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

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          <p>
            <s xml:id="echoid-s11546" xml:space="preserve">
              <pb o="450" file="0470" n="470" rhead="GEOMETRIÆ"/>
            poſita ex, O℟, & </s>
            <s xml:id="echoid-s11547" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s11548" xml:space="preserve">℟A, & </s>
            <s xml:id="echoid-s11549" xml:space="preserve">ſub, ℟A, & </s>
            <s xml:id="echoid-s11550" xml:space="preserve">quia horum rectangulo-
              <lb/>
              <note position="left" xlink:label="note-0470-01" xlink:href="note-0470-01a" xml:space="preserve">5. 1. 2.</note>
            rum altitudines ſunt æquales, ideò trilineum, A℟Q, ad trilineum,
              <lb/>
            ℟QY, erit vt, O℟, cum {2/3}. </s>
            <s xml:id="echoid-s11551" xml:space="preserve">℟A, ad, O℟, cum tertia parte, ℟A,
              <lb/>
            quod oſtendcre opus erat.</s>
            <s xml:id="echoid-s11552" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1062" type="section" level="1" n="638">
          <head xml:id="echoid-head668" xml:space="preserve">THEOREMA XIV. PROPOS. XIV.</head>
          <p>
            <s xml:id="echoid-s11553" xml:space="preserve">SI duæ rectæ lineę ducantur, quarum altera parabolam
              <lb/>
            tangat, altera verò ducta axi, vel diametro eiuſdem
              <lb/>
            æquidiſtans, eandem ſecet, iuncto verò puncto contactus
              <lb/>
            cum hoc ſectionis puncto, rurſus ab hoc puncto ad latus
              <lb/>
            illi oppoſitum in facto triangulo recta producatur, quæ
              <lb/>
            curuam ſecabit parabolæ, à quo ſectionis puncto ducatur
              <lb/>
            axi, vel diametro parallela quouſq; </s>
            <s xml:id="echoid-s11554" xml:space="preserve">incidat in tangentem:
              <lb/>
            </s>
            <s xml:id="echoid-s11555" xml:space="preserve">Triangulum ſub eductis ad ſecantem à puncto contactus,
              <lb/>
            ad portionem parabolæ eiſdem interceptam erit, vt qua-
              <lb/>
            dratum totius tangentis ad rectangulum ſub eadem, & </s>
            <s xml:id="echoid-s11556" xml:space="preserve">ſub
              <lb/>
            illius abſciſſa per eam verſus punctum conta ctus per ſecũ-
              <lb/>
            dò ductam axi, vel diametro parallelam, vna cum {1/3}. </s>
            <s xml:id="echoid-s11557" xml:space="preserve">qua-
              <lb/>
            drati differentiæ dictarum tangentium.</s>
            <s xml:id="echoid-s11558" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11559" xml:space="preserve">Sit parabola curua, BIA, quam tangat, DA, in puncto, ADB,
              <lb/>
            vero axi, vel diametro eiuſdem parallel a eandem ſecet in puncto,
              <lb/>
            B, iunctis verò, BA, à puncto, A, ducatur intra triangulum, ABD,
              <lb/>
            adlatus oppoſitum, BD, vtcumq; </s>
            <s xml:id="echoid-s11560" xml:space="preserve">AC, ſecans curuam, AIB, in, I,
              <lb/>
            à quo verſus tangentem, AD, ducatur, IE, axi, vel diametro iam
              <lb/>
            dicto æquidiſtans. </s>
            <s xml:id="echoid-s11561" xml:space="preserve">Dico igitur triangulum, ABC, ad trilineum,
              <lb/>
            ABI, eſſe vt quadratum, DA, ad rectangulum, DAE, vna cum
              <lb/>
            {1/3}. </s>
            <s xml:id="echoid-s11562" xml:space="preserve">quadrati, DE. </s>
            <s xml:id="echoid-s11563" xml:space="preserve">Exponatur parallelogrammum, FP, cuius an-
              <lb/>
            gulus, OPH, ſit æqualis angulo, ADB, &</s>
            <s xml:id="echoid-s11564" xml:space="preserve">, OP, æqualis ipſi, AD,
              <lb/>
            &</s>
            <s xml:id="echoid-s11565" xml:space="preserve">, HP, ipſi, BD, abſcindatur deinde ab, OP, verſus, O, ipſa, ON,
              <lb/>
            æqualis ipſi, AE, & </s>
            <s xml:id="echoid-s11566" xml:space="preserve">per, N, ducatur, GN, parallela ipſi, HP, ſe-
              <lb/>
            cans iungentem, HO, in, M, (ſint enim iuncta, H, O, puncta re-
              <lb/>
            cta, HO,) ſit verò regula, HP. </s>
            <s xml:id="echoid-s11567" xml:space="preserve">Quia ergo, BD, ad, DC, eſt vt, D
              <lb/>
              <note position="left" xlink:label="note-0470-02" xlink:href="note-0470-02a" xml:space="preserve">Corol. 9.
                <lb/>
              huius ad
                <lb/>
              poſteriorẽ
                <lb/>
              demonſt.
                <lb/>
              10. 1. 2.</note>
            A, ad, AE, per conuerſionem rationis, & </s>
            <s xml:id="echoid-s11568" xml:space="preserve">conuertendo, CB, ad, B
              <lb/>
            D, erit vt, ED, ad, DA, ideſt vt, NP, ad, PO, ideſt vt omnia qua-
              <lb/>
            drata, GP, ad omnia quadrata, FP, regula, HP, ſed vt, CB, ad, B
              <lb/>
            D, ſic triangulus, ABC, ad triangulum, ABD, ergo vt omnia qua-
              <lb/>
            drata, GP, ad omnia quadrata, FP, ſic erit triangulus, ABC, ad
              <lb/>
            triangulum, ABD, quod ſerua,</s>
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