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

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          <p>
            <s xml:id="echoid-s9892" xml:space="preserve">
              <pb o="385" file="0405" n="405" rhead="LIBER V."/>
            integræ hyperbolæ, ſiat autem parallelogram mum ſub di-
              <lb/>
            cta baſi, & </s>
            <s xml:id="echoid-s9893" xml:space="preserve">axi, vel diametio, in angulo ab eiſdon comen-
              <lb/>
            to, ſumpta baſi pro regula: </s>
            <s xml:id="echoid-s9894" xml:space="preserve">Omnia quadrata dicti paralle.
              <lb/>
            </s>
            <s xml:id="echoid-s9895" xml:space="preserve">logrammi ad omnia quadrata trilinei extia hypeibolam
              <lb/>
            conſtituti, erunt vt idem parallelogran n@un ad ſuiieli-
              <lb/>
            quum ab eodem dempta ſemil yperbola, vna cum exceſſu,
              <lb/>
            quo dicta ſemihyperbola ſuperat. </s>
            <s xml:id="echoid-s9896" xml:space="preserve">dicti parallel@ gram-
              <lb/>
            mi, cum {1/6}. </s>
            <s xml:id="echoid-s9897" xml:space="preserve">parallelogrammi ſub targente hyperbolan, & </s>
            <s xml:id="echoid-s9898" xml:space="preserve">
              <lb/>
            axis, vel diametri hyperbolæ ea poitione, ad quam teli-
              <lb/>
            qua ſit, vt integra axis, vel dian eter ad eiuſdem latus
              <lb/>
            tranſuerſum.</s>
            <s xml:id="echoid-s9899" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9900" xml:space="preserve">Sit ergo axis, vel diameter hyperbolę, BE, cuius
              <lb/>
              <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a" number="276">
                <image file="0405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0405-01"/>
              </figure>
            dimidia, BED, latus tranſuerſum, AB, & </s>
            <s xml:id="echoid-s9901" xml:space="preserve">in angu-
              <lb/>
            lo, BED, ſub, BE, ED, conſticutum parallelogrã-
              <lb/>
            mum, GE, ſit autem, vt, EB, ad, BA, ita, EH, ad, H
              <lb/>
            B, & </s>
            <s xml:id="echoid-s9902" xml:space="preserve">per, H, ducta, HM, parallela ipſi, ED, quę ſu-
              <lb/>
            matur, pro regula, ita vt ſit conſtitutum parallelo-
              <lb/>
            grammum ſub, HB, & </s>
            <s xml:id="echoid-s9903" xml:space="preserve">ſub, BG, quæ erit tangens
              <lb/>
            hyperbolam in puncto, B. </s>
            <s xml:id="echoid-s9904" xml:space="preserve">Dico gitur omnia qua-
              <lb/>
            drata, BD, ad omnia quadrata trilinei, BGD, eſſe
              <lb/>
            ut, BD, ad ſui reliquum, dempto ab eodem ſemihyperbola, BE
              <lb/>
            D, vna cum exceſſu, quo ipſa ſuperat {1/3}. </s>
            <s xml:id="echoid-s9905" xml:space="preserve">dicti paralle ogran mi, B
              <lb/>
            D, cum {1/6}. </s>
            <s xml:id="echoid-s9906" xml:space="preserve">B M. </s>
            <s xml:id="echoid-s9907" xml:space="preserve">Nam omnia quadrata, BD, ad rectangula ſub, B
              <lb/>
            D, & </s>
            <s xml:id="echoid-s9908" xml:space="preserve">ſemihyperbola, BED, ſunt vt, BD, adiplam, BED, rectan-
              <lb/>
              <note position="right" xlink:label="note-0405-01" xlink:href="note-0405-01a" xml:space="preserve">Coroll. 1.
                <lb/>
              26. 2.</note>
            gula verò ſub, BD,, & </s>
            <s xml:id="echoid-s9909" xml:space="preserve">BED, æquantur rectangulis ſub, BOD, B
              <lb/>
            ED, ſimul cum omnibus quadratis, BED, ergo omnia quadrata,
              <lb/>
            BD, ad rectangula ſub, BGD, BED, cum ommbus quadratis, BE
              <lb/>
              <note position="right" xlink:label="note-0405-02" xlink:href="note-0405-02a" xml:space="preserve">C Co. 23.
                <lb/>
              l. 2.</note>
            D, erunt vt, BD, ad, BED; </s>
            <s xml:id="echoid-s9910" xml:space="preserve">ſunt autem omnia quadrata, BD, ad
              <lb/>
            omnia quadrata, BED, vt, AE, ad compoſitam ex {1/2}. </s>
            <s xml:id="echoid-s9911" xml:space="preserve">AB, & </s>
            <s xml:id="echoid-s9912" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9913" xml:space="preserve">B
              <lb/>
            E, .</s>
            <s xml:id="echoid-s9914" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9915" xml:space="preserve">vt, BE, ad compoſitam ex {1/2}. </s>
            <s xml:id="echoid-s9916" xml:space="preserve">BH, & </s>
            <s xml:id="echoid-s9917" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9918" xml:space="preserve">HE, quia, AE, BE,
              <lb/>
            proportionaliter diuiduntur in punctis, B, H, .</s>
            <s xml:id="echoid-s9919" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9920" xml:space="preserve">vt parallelogram-
              <lb/>
              <note position="right" xlink:label="note-0405-03" xlink:href="note-0405-03a" xml:space="preserve">I. huius</note>
            mum, BD, ad compofitum ex {1/2}. </s>
            <s xml:id="echoid-s9921" xml:space="preserve">BM, & </s>
            <s xml:id="echoid-s9922" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9923" xml:space="preserve">HD, .</s>
            <s xml:id="echoid-s9924" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9925" xml:space="preserve">vt, BD, ad
              <lb/>
            compoſitum ex {1/3}. </s>
            <s xml:id="echoid-s9926" xml:space="preserve">BD, & </s>
            <s xml:id="echoid-s9927" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s9928" xml:space="preserve">BM, ergo omnia quadrata, BD, ad
              <lb/>
            rectangula ſub, BGD, BED, erunt vt, BD, ad exceſſum, quo ſe-
              <lb/>
              <note position="right" xlink:label="note-0405-04" xlink:href="note-0405-04a" xml:space="preserve">5. l. 2.</note>
            mihyperbola ſuperat {1/3}. </s>
            <s xml:id="echoid-s9929" xml:space="preserve">BD, cum {1/6}. </s>
            <s xml:id="echoid-s9930" xml:space="preserve">BM, erant autem omnia qua-
              <lb/>
            drata, BD, ad rectangula ſub, IGD, BED, vna cum ommb. </s>
            <s xml:id="echoid-s9931" xml:space="preserve">qua-
              <lb/>
            dratis, BED, vt, BD, ad, BED, ergo omnia quadrata, BD, ad
              <lb/>
            rectangula bis ſub, BGD, BED, vna cum omnibus quadratis, </s>
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