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

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          <pb o="301" file="0321" n="321" rhead="LIBER IV."/>
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        <div xml:id="echoid-div720" type="section" level="1" n="424">
          <head xml:id="echoid-head444" xml:space="preserve">THEOREMA XII. PROPOS. XIII.</head>
          <p>
            <s xml:id="echoid-s7277" xml:space="preserve">SIab extremo puncto baſis datæ parabolæ ducatur vſq; </s>
            <s xml:id="echoid-s7278" xml:space="preserve">ad
              <lb/>
            curuam parabolæ ſupra, vel infra baſim (indefinitè
              <lb/>
            producta ipſa curua) recta linea: </s>
            <s xml:id="echoid-s7279" xml:space="preserve">Data parabola ad ſegmen-
              <lb/>
            ta ſub ductis lineis, & </s>
            <s xml:id="echoid-s7280" xml:space="preserve">curua ab ijſdem abſciſſa comprehen-
              <lb/>
            ſa, ſingillatim ſumpta, erit vt cubus baſis ipſius datæpara-
              <lb/>
            bolæ ad cubum rectæ lineæ dicto puncto interceptæ, & </s>
            <s xml:id="echoid-s7281" xml:space="preserve">alio
              <lb/>
            puncto eiuſdem baſis productæ, ſi opus ſit, in quod cadit
              <lb/>
            recta linea, quæ ducitur ab alio extremo puncto baſis re-
              <lb/>
            ſecti ſegmenti parallela axi, vel diametro ipſius datæ pa-
              <lb/>
            rabolæ.</s>
            <s xml:id="echoid-s7282" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7283" xml:space="preserve">Sit ergo data parabola, HNB, inbaſi, HB, ſumpto autem vno
              <lb/>
            extremorum punctorum, H, B, ipſius baſis, H B, vtipſum, H, ab
              <lb/>
            eo ducatur vtcunq; </s>
            <s xml:id="echoid-s7284" xml:space="preserve">recta linea, HA, ſupra baſim, HB, & </s>
            <s xml:id="echoid-s7285" xml:space="preserve">indefi-
              <lb/>
            nitè producta curua, NAB, alia, HV, ſubterbàſim, vt ſint con-
              <lb/>
            ſtituta ſegmenta, ANH, VBNH, ſit autem axis, vel diameter,
              <lb/>
            NO, cui parallelæ ducantur per puncta, AV, verſus baſim, HB,
              <lb/>
              <figure xlink:label="fig-0321-01" xlink:href="fig-0321-01a" number="215">
                <image file="0321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0321-01"/>
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            productam, ſi opus ſit, occur-
              <lb/>
            rentes illi in punctis, X, C.
              <lb/>
            </s>
            <s xml:id="echoid-s7286" xml:space="preserve">Dico ergo parabolam, HNB,
              <lb/>
            ad ſegmentum, HN.</s>
            <s xml:id="echoid-s7287" xml:space="preserve">A, eſſe vt
              <lb/>
            cubus, HB, ad cubum, HC. </s>
            <s xml:id="echoid-s7288" xml:space="preserve">
              <lb/>
            Eandem verò ad ſegmentum,
              <lb/>
            HNBV, eſſe vt cubum, BH,
              <lb/>
            ad cubum, HX, iungantur
              <lb/>
            puncta, B, A; </s>
            <s xml:id="echoid-s7289" xml:space="preserve">B, N; </s>
            <s xml:id="echoid-s7290" xml:space="preserve">N, H,
              <lb/>
            & </s>
            <s xml:id="echoid-s7291" xml:space="preserve">ſit, CE, tertia proportiona-
              <lb/>
            lis duarum, quarum prima eſt
              <lb/>
            tripla, CH, ſecunda autem ipſa, BC. </s>
            <s xml:id="echoid-s7292" xml:space="preserve">Quoniam ergo triangula,
              <lb/>
              <note position="right" xlink:label="note-0321-01" xlink:href="note-0321-01a" xml:space="preserve">Coroll.1.
                <lb/>
              19.huius.</note>
            NBH, BAH, ſunt in eadem baſi, BH, erunt inter ſe, vt altitu-
              <lb/>
            dines, vel vt lineæ, quæ a verticibus, NA, ad baſes ductæ cum
              <lb/>
            eiſdem æqualiter inclinantur .</s>
            <s xml:id="echoid-s7293" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7294" xml:space="preserve">triangulum, HNB, ad triangu-
              <lb/>
            lum, HAB, erit vt, NO, ad, AC, .</s>
            <s xml:id="echoid-s7295" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7296" xml:space="preserve">vt rectangulum, HOB,
              <lb/>
            ad rectangulum, HCB. </s>
            <s xml:id="echoid-s7297" xml:space="preserve">Inſuper triangulum, HNB, ad portion-
              <lb/>
              <note position="right" xlink:label="note-0321-02" xlink:href="note-0321-02a" xml:space="preserve">Defin.12.
                <lb/>
              l.1.</note>
            culam, ASB, habet rationem compoſitam ex ratione trianguli,
              <lb/>
            HNB, ad triangulum, HAB, .</s>
            <s xml:id="echoid-s7298" xml:space="preserve">i. </s>
            <s xml:id="echoid-s7299" xml:space="preserve">ex ratione rectanguli, HOB,
              <lb/>
            ad rectangulum, HCB, & </s>
            <s xml:id="echoid-s7300" xml:space="preserve">ex ratione trianguli, HAB, ad </s>
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