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

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          <p>
            <s xml:id="echoid-s8363" xml:space="preserve">
              <pb o="339" file="0359" n="359" rhead="LIBER IV."/>
            G, ad parallelepipedum ſub, BG, & </s>
            <s xml:id="echoid-s8364" xml:space="preserve">quadrato, HF; </s>
            <s xml:id="echoid-s8365" xml:space="preserve">item omnia
              <lb/>
            quadrata, AF, ad omnia quadrata, AG, ſunt vt quadratum, FH,
              <lb/>
            ad quadratum, HG, .</s>
            <s xml:id="echoid-s8366" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s8367" xml:space="preserve">ſumpta, BG, communi altitudine, vt pa-
              <lb/>
            rallelepipedum ſub, BG, & </s>
            <s xml:id="echoid-s8368" xml:space="preserve">quadrato, FH, ad parallelepipedum
              <lb/>
            ſub, BG, & </s>
            <s xml:id="echoid-s8369" xml:space="preserve">quadrato, HG: </s>
            <s xml:id="echoid-s8370" xml:space="preserve">_Tandem omnia quadrata,_ AG, dupla
              <lb/>
            ſunt omnium quadratorum ſemiparabolæ, BHG, ergo, ex æquali,
              <lb/>
            omnia quadrata figuræ, CBHF, demptis omnibus quadratis trili-
              <lb/>
            nei, BCE, ad omnia quadrata ſemiparabolæ, BHG, erunt vt pa-
              <lb/>
            rallelepipedum ſub, BG, & </s>
            <s xml:id="echoid-s8371" xml:space="preserve">his ſpatijs .</s>
            <s xml:id="echoid-s8372" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s8373" xml:space="preserve">quadrato, FG, {1/2}. </s>
            <s xml:id="echoid-s8374" xml:space="preserve">quadra-
              <lb/>
            ti, GH, & </s>
            <s xml:id="echoid-s8375" xml:space="preserve">rectangulo ſub, FG, & </s>
            <s xml:id="echoid-s8376" xml:space="preserve">ſexquitertia, GH, ab eodem
              <lb/>
            dempto {1/6}. </s>
            <s xml:id="echoid-s8377" xml:space="preserve">parallelepipedi ſub, CE, & </s>
            <s xml:id="echoid-s8378" xml:space="preserve">quadrato, FG, ad dimidium
              <lb/>
            parallelepipedi ſub, BG, & </s>
            <s xml:id="echoid-s8379" xml:space="preserve">quadrato, GH, quod erat demonſtran-
              <lb/>
            dum.</s>
            <s xml:id="echoid-s8380" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div817" type="section" level="1" n="484">
          <head xml:id="echoid-head504" xml:space="preserve">THEOREMA XLIV. PROP. XLVI.</head>
          <p>
            <s xml:id="echoid-s8381" xml:space="preserve">IN parabola ducta axi, vel diametro æquidiſtanter rect@
              <lb/>
            linea, ſi deinde fiat parallelogrammum ſub eadem du-
              <lb/>
            cta, & </s>
            <s xml:id="echoid-s8382" xml:space="preserve">ſub baſi, angulum habens æqualem angulo i
              <gap/>
              <lb/>
            tionis eiuſdem ductæ ad baſim, regula ſumpta baſi. </s>
            <s xml:id="echoid-s8383" xml:space="preserve">Re
              <gap/>
              <lb/>
            gula ſub parallelogram
              <gap/>
            , in quæ dictum parallelogram-
              <lb/>
            mum diuiditur à ducta linea, ſunt dupla rectangulorum
              <lb/>
            ſub portionibus fruſti parabolæ, dicto parallelogrammo in-
              <lb/>
            cluſæ, per eandem ductam conſtituris.</s>
            <s xml:id="echoid-s8384" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8385" xml:space="preserve">Sit parabola, AZG, in baſi, ZG, circa axim, vel diametrum,
              <lb/>
              <figure xlink:label="fig-0359-01" xlink:href="fig-0359-01a" number="243">
                <image file="0359-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0359-01"/>
              </figure>
            AQ, cui parallela ducatur vt-
              <lb/>
            cumque recta, DP, fiat autem
              <lb/>
            parallelogrammum ſub, ZQ,
              <lb/>
            DP, angulum habens æqualẽ
              <lb/>
            angulo inclinationis, DP, ad
              <lb/>
            ZG, .</s>
            <s xml:id="echoid-s8386" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8387" xml:space="preserve">angulo, qui ſit, DPG,
              <lb/>
            vtcunque exduobus, DPG,
              <lb/>
            DPZ, ſit autem hoc paralle.
              <lb/>
            </s>
            <s xml:id="echoid-s8388" xml:space="preserve">logrammum, HG, regula ve. </s>
            <s xml:id="echoid-s8389" xml:space="preserve">
              <lb/>
            ro, HG. </s>
            <s xml:id="echoid-s8390" xml:space="preserve">Dico ergo, rectãgula
              <lb/>
            ſub, HP, PE, dupla eſſe rectãgulorũ ſub portionibus, BDPZ, DGP. </s>
            <s xml:id="echoid-s8391" xml:space="preserve">
              <lb/>
            Sumpto ergo vtcunq; </s>
            <s xml:id="echoid-s8392" xml:space="preserve">in, DP, puncto, T, per, T, ducatur, RF, ipſi,
              <lb/>
            ZG, æquidiſtans ſecanſq; </s>
            <s xml:id="echoid-s8393" xml:space="preserve">curuam parabolæ in, SI, &</s>
            <s xml:id="echoid-s8394" xml:space="preserve">, AQ, in, O. </s>
            <s xml:id="echoid-s8395" xml:space="preserve">
              <lb/>
            Rectangulum ergo, ZPQ, ad rectangulum, STI, habet </s>
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