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

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          <p>
            <s xml:id="echoid-s8395" xml:space="preserve">
              <pb o="340" file="0360" n="360" rhead="GEOMETRIÆ"/>
            compoſitam ex ea, quam habet rectangulum, ZPG, ad rectangu-
              <lb/>
            lum, ZQG, ideſt ex ea, quam habet, DP, ad, AQ, & </s>
            <s xml:id="echoid-s8396" xml:space="preserve">ex ra-
              <lb/>
            tione rectanguli, ZQG, ad rectangulum, SOI, vel quadra-
              <lb/>
            ti, QG, ad quadratum, OI, ideſt ex ea, quam habet, QA, ad,
              <lb/>
              <note position="left" xlink:label="note-0360-01" xlink:href="note-0360-01a" xml:space="preserve">3. huius.</note>
            AO, & </s>
            <s xml:id="echoid-s8397" xml:space="preserve">ex ratione rectanguli, SOI, ad rectangulum, STI, ideſt
              <lb/>
            ex ratione, AO, ad, DT, ergo rectangulum, ZPG, vel, RTF, ad
              <lb/>
            rectangulum, STI, erit vt, PD, ad DT, abſciſſam. </s>
            <s xml:id="echoid-s8398" xml:space="preserve">Et quoniam,
              <lb/>
              <note position="left" xlink:label="note-0360-02" xlink:href="note-0360-02a" xml:space="preserve">3. huius.</note>
            HG, eſt parallelogrammum in eadem baſi, & </s>
            <s xml:id="echoid-s8399" xml:space="preserve">altitudine cum fruſto,
              <lb/>
            BZGD, & </s>
            <s xml:id="echoid-s8400" xml:space="preserve">per punctum, T, vtcunq. </s>
            <s xml:id="echoid-s8401" xml:space="preserve">ſumptum ducta, BP, regulæ
              <lb/>
            parallela, quę eſt baſis, ZG, inuentũ eſt rectangulũ BTP, ad rectan-
              <lb/>
            gulũ, STI, eſſe vt, PD, ad DT; </s>
            <s xml:id="echoid-s8402" xml:space="preserve">quatuor ergo horum magnitudinum
              <lb/>
              <note position="left" xlink:label="note-0360-03" xlink:href="note-0360-03a" xml:space="preserve">Yux. Cor.
                <lb/>
              3. 26.l. 2.</note>
            ordinibus conſtructis, iuxta has quatuor magnitudines, quę inuentę
              <lb/>
            ſunt eſſe proportionales, & </s>
            <s xml:id="echoid-s8403" xml:space="preserve">hoc modo ſolito, reperimus rectangula
              <lb/>
            ſub, HP, PE, ad rectangula ſub portionibus, BZPD, DGP, eſſe vt
              <lb/>
            maximę abſciſſarum, DP, ad omnes abſciſſas, DP, recti, vel eiuſdẽ
              <lb/>
              <note position="left" xlink:label="note-0360-04" xlink:href="note-0360-04a" xml:space="preserve">Corol. 2.
                <lb/>
                <gap/>
              2.</note>
            obliqui tranſitus .</s>
            <s xml:id="echoid-s8404" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8405" xml:space="preserve">eſſe eorum dupla, quod oſtendere opus erat.</s>
            <s xml:id="echoid-s8406" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div819" type="section" level="1" n="485">
          <head xml:id="echoid-head505" xml:space="preserve">THEOREMA XLV. PROP. XLVII.</head>
          <p>
            <s xml:id="echoid-s8407" xml:space="preserve">IN anteced. </s>
            <s xml:id="echoid-s8408" xml:space="preserve">figura oſtendemus, regula eadem, ZG, omnia
              <lb/>
            quadrata, DG, ad omnia quadrata, DPG, eſſe vt, ZP,
              <lb/>
            ad compoſitam ex {1/3}. </s>
            <s xml:id="echoid-s8409" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8410" xml:space="preserve">{1/2}. </s>
            <s xml:id="echoid-s8411" xml:space="preserve">PCOmnia verò quadrata,
              <lb/>
            DC, ad omnia quadrata trilinei, DGE, eſſe vt, ZP, ad ſui
              <lb/>
            reliquum, demptis ab eadem {2/3}. </s>
            <s xml:id="echoid-s8412" xml:space="preserve">ZP, cum {1/6}, PG.</s>
            <s xml:id="echoid-s8413" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8414" xml:space="preserve">Rectangula enim ſub, HP, PE, adrectangula ſub, HP, & </s>
            <s xml:id="echoid-s8415" xml:space="preserve">por-
              <lb/>
              <note position="left" xlink:label="note-0360-05" xlink:href="note-0360-05a" xml:space="preserve">Coroll. 1.
                <lb/>
              26.I. 2.</note>
            tione, DPG, ſunt vt, EP, ad portionem, DPG, .</s>
            <s xml:id="echoid-s8416" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s8417" xml:space="preserve">vt, ZP, ad
              <lb/>
            compoſitam ex {1/4}. </s>
            <s xml:id="echoid-s8418" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8419" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8420" xml:space="preserve">PG; </s>
            <s xml:id="echoid-s8421" xml:space="preserve">eadem autem rectangula ſub, H
              <lb/>
            P, PE, ſunt dupla rectangulorum ſub portionibus, DBZP, DP
              <lb/>
              <note position="left" xlink:label="note-0360-06" xlink:href="note-0360-06a" xml:space="preserve">3. huius.</note>
            G, .</s>
            <s xml:id="echoid-s8422" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8423" xml:space="preserve">ſunt ad illa, vt, ZP, ad {1/2}. </s>
            <s xml:id="echoid-s8424" xml:space="preserve">ZP, ergo ad reſiduum rectangulo-
              <lb/>
              <note position="left" xlink:label="note-0360-07" xlink:href="note-0360-07a" xml:space="preserve">Ex antec.</note>
              <figure xlink:label="fig-0360-01" xlink:href="fig-0360-01a" number="244">
                <image file="0360-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0360-01"/>
              </figure>
            rum ſub, HP, &</s>
            <s xml:id="echoid-s8425" xml:space="preserve">, DPG, dem-
              <lb/>
            ptis rectangulis ſub portioni-
              <lb/>
            bus, DBZP, DGP, ideſt
              <lb/>
              <note position="left" xlink:label="note-0360-08" xlink:href="note-0360-08a" xml:space="preserve">Jux. A. 23.
                <lb/>
              l. 2.</note>
            ad rectangula ſub trilineo, D
              <lb/>
            P G, & </s>
            <s xml:id="echoid-s8426" xml:space="preserve">trilineo, BHZ, .</s>
            <s xml:id="echoid-s8427" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8428" xml:space="preserve">tri-
              <lb/>
            lineo, DEG, erunt vt, ZP,
              <lb/>
            ad {1/6}. </s>
            <s xml:id="echoid-s8429" xml:space="preserve">PG, .</s>
            <s xml:id="echoid-s8430" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8431" xml:space="preserve">ſumpta, PG, cõ-
              <lb/>
            muni altitudine, vt rectangu-
              <lb/>
            lum, ZPG, ad rectangulum
              <lb/>
            ſub, PG, & </s>
            <s xml:id="echoid-s8432" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8433" xml:space="preserve">PG, .</s>
            <s xml:id="echoid-s8434" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8435" xml:space="preserve">ad {1/6}.
              <lb/>
            </s>
            <s xml:id="echoid-s8436" xml:space="preserve">quadrati, PG, ſunt autem omnia quadrata, DG, ad rectangula </s>
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