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

Table of Notes

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          <p>
            <s xml:id="echoid-s8500" xml:space="preserve">
              <pb o="342" file="0362" n="362" rhead="GEO METRIÆ"/>
            ra bolæ, AQG, habent rationem compoſitam ex ea, quam habent
              <lb/>
            omnia quadrata, DPG, ad omnia quadrata, DG, ideſt ex ratione
              <lb/>
            compoſitę ex {1/2}. </s>
            <s xml:id="echoid-s8501" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8502" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8503" xml:space="preserve">PG, ad, ZP, & </s>
            <s xml:id="echoid-s8504" xml:space="preserve">ex ea, quam habent om-
              <lb/>
              <note position="left" xlink:label="note-0362-01" xlink:href="note-0362-01a" xml:space="preserve">Exantec.</note>
            nia quadrata, DG, ad omnia quadrata, AG, .</s>
            <s xml:id="echoid-s8505" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8506" xml:space="preserve">ex ratione paralle-
              <lb/>
            lepipedi ſub, DP, & </s>
            <s xml:id="echoid-s8507" xml:space="preserve">quadrato, PG, ad parallel epipedum ſub, AQ,
              <lb/>
            & </s>
            <s xml:id="echoid-s8508" xml:space="preserve">quadrato, QG, & </s>
            <s xml:id="echoid-s8509" xml:space="preserve">tandem ex ea, quam habent omnia quadrata,
              <lb/>
            A G, ad omnia quadrata ſemiparabolæ, AQG, .</s>
            <s xml:id="echoid-s8510" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8511" xml:space="preserve">ex ratione paral-
              <lb/>
            lelepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8512" xml:space="preserve">quadrato, QG, ad eiuſdem dimidium: </s>
            <s xml:id="echoid-s8513" xml:space="preserve">Duæ
              <lb/>
              <note position="left" xlink:label="note-0362-02" xlink:href="note-0362-02a" xml:space="preserve">Effcitur ex
                <lb/>
              x1. l. 2.</note>
            autem rationes parallelepipedi ſub, DP, & </s>
            <s xml:id="echoid-s8514" xml:space="preserve">quadrato, PG, ad paral-
              <lb/>
            lelepipedum ſub, AQ, & </s>
            <s xml:id="echoid-s8515" xml:space="preserve">quadrato, QG, & </s>
            <s xml:id="echoid-s8516" xml:space="preserve">ratio huius ad eiuſdem
              <lb/>
              <note position="left" xlink:label="note-0362-03" xlink:href="note-0362-03a" xml:space="preserve">21. huius.</note>
            dimidium, conſiciunt rationem parallelepipedi ſub, DP, & </s>
            <s xml:id="echoid-s8517" xml:space="preserve">quadra-
              <lb/>
            to, PG, ad {1/2}. </s>
            <s xml:id="echoid-s8518" xml:space="preserve">parallelepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8519" xml:space="preserve">quadrato, QG, ergo om-
              <lb/>
              <note position="left" xlink:label="note-0362-04" xlink:href="note-0362-04a" xml:space="preserve">Defin. 12,
                <lb/>
              l. 1.</note>
            nia quadrata, D & </s>
            <s xml:id="echoid-s8520" xml:space="preserve">G, ad omnia quadrata ſemiparabolæ, AQG, ha-
              <lb/>
            bent rationem compoſitam ex ratione rectæ compoſitæ ex {1/3}. </s>
            <s xml:id="echoid-s8521" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8522" xml:space="preserve">
              <lb/>
            {1/2}. </s>
            <s xml:id="echoid-s8523" xml:space="preserve">PG, ad, ZP, & </s>
            <s xml:id="echoid-s8524" xml:space="preserve">ex ratione parallelepipedi ſub, DP, & </s>
            <s xml:id="echoid-s8525" xml:space="preserve">quadrato,
              <lb/>
            P G, ad {1/2}. </s>
            <s xml:id="echoid-s8526" xml:space="preserve">parallelepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8527" xml:space="preserve">quadrato, QG, vt dictum eſt.</s>
            <s xml:id="echoid-s8528" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8529" xml:space="preserve">Inſuper omnia quadrata trilinei, AXG, ad omnia quadrata trili-
              <lb/>
            nei, DEG, habent rationem compoſitam ex ratione omnium qua-
              <lb/>
              <note position="left" xlink:label="note-0362-05" xlink:href="note-0362-05a" xml:space="preserve">30. huius.</note>
            dratorum, AXG, ad omnia quadrata, AG, .</s>
            <s xml:id="echoid-s8530" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8531" xml:space="preserve">ſubſexcupla .</s>
            <s xml:id="echoid-s8532" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8533" xml:space="preserve">ex ra-
              <lb/>
            tione {1/6}. </s>
            <s xml:id="echoid-s8534" xml:space="preserve">parallelepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8535" xml:space="preserve">quadrato, QG, ad idem paral-
              <lb/>
            lelepipedum, & </s>
            <s xml:id="echoid-s8536" xml:space="preserve">ex ratione omnium quadratorum, AG, ad omnia
              <lb/>
            quadrata, DG, .</s>
            <s xml:id="echoid-s8537" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8538" xml:space="preserve">parallelepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8539" xml:space="preserve">quadrato, QG, ad
              <lb/>
            parallelepipedum ſub, DP, & </s>
            <s xml:id="echoid-s8540" xml:space="preserve">quadrato, PG, quæ duæ rationes cõ-
              <lb/>
            ficiunt rationem {1/6}. </s>
            <s xml:id="echoid-s8541" xml:space="preserve">parallepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8542" xml:space="preserve">quadrato, QG, ad pa-
              <lb/>
            rallelepipedum ſub, DP, & </s>
            <s xml:id="echoid-s8543" xml:space="preserve">quadrato, PG, & </s>
            <s xml:id="echoid-s8544" xml:space="preserve">tandem ex ratione
              <lb/>
            omnium quadratorum, DG, ad omnia quadrata trilinei, DEG, .</s>
            <s xml:id="echoid-s8545" xml:space="preserve">i.
              <lb/>
            </s>
            <s xml:id="echoid-s8546" xml:space="preserve">
              <note position="left" xlink:label="note-0362-06" xlink:href="note-0362-06a" xml:space="preserve">47. huius.</note>
            ex ea, quam habet, ZP, ad reſiduum, ab eadem, ZP, demptis {2/3}. </s>
            <s xml:id="echoid-s8547" xml:space="preserve">ZP,
              <lb/>
            cum {1/6}. </s>
            <s xml:id="echoid-s8548" xml:space="preserve">PG, ergo omnia quadrata trilinei, AXG, ad omnia quadra-
              <lb/>
            ta trilinei, DEG, habent rationem compoſitam ex ea, quam habet
              <lb/>
            {1/6}. </s>
            <s xml:id="echoid-s8549" xml:space="preserve">parallelepipedi, ſub, AQ, & </s>
            <s xml:id="echoid-s8550" xml:space="preserve">quadrato, QG, ad parallelepipedum
              <lb/>
            ſub, DP, & </s>
            <s xml:id="echoid-s8551" xml:space="preserve">quadrato, PG, & </s>
            <s xml:id="echoid-s8552" xml:space="preserve">ex ea, quam habet, ZP, ad ſui reſi-
              <lb/>
            duum, demptis ab ea {2/3}. </s>
            <s xml:id="echoid-s8553" xml:space="preserve">ZP, cum {1/6}. </s>
            <s xml:id="echoid-s8554" xml:space="preserve">PG, quæ oſtendere oportebat.</s>
            <s xml:id="echoid-s8555" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div824" type="section" level="1" n="487">
          <head xml:id="echoid-head507" xml:space="preserve">THEOREMA XLVII. PROPOS. XLIX.</head>
          <p>
            <s xml:id="echoid-s8556" xml:space="preserve">IN eadem figura Propoſ. </s>
            <s xml:id="echoid-s8557" xml:space="preserve">46. </s>
            <s xml:id="echoid-s8558" xml:space="preserve">oſtendemus, producta, PD,
              <lb/>
            verſus, AX, cui occurrat in, C, omnia quadrata trilinei,
              <lb/>
            D GP, ad omnia quadrata figuræ, CAZP, demptis omni-
              <lb/>
            bus quadratis trilinei, ACD, habere rationem compoſitam
              <lb/>
            ex ea, quam habet compoſita ex {1/3}. </s>
            <s xml:id="echoid-s8559" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8560" xml:space="preserve">{1/6}, PG, ad ZP, & </s>
            <s xml:id="echoid-s8561" xml:space="preserve">
              <lb/>
            ex ratione parallelepipedi ſub, DP, & </s>
            <s xml:id="echoid-s8562" xml:space="preserve">quadrato, PG, ad </s>
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