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

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            <s xml:id="echoid-s6286" xml:space="preserve">
              <pb o="252" file="0272" n="272" rhead="GEOMETRIÆ"/>
            BOMFH, erunt vt, AN, ad figuram, BDMO, quod oſtendere
              <lb/>
            oportebat. </s>
            <s xml:id="echoid-s6287" xml:space="preserve">Per hanc autem, & </s>
            <s xml:id="echoid-s6288" xml:space="preserve">antecedentem Propoſit. </s>
            <s xml:id="echoid-s6289" xml:space="preserve">vniuerſa-
              <lb/>
            lius oſtenduntur Propoſ. </s>
            <s xml:id="echoid-s6290" xml:space="preserve">15. </s>
            <s xml:id="echoid-s6291" xml:space="preserve">16. </s>
            <s xml:id="echoid-s6292" xml:space="preserve">necnon Corollaria Prop. </s>
            <s xml:id="echoid-s6293" xml:space="preserve">19. </s>
            <s xml:id="echoid-s6294" xml:space="preserve">& </s>
            <s xml:id="echoid-s6295" xml:space="preserve">20.</s>
            <s xml:id="echoid-s6296" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div604" type="section" level="1" n="351">
          <head xml:id="echoid-head368" xml:space="preserve">THEOREMA XXX. PROPOS. XXXI.</head>
          <p>
            <s xml:id="echoid-s6297" xml:space="preserve">SI parallelogrammum fuerit ellipſi circumſcriptum, ita ta-
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            men, vt eiuſdem latera non tangant ellipſim in extremis
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            punctis axium eiuſdem; </s>
            <s xml:id="echoid-s6298" xml:space="preserve">portiones coalternè tangentes erunt
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            æquales; </s>
            <s xml:id="echoid-s6299" xml:space="preserve">& </s>
            <s xml:id="echoid-s6300" xml:space="preserve">ſi duabus oppoſitis tangentibus ducantur paral-
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            lelę abſcindentes à reliquis coalternis tangentibus rectas li-
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            neas æquales, ſumptas verſus puncta contactuum; </s>
            <s xml:id="echoid-s6301" xml:space="preserve">rectangu-
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            lum, quod continetur ſub vnius parallelarum ea parte, quæ
              <lb/>
            manet intra curuam ellipſis, & </s>
            <s xml:id="echoid-s6302" xml:space="preserve">tangentem ex ea parte, & </s>
            <s xml:id="echoid-s6303" xml:space="preserve">ſub
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            reliqua illi in directum manente intra ellipſim, erit æquale
              <lb/>
            rectangulo ad coalternam tangentem ſimiliter ſumpto.</s>
            <s xml:id="echoid-s6304" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6305" xml:space="preserve">Sit ergo ellipſis, BDMG, cui ſit circumſcriptum parallelogram-
              <lb/>
            mum, AR, ita tamen, vt puncta contactuum non ſint puncta ex-
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            trema axium eiuſdem, tangant autem in punctis, BDMG, & </s>
            <s xml:id="echoid-s6306" xml:space="preserve">iun-
              <lb/>
            gantur, BM, DG, & </s>
            <s xml:id="echoid-s6307" xml:space="preserve">quoniam, AC, FR, ſunt tangentes paralle-
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            læ, vt etiam, AF, CR, ideò, BM, GD, per centrum ellipſis tran-
              <lb/>
              <note position="left" xlink:label="note-0272-01" xlink:href="note-0272-01a" xml:space="preserve">Elicitur
                <lb/>
              ex 27. 2.
                <lb/>
              Con.</note>
            ſibunt, ſit earum communis ſectio punctum, S, ergo, S, erit centrum
              <lb/>
              <figure xlink:label="fig-0272-01" xlink:href="fig-0272-01a" number="169">
                <image file="0272-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0272-01"/>
              </figure>
            ellipſis, cum, BM, GD, ſint diametri.
              <lb/>
            </s>
            <s xml:id="echoid-s6308" xml:space="preserve">Dico ergo portiones laterum parallelo-
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            grammi, AR, coalternè tangentes eſſe
              <lb/>
            æquales. </s>
            <s xml:id="echoid-s6309" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s6310" xml:space="preserve">AD, ipſi, GR, AB, ipſi, M
              <lb/>
            R, BC, ipſi, FM, &</s>
            <s xml:id="echoid-s6311" xml:space="preserve">, CG, ipſi, DF; </s>
            <s xml:id="echoid-s6312" xml:space="preserve">
              <lb/>
            iungantur, BG, DM; </s>
            <s xml:id="echoid-s6313" xml:space="preserve">in triangulis ergo,
              <lb/>
            BSG, DSM, latus, BS, æquatur late-
              <lb/>
            ri, SM, & </s>
            <s xml:id="echoid-s6314" xml:space="preserve">latus, GS, lateri, SD, item
              <lb/>
            angulus; </s>
            <s xml:id="echoid-s6315" xml:space="preserve">BSG, angulo, DSM, ergo ba-
              <lb/>
              <note position="left" xlink:label="note-0272-02" xlink:href="note-0272-02a" xml:space="preserve">4. 1. Elem.</note>
            ſis, BG, æquatur baſi, DM, & </s>
            <s xml:id="echoid-s6316" xml:space="preserve">angulus,
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            SBG, angulo, SMD, &</s>
            <s xml:id="echoid-s6317" xml:space="preserve">, SGB, ipſi, S
              <lb/>
            DM, totus autem angulus, CBS, æquatur toti, FMS, ſibi coal-
              <lb/>
            terno, ergo reliquus angulus, CBG, æquatur reliquo angulo, DM
              <lb/>
            F, & </s>
            <s xml:id="echoid-s6318" xml:space="preserve">ſimiliter probabimus angulum, BGC, æquari angulo, MD
              <lb/>
            F, ergo reliquus, BCG, æquabitur reliquo, DFM, (qui etiam ſunt
              <lb/>
            æquales, quia ſunt anguli oppoſiti parallelogrammi, AR,) & </s>
            <s xml:id="echoid-s6319" xml:space="preserve">ideò
              <lb/>
            trianguli, BCG, DFM, erunt æquianguli, &</s>
            <s xml:id="echoid-s6320" xml:space="preserve">, BG, DM, </s>
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