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

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          <pb o="217" file="0237" n="237" rhead="LIBER III."/>
        </div>
        <div xml:id="echoid-div541" type="section" level="1" n="324">
          <head xml:id="echoid-head341" xml:space="preserve">THEOREMA XII. PROPOS. XIII.</head>
          <p>
            <s xml:id="echoid-s5276" xml:space="preserve">SI circulum, vel ellipſim duæ rectæ lineæ in terminis
              <lb/>
            coniugatarum diametrorum tetigetint inter ſe conue-
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            nientes, eiſdem diametris ductis. </s>
            <s xml:id="echoid-s5277" xml:space="preserve">Omnia quadrata conſti-
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            tuti parallelogrammiad omnia quadrata trilinei à dictis tan-
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            gentibus, & </s>
            <s xml:id="echoid-s5278" xml:space="preserve">ab incluſa curua comprehenſi, regula altera
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            diametrorum, erunt vt dictum parallelogrammum ad ſui
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            reliquum, dempto quadrante circuli, vel ellipſis iam dictæ,
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            quod inſcribitur prædicto parallelogrammo, ſimul cum ex-
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            ceſſu dicti quadrantis ſuper duas tertias iam dicti parallelo-
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            grammi, quæ ratio erit proximè, vt 21. </s>
            <s xml:id="echoid-s5279" xml:space="preserve">ad 2.</s>
            <s xml:id="echoid-s5280" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5281" xml:space="preserve">Sit circulus, vel ellipſis, ABCD, cuius diametri coniugatæ, A
              <lb/>
            C, BD, in quorum terminis, C, D, duæ rectæ lineę ipſum tangen-
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            tesinter ſe conueniant in, V. </s>
            <s xml:id="echoid-s5282" xml:space="preserve">Dico ergo (ſumpta regula qualibet
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            diametrorum, vt, BD,) quod omnia quadrata parallelogrammi, O
              <lb/>
            V, ad omnia quadrata trilinei, DCV, duabus tangentibus, DV,
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            VC, & </s>
            <s xml:id="echoid-s5283" xml:space="preserve">ab ijs incluſa curua, DC, comprehenſi ſunt, vt idem paral-
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              <figure xlink:label="fig-0237-01" xlink:href="fig-0237-01a" number="148">
                <image file="0237-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0237-01"/>
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            lelogrammum, OV, ad ſui reliquum
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            dempto quadrante, OCD, circuli,
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            vel ellipſis, ABCD, ſimul cum eo
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            ſpatio, quo idem quadrans excedit
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            duas tertias parallelogrammi, OV.
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            </s>
            <s xml:id="echoid-s5284" xml:space="preserve">Sumatur intra, OC, vtcunque pun
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            ctum, E, & </s>
            <s xml:id="echoid-s5285" xml:space="preserve">per, E, ducatur ipſi, B
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            D, parallela, EF, ſecans curuam, D
              <lb/>
            C, in, I. </s>
            <s xml:id="echoid-s5286" xml:space="preserve">Omnia ergo quadrata pa-
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            rallelogrammi, OV, ad rectangula
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            ſub parallelogrammo, OV, & </s>
            <s xml:id="echoid-s5287" xml:space="preserve">ſemi-
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            portione, OCD, ſunt vt parallelo
              <lb/>
              <note position="right" xlink:label="note-0237-01" xlink:href="note-0237-01a" xml:space="preserve">Coroll.1.
                <lb/>
              26.lib.2.</note>
            grammum, OV, ad eandem ſemiportionem, OCD; </s>
            <s xml:id="echoid-s5288" xml:space="preserve">ſed eadem ad
              <lb/>
              <note position="right" xlink:label="note-0237-02" xlink:href="note-0237-02a" xml:space="preserve">Coroll.1.
                <lb/>
              huius.</note>
            omnia quadrata ſemiportionis, OCD, ſunt ſexquialtera, ergo ad
              <lb/>
            reſiduum erunt vt parallelogrammum, OV, ad reſiduum ſemipor-
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            tionis, OCD, demptis ab ea, {2/3}, parallelogrammi, OV, quarum
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            idem parallelogrammum, OV, eſt ſexquialterum; </s>
            <s xml:id="echoid-s5289" xml:space="preserve">reſiduum autem
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            rectangulorum ſub parallelogrammo, OV, & </s>
            <s xml:id="echoid-s5290" xml:space="preserve">ſemiportione, OC
              <lb/>
            D, demptis omnibus quadratis ſemiportionis, OCD, ſunt rectan-
              <lb/>
              <note position="right" xlink:label="note-0237-03" xlink:href="note-0237-03a" xml:space="preserve">Vide ibid.
                <lb/>
              dicta.</note>
            gula ſub ſemiportione, OCD, & </s>
            <s xml:id="echoid-s5291" xml:space="preserve">trilineo, CDV, nam veluti </s>
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