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

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            <s xml:id="echoid-s4249" xml:space="preserve">
              <pb o="173" file="0193" n="193" rhead="LIBER II."/>
            Theorematis, nam, vt alibi oſtendimus, ſi ſupponamus ipſi, BE, adiun-
              <lb/>
            girectam, EM, æqualem ipſi, DE, & </s>
            <s xml:id="echoid-s4250" xml:space="preserve">BE, eſſe æqualem ipſi, EF, om-
              <lb/>
            nes lineæ trapezij, ADEC, erunt æquales omnibus abſciſſis ipſius, BE,
              <lb/>
            (quæ ſit propoſita linea) adiuncta tamen, EM, & </s>
            <s xml:id="echoid-s4251" xml:space="preserve">omnes lineæ triangu-
              <lb/>
            li, CEF, (intellige ſemper regulam, DF,) erunt æquales reſiduis om-
              <lb/>
            nium abſciſſarum prop@ſitæ lineæ, BE, item omnes lineæ, AE, erunt
              <lb/>
            æquales ijs, quæ adiunguntur maximis abſciſſarum, BE, nam earum ſin-
              <lb/>
            gulæ ſunt æquales ipſi, DE, vel, EM, & </s>
            <s xml:id="echoid-s4252" xml:space="preserve">omnes lineæ, EC, maximis
              <lb/>
            abſciſſarum, BE, pariter æquales erunt, vnde patet propoſitum. </s>
            <s xml:id="echoid-s4253" xml:space="preserve">Exſe-
              <lb/>
            cunda verò parte conſimili ratione colligemus rectangula ſub maximis
              <lb/>
            abſciſſ rum propoſitæ lineæ, vt, BE, adiuncta quadam, vt, EM, & </s>
            <s xml:id="echoid-s4254" xml:space="preserve">
              <lb/>
            ſub maximis abſciſſarum eiuſdem propoſitæ, BE, ad rectangula ſub om-
              <lb/>
            nibus abſciſſis, ſumptis verſus, E, eiuſdem propoſitæ, BE, adiuncta,
              <lb/>
            EM, & </s>
            <s xml:id="echoid-s4255" xml:space="preserve">ſub eiuſdem omnibus abſciſſis propoſitæ, BE, eſſe vt compoſita
              <lb/>
            ex propoſita, & </s>
            <s xml:id="echoid-s4256" xml:space="preserve">adiecta .</s>
            <s xml:id="echoid-s4257" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s4258" xml:space="preserve">vt, BM, ad compoſitam ex, {1/2}, adiectæ, quæ
              <lb/>
            eſt, ME, & </s>
            <s xml:id="echoid-s4259" xml:space="preserve">{1/3}, propoſitæ, quæ eſt, BE.</s>
            <s xml:id="echoid-s4260" xml:space="preserve"/>
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        <div xml:id="echoid-div430" type="section" level="1" n="260">
          <head xml:id="echoid-head275" xml:space="preserve">THEOREMA XXXI. PROPOS. XXXI.</head>
          <p>
            <s xml:id="echoid-s4261" xml:space="preserve">EXpoſita Propoſit. </s>
            <s xml:id="echoid-s4262" xml:space="preserve">antecedentis figura, & </s>
            <s xml:id="echoid-s4263" xml:space="preserve">intra parallelas,
              <lb/>
            AC, DF, eiſdem æquidiſtanter ducta recta linea, H
              <lb/>
            O, quæ ſecet, BE, in, M, &</s>
            <s xml:id="echoid-s4264" xml:space="preserve">, CE, in, N, oſtendemus, re-
              <lb/>
            gula eadem, DF, rectangula ſub parallelogrammis, AO, O
              <lb/>
            B, ad rectangula ſub trapezijs, HACN, MBCN, eſſe vt
              <lb/>
            rectangulum, HOM, ad rectangulum ſub, HO, MN, cum
              <lb/>
            rectangulo ſub compoſita ex, {1/2}, HM, &</s>
            <s xml:id="echoid-s4265" xml:space="preserve">, {1/5}, NO, & </s>
            <s xml:id="echoid-s4266" xml:space="preserve">ſub, NO.</s>
            <s xml:id="echoid-s4267" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4268" xml:space="preserve">Rectangula enim ſub parallelogram-
              <lb/>
              <figure xlink:label="fig-0193-01" xlink:href="fig-0193-01a" number="113">
                <image file="0193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0193-01"/>
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            mis, AO, OB, ad rectangula ſub paral-
              <lb/>
              <note position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve">3. huius.</note>
            lelogrammis, AM, MC, ſunt vt re-
              <lb/>
            ctangulum, HOM, ad rectangulum,
              <lb/>
              <note position="right" xlink:label="note-0193-02" xlink:href="note-0193-02a" xml:space="preserve">Coroll. 1.
                <lb/>
              26. huius.</note>
            HMO, rectangula verò ſub, AM, M
              <lb/>
            C, ad rectangula ſub parallelogrammo,
              <lb/>
            AM, & </s>
            <s xml:id="echoid-s4269" xml:space="preserve">trapezio, BMNC, ſunt vt, B
              <lb/>
              <note position="right" xlink:label="note-0193-03" xlink:href="note-0193-03a" xml:space="preserve">20. huius.</note>
            O, ad trapezium, BMNC, .</s>
            <s xml:id="echoid-s4270" xml:space="preserve">i. </s>
            <s xml:id="echoid-s4271" xml:space="preserve">vt, MO,
              <lb/>
            ad, MN, cum, {1/2}, NO, vel vt rectan-
              <lb/>
              <note position="right" xlink:label="note-0193-04" xlink:href="note-0193-04a" xml:space="preserve">5. huius.</note>
            gulum, HMO, ad rectangulum ſub, H
              <lb/>
            M, & </s>
            <s xml:id="echoid-s4272" xml:space="preserve">ſub compoſita ex, MN, &</s>
            <s xml:id="echoid-s4273" xml:space="preserve">, {1/2}, N
              <lb/>
            O, ergo, ex æquali, rectangula ſub, AO, OB, ad rectangula ſub,
              <lb/>
            AM, & </s>
            <s xml:id="echoid-s4274" xml:space="preserve">trapezio, BMNC, ſunt vt rectangulum, HOM, ad rectan-
              <lb/>
            gulum ſub, HM, & </s>
            <s xml:id="echoid-s4275" xml:space="preserve">compoſita ex, MN, &</s>
            <s xml:id="echoid-s4276" xml:space="preserve">, {1/2}, NO, quod ſerua.</s>
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