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

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        <div xml:id="echoid-div463" type="section" level="1" n="280">
          <pb o="186" file="0206" n="206" rhead="GEOMETRI Æ"/>
        </div>
        <div xml:id="echoid-div464" type="section" level="1" n="281">
          <head xml:id="echoid-head297" xml:space="preserve">L. SECTIO XI.</head>
          <p style="it">
            <s xml:id="echoid-s4561" xml:space="preserve">_I_N Prop. </s>
            <s xml:id="echoid-s4562" xml:space="preserve">28. </s>
            <s xml:id="echoid-s4563" xml:space="preserve">habetur cylindricum in ea dem baſi, & </s>
            <s xml:id="echoid-s4564" xml:space="preserve">altitudine cum
              <lb/>
            fruſto conici conſtitutum, ad idem, eſſe (ſumptis duabus homologis
              <lb/>
            in oppoſitis fruſti conici baſibus) vt quadratum maioris dictarum homo-
              <lb/>
            logarum ad rectangulum ſub dictis homologis vna cum, {1/3}, quadrati dif-
              <lb/>
            ferentiæ earumdem homologarum. </s>
            <s xml:id="echoid-s4565" xml:space="preserve">Sit eylindricus, AC, in baſi figura
              <lb/>
            quacumque plana, BC, in eadem autem baſi, & </s>
            <s xml:id="echoid-s4566" xml:space="preserve">altitudine ſit fruſtum
              <lb/>
            conici, EBCI, ſic tamen ſe habens, vt ducto plano per latera cylindri-
              <lb/>
              <figure xlink:label="fig-0206-01" xlink:href="fig-0206-01a" number="123">
                <image file="0206-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0206-01"/>
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            ci, AC, idemtranſeat per latera fruſti conici
              <lb/>
            BEIC, ſit autem ductum tale planum, quod
              <lb/>
            faciat in cylindrico, AC, parallelogram-
              <lb/>
            mum, AC, & </s>
            <s xml:id="echoid-s4567" xml:space="preserve">in fruſto, BEIC, trapezium,
              <lb/>
            BEIC, erunt igitur rectæ, BC, EI, lineæ
              <lb/>
            oppoſitarum baſium fructi inter ſe bomologæ,
              <lb/>
              <note position="left" xlink:label="note-0206-01" xlink:href="note-0206-01a" xml:space="preserve">_Corol. 21._
                <lb/>
              _lib. 1._</note>
            & </s>
            <s xml:id="echoid-s4568" xml:space="preserve">quia cylindricus, AC, eſt ſolidum ſimi-
              <lb/>
            lare genitum ex, AC, iuxta regulam, BC,
              <lb/>
              <note position="left" xlink:label="note-0206-02" xlink:href="note-0206-02a" xml:space="preserve">_Coroll. 3._
                <lb/>
              _34. huius._
                <lb/>
              _33. huius._
                <lb/>
              _27. huius._</note>
            & </s>
            <s xml:id="echoid-s4569" xml:space="preserve">fruſtum, EBCI, eſt ſolidum prædicto ſimilare genitum ex trapezio,
              <lb/>
            EBCI, ſunt autem h æc ſolida ſimilaria, vt omnia eorumdem quadrata,
              <lb/>
            & </s>
            <s xml:id="echoid-s4570" xml:space="preserve">omnia quadrata, AC, regula, BC, ad omnia quadrata trapezij, E
              <lb/>
            BCI, regula eadem ſunt, vt quadratum, BC, ad rectangulum ſub, BC,
              <lb/>
            EI, vna cum, _{1/3},_ quadrati differentiæ earumdem, ergo cylindricus, A
              <lb/>
            C, ad fruſtum conicum, EBCI, & </s>
            <s xml:id="echoid-s4571" xml:space="preserve">ad quoduis aliud in eadem baſi, & </s>
            <s xml:id="echoid-s4572" xml:space="preserve">al-
              <lb/>
            titudine cum hoc conſtitutum (quo niam exiſtet huic æquale) erit vt qua-
              <lb/>
              <note position="left" xlink:label="note-0206-03" xlink:href="note-0206-03a" xml:space="preserve">_K. Huius._
                <lb/>
              _Coroll._
                <lb/>
              _Gener._</note>
            dratum, BC, ad rectangulum ſub, BC, EI, vna cum, _{1/3}_, quadrati dif-
              <lb/>
            ferentiæ earu mdem, BC, EI, quæ ſunt duarum oppoſitarum baſium, E
              <lb/>
            I, BC, bomologæ vtcumque ſumptæ, nam planum eadem ſolida ſecans
              <lb/>
              <note position="left" xlink:label="note-0206-04" xlink:href="note-0206-04a" xml:space="preserve">_Corol. 21._
                <lb/>
              _lib. I._</note>
            ductum eſt vtcumque, dummodo per eorumdem latera tranſeat.</s>
            <s xml:id="echoid-s4573" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div466" type="section" level="1" n="282">
          <head xml:id="echoid-head298" xml:space="preserve">M. SECTIO XII.</head>
          <p style="it">
            <s xml:id="echoid-s4574" xml:space="preserve">_H_Inc pátet ſi in eadem baſi, BC, figura, fuerit conicus, & </s>
            <s xml:id="echoid-s4575" xml:space="preserve">eadem
              <lb/>
            altitudine cum fruſto, ideſt cum cylindrico, AC, qui ſit conicus,
              <lb/>
              <note position="left" xlink:label="note-0206-05" xlink:href="note-0206-05a" xml:space="preserve">_I. Huius._
                <lb/>
              _Corollar._
                <lb/>
              _Gener._</note>
            BOC, quod hic erit, _{1/3}_, cylindrici, AC, & </s>
            <s xml:id="echoid-s4576" xml:space="preserve">ideò ad fruſtum, EBCI, erit
              <lb/>
            vt, _{1/3}_, quadrati, BC, ad rect angulum ſub, BC, EI, vna cum, _{1/3}_, qua-
              <lb/>
            drati differentiæ, BC, EI, ideſt vtt otum quadr atum, BC, ad rectangu-
              <lb/>
            lum ſub, BC, & </s>
            <s xml:id="echoid-s4577" xml:space="preserve">tripla, EI, vna cumtoto quadrato differentiæ earum-
              <lb/>
            dem, BC, EI. </s>
            <s xml:id="echoid-s4578" xml:space="preserve">Vide igitur quam ſit amplior hæc demonſtratio ea, qua
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            alij oſtenderunt cylindrum eſſe triplum coni, & </s>
            <s xml:id="echoid-s4579" xml:space="preserve">priſma piramidis in ea-
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            dem baſi, & </s>
            <s xml:id="echoid-s4580" xml:space="preserve">al
              <unsure/>
            titudine cum ipſo conſtitute, nam ad tot varia ſolida </s>
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