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

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        <div xml:id="echoid-div1069" type="section" level="1" n="641">
          <pb o="457" file="0477" n="477" rhead="LIBER VI."/>
        </div>
        <div xml:id="echoid-div1071" type="section" level="1" n="642">
          <head xml:id="echoid-head672" xml:space="preserve">THEOREMA XVII. PROPOS. XVII.</head>
          <p>
            <s xml:id="echoid-s11697" xml:space="preserve">COmpræhenſ@m ſpatium ſub ſpirali, quæ eſt minor ea,
              <lb/>
            quæ ſub vna reuolutione fit, nec habet terminum
              <lb/>
            initium ſpiralis, & </s>
            <s xml:id="echoid-s11698" xml:space="preserve">rectis, quæ à terminis ipſius in reuolu-
              <lb/>
            tionis initium ducuntur ad ſectorem habentem radium æ-
              <lb/>
            qualem maiori earum, quę à termino ad initium reuolutio-
              <lb/>
            nis ducitur, arcum verò, qui intercipitur inter duas rectas
              <lb/>
            ſecundum eaſdem partem ſpiralis; </s>
            <s xml:id="echoid-s11699" xml:space="preserve">habet eandem rationẽ,
              <lb/>
            quam rectangulum compræhenſum ſub rectis à terminis
              <lb/>
            ad initium reuolutionis ductis, vna cum tertia parte qua-
              <lb/>
            drati exceſſus, quo maior dictarum linearum ſuperat mi-
              <lb/>
            norem, ad quadratum maioris earundem.</s>
            <s xml:id="echoid-s11700" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11701" xml:space="preserve">In eadem antecedentis figura ſupponamus arſumptam, IS, por-
              <lb/>
            tionem ſpiralis in vna reuolutione genitæ, quæ non habcat termi-
              <lb/>
            num initium talis ſpiralis, a cuius extremis punctis, I, S, ſint du-
              <lb/>
            ctæ ad, A, initium reuolutionis ipſæ, SA, IA, & </s>
            <s xml:id="echoid-s11702" xml:space="preserve">ſit ſector, IAR,
              <lb/>
            cuius ſemidiameter ſit æqual s maiori ductarum, IA, AS, nempè
              <lb/>
            ipſi, IA. </s>
            <s xml:id="echoid-s11703" xml:space="preserve">Dico ſectorem, IAR, ad trilineum, IAS, eſſe vt quadra-
              <lb/>
            tum, RA, ad rectangulum, RAS, vna cum {1/3}. </s>
            <s xml:id="echoid-s11704" xml:space="preserve">quadrati, RS, (vta-
              <lb/>
            mur conſtructis in eadem figura) Sector igitur, AIRεKA, eſt æqua-
              <lb/>
            lis triangulo, LZ {12/ }, vt in antecedenti oſtenſum eſt, eodem modo
              <lb/>
            probabimus triangulum, L+{13/ }, eſſe æqualem ſectori, ARεKA,
              <lb/>
            ergo reliquus triangulus, LZ+, erit æqualis reliquo ſectori, IAR;
              <lb/>
            </s>
            <s xml:id="echoid-s11705" xml:space="preserve">ſimiliter iuxta antecedentem oſtendemus ſpatium, AISMGA, eſſe
              <lb/>
            æqualem trilineo, LΖΩ, & </s>
            <s xml:id="echoid-s11706" xml:space="preserve">ſpatium, ASMGA, eſſe æqualem tri-
              <lb/>
              <note position="right" xlink:label="note-0477-01" xlink:href="note-0477-01a" xml:space="preserve">14. huius.</note>
            lineo, LVΩ ergo reliquum ſpatium, IAS, erit æquale trilineo, LZ
              <lb/>
            V, ergo ſector, IAR, ad trilineum, LZV, erit vt triangulus LZ+,
              <lb/>
            ad trilineum, LZV, ideſt vt quadratum, L4, ad rectangulum ſub,
              <lb/>
            4L3, cum {1/3}, quadrati, 34, ideſt vt quadratum, IA, vel, RA, ad re-
              <lb/>
            ctangulum ſub, RA, A
              <emph style="sub">c</emph>
            , vna cum {1/3}. </s>
            <s xml:id="echoid-s11707" xml:space="preserve">quadrati, RS, quod oſtende-
              <lb/>
            re opus erat.</s>
            <s xml:id="echoid-s11708" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1073" type="section" level="1" n="643">
          <head xml:id="echoid-head673" xml:space="preserve">THEOREMA XVIII. PROPOS. XVIII.</head>
          <p>
            <s xml:id="echoid-s11709" xml:space="preserve">TRilineum, IRS, ad trilineum, ISX, erit vt, SA, cum
              <lb/>
            {2/3}. </s>
            <s xml:id="echoid-s11710" xml:space="preserve">SR, ad, SA, cum {1/3}. </s>
            <s xml:id="echoid-s11711" xml:space="preserve">SR.</s>
            <s xml:id="echoid-s11712" xml:space="preserve"/>
          </p>
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