AERB, erunt &</s>
            <s xml:id="echoid-s4059" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4060" xml:space="preserve">quales omnibus lineis trapeZij, AIDB,
              <lb/>
            regula, AB, & </s>
            <s xml:id="echoid-s4061" xml:space="preserve">conſequenter ipſa trapezia erunt &</s>
            <s xml:id="echoid-s4062" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4063" xml:space="preserve">qualia, & </s>
            <s xml:id="echoid-s4064" xml:space="preserve">omnia
              <lb/>
            eorundem quadrata pariter &</s>
            <s xml:id="echoid-s4065" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4066" xml:space="preserve">qualia, quod oſtendere opus erat.</s>
            <s xml:id="echoid-s4067" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div418" type="section" level="1" n="254">
          <head xml:id="echoid-head269" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXVIII:</head>
          <p>
            <s xml:id="echoid-s4068" xml:space="preserve">SI parallelogrammum, & </s>
            <s xml:id="echoid-s4069" xml:space="preserve">trapezium habuerint commu-
              <lb/>
            nem baſim vnum ęquidiſtantium laterum trapezij, quod
              <lb/>
            ſit ſumptum pro regula; </s>
            <s xml:id="echoid-s4070" xml:space="preserve">Omnia quadrata parallelogrammi
              <lb/>
            ad omnia quadrata trapezij erunt, vt quadratum dictæ baſis
              <lb/>
            ad rectangulum ſub parallelis lateribus trapezij, cum, {1/3},
              <lb/>
            quadrati differentiæ dictorum laterum &</s>
            <s xml:id="echoid-s4071" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4072" xml:space="preserve">quidiſtantium.</s>
            <s xml:id="echoid-s4073" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4074" xml:space="preserve">Sit parallelogrammum, AC, & </s>
            <s xml:id="echoid-s4075" xml:space="preserve">trapezium, IBCO, cuius late-
              <lb/>
            rum &</s>
            <s xml:id="echoid-s4076" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4077" xml:space="preserve">quidiſtantium alterum, vt, BC, ſit communis baſis ipſi, & </s>
            <s xml:id="echoid-s4078" xml:space="preserve">
              <lb/>
            trapezio, & </s>
            <s xml:id="echoid-s4079" xml:space="preserve">regula. </s>
            <s xml:id="echoid-s4080" xml:space="preserve">Dico ergo omnia quadrata, AC, ad omnia qua-
              <lb/>
              <figure xlink:label="fig-0188-02" xlink:href="fig-0188-02a" number="109">
                <image file="0188-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0188-02"/>
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            drata trapezij, IBCO, eſſe vt quadratum,
              <lb/>
            BC, ad rectangulum ſub, BC, IO, vna
              <lb/>
            cum, {1/3}, quadrati differentiæ ipſarum, B
              <lb/>
            CIO. </s>
            <s xml:id="echoid-s4081" xml:space="preserve">Sumatur in, DA, ipſa, ED, &</s>
            <s xml:id="echoid-s4082" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4083" xml:space="preserve">-
              <lb/>
            qualis ipſi, IO, & </s>
            <s xml:id="echoid-s4084" xml:space="preserve">iungatur, BE, & </s>
            <s xml:id="echoid-s4085" xml:space="preserve">per,
              <lb/>
            E, ipſis, AB, DC, parallela ducatur, E
              <lb/>
              <note position="left" xlink:label="note-0188-01" xlink:href="note-0188-01a" xml:space="preserve">PerD. Co
                <lb/>
              toll. 23.
                <lb/>
              huius.</note>
            M: </s>
            <s xml:id="echoid-s4086" xml:space="preserve">Omnia ergo quadrata trapezij, EBC
              <lb/>
            D, perlineam, EM, diuiduntur in omnia
              <lb/>
            quadrata trianguli, EBM, & </s>
            <s xml:id="echoid-s4087" xml:space="preserve">in omnia
              <lb/>
            quadrata, MD, & </s>
            <s xml:id="echoid-s4088" xml:space="preserve">in rectangula ſub tri-
              <lb/>
            angulo, EBM, &</s>
            <s xml:id="echoid-s4089" xml:space="preserve">, EC, bis ſumpta; </s>
            <s xml:id="echoid-s4090" xml:space="preserve">ad horum ergo ſingula com-
              <lb/>
            paremus omnia quadrata, AC. </s>
            <s xml:id="echoid-s4091" xml:space="preserve">Igitur omnia quadrata, AC, ad
              <lb/>
              <note position="left" xlink:label="note-0188-02" xlink:href="note-0188-02a" xml:space="preserve">9. huius.</note>
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