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

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        <div xml:id="echoid-div776" type="section" level="1" n="458">
          <pb o="323" file="0343" n="343" rhead="LIBER IV."/>
        </div>
        <div xml:id="echoid-div777" type="section" level="1" n="459">
          <head xml:id="echoid-head479" xml:space="preserve">THEOREMA XXVIII. PROPOS. XXX.</head>
          <p>
            <s xml:id="echoid-s7758" xml:space="preserve">SI duæ rectæ lineæ ducantur, quarum altera parabolam
              <lb/>
            tangat, altera verò axi, vel diametro parabolæ æqui-
              <lb/>
            diſtanter ducta eandem ſecet, in idem punctum concurren-
              <lb/>
            tes: </s>
            <s xml:id="echoid-s7759" xml:space="preserve">Omnia quadrata parallelogrammi, regula tangente,
              <lb/>
            erunt ſexcupla omnium quadratorum trilinei ſub dictis tan-
              <lb/>
            gente, & </s>
            <s xml:id="echoid-s7760" xml:space="preserve">ſecante, & </s>
            <s xml:id="echoid-s7761" xml:space="preserve">curua parabolæ ab ijſdem incluſa,
              <lb/>
            comprehenſi.</s>
            <s xml:id="echoid-s7762" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7763" xml:space="preserve">Sit ſemiparabola, ACB, quam tangat lineæ, AD, & </s>
            <s xml:id="echoid-s7764" xml:space="preserve">à pun-
              <lb/>
            cto, A, ducta, AB, axis, vel diameter, integræ parabolæ, dein-
              <lb/>
            de agatur vtcunque, DC, parallela axi, vel diametro, AB, ſecans
              <lb/>
            curuam parabolæ in, C, & </s>
            <s xml:id="echoid-s7765" xml:space="preserve">occurrens tangenti, AD, in, D, duca-
              <lb/>
            tur tandem à puncto, C, ipſi, AD, æquidiſtans, CB, ſecans, AB,
              <lb/>
            in, B, regula autem ſit, AD. </s>
            <s xml:id="echoid-s7766" xml:space="preserve">Dico ergo omnia quadrata parallelo-
              <lb/>
              <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a" number="230">
                <image file="0343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0343-01"/>
              </figure>
            grammi, AC, eſſe omnium quadratorum
              <lb/>
            trilinei, ADC, ſexcupla: </s>
            <s xml:id="echoid-s7767" xml:space="preserve">Omnia enim
              <lb/>
            quadrata, AC, ad rectangula ſub, AC,
              <lb/>
              <note position="right" xlink:label="note-0343-01" xlink:href="note-0343-01a" xml:space="preserve">Coroll. 1.
                <lb/>
              26. l. 2.</note>
            & </s>
            <s xml:id="echoid-s7768" xml:space="preserve">ſemiparabola, ABC, ſunt vt, AC, ad
              <lb/>
            ſemiparabolam, ABC, .</s>
            <s xml:id="echoid-s7769" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7770" xml:space="preserve">ſexquialtera
              <lb/>
            .</s>
            <s xml:id="echoid-s7771" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7772" xml:space="preserve">vt 6. </s>
            <s xml:id="echoid-s7773" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s7774" xml:space="preserve">omnia autem quadrata, AC,
              <lb/>
              <note position="right" xlink:label="note-0343-02" xlink:href="note-0343-02a" xml:space="preserve">21. huius.</note>
            ad omnia quadrata ſemiparabolę ABC,
              <lb/>
            ſunt dupla .</s>
            <s xml:id="echoid-s7775" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7776" xml:space="preserve">vt 6. </s>
            <s xml:id="echoid-s7777" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s7778" xml:space="preserve">ergo omnia qua-
              <lb/>
              <note position="right" xlink:label="note-0343-03" xlink:href="note-0343-03a" xml:space="preserve">Coroll.
                <lb/>
              23. l. 2.</note>
            drata, AC, ad reſiduum demptis omni-
              <lb/>
            bus quadratis ſemiparabolæ, ABC, à re-
              <lb/>
            ctangulis ſub, AC, & </s>
            <s xml:id="echoid-s7779" xml:space="preserve">ſemiparabola, ABC, .</s>
            <s xml:id="echoid-s7780" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7781" xml:space="preserve">ad rectangula ſub
              <lb/>
            ſemiparabola, ABC, & </s>
            <s xml:id="echoid-s7782" xml:space="preserve">trilineo, ADC, erunt in ratione ſex-
              <lb/>
            cuplaideſt vt 6. </s>
            <s xml:id="echoid-s7783" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s7784" xml:space="preserve">ad eadem verò bis ſumpta, vt 6. </s>
            <s xml:id="echoid-s7785" xml:space="preserve">ad 2. </s>
            <s xml:id="echoid-s7786" xml:space="preserve">quo-
              <lb/>
            niam verò omnia quadrata, AC, ad omnia quadrata ſemipara-
              <lb/>
            bolæ, ABC, ſum vt 6. </s>
            <s xml:id="echoid-s7787" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s7788" xml:space="preserve">vt dictum eſt, ideò omnia quadrata,
              <lb/>
            AC, ad omnia quadrata ſemiparabolæ, ABC, cum rectangulis
              <lb/>
            bis ſub ſemiparabola, ABC, & </s>
            <s xml:id="echoid-s7789" xml:space="preserve">trilineo, ADC, erunt vt 6. </s>
            <s xml:id="echoid-s7790" xml:space="preserve">ad 5.
              <lb/>
            </s>
            <s xml:id="echoid-s7791" xml:space="preserve">ergo ad reliquum ſcilicet ad omnia quadrata trilinei, ADC, omnia
              <lb/>
              <note position="right" xlink:label="note-0343-04" xlink:href="note-0343-04a" xml:space="preserve">D. 23. l. 2.</note>
            quadrata, AC, erunt vt 6. </s>
            <s xml:id="echoid-s7792" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s7793" xml:space="preserve">ideſt erunt eorundem ſexcupla,
              <lb/>
            quod oſtendere oportebat.</s>
            <s xml:id="echoid-s7794" xml:space="preserve"/>
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