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

Table of contents

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[461.] B. SECTIO II.
Page: 344 (324)
[462.] C. SECTIO III.
Page: 344 (324)
[463.] D. SECTIO IV.
Page: 344 (324)
[464.] E. SECTIO V.
Page: 345 (325)
[465.] THEOREMA XXIX. PROPOS. XXXI.
Page: 345 (325)
[466.] THEOREMA XXX. PROPOS. XXXII.
Page: 346 (326)
[467.] COROLLARIVM.
Page: 347 (327)
[468.] THEOREMA XXXI. PROPOS. XXXIII.
Page: 347 (327)
[469.] THEOREMA XXXII. PROPOS. XXXIV.
Page: 348 (328)
[470.] COROLLARIVM.
Page: 349 (329)
[471.] THEOREMA XXXIII. PROPOS. XXXV.
Page: 349 (329)
[472.] COROLLARIVM.
Page: 350 (330)
[473.] THEOREMA XXXIV. PROPOS. XXXVI.
Page: 351 (331)
[474.] THEOREMA XXXV. PROPOS. XXXVII.
Page: 352 (332)
[475.] THEOREMA XXXVI. PROP. XXXVIII.
Page: 353 (333)
[476.] THEOREMA XXXVII. PROP. XXXIX.
Page: 354 (334)
[477.] THEOREMA XXXVIII. PROP. XL.
Page: 354 (334)
[478.] COROLLARIVM.
Page: 354 (334)
[479.] THEOREMA XXXIX. PROPOS. XLI
Page: 355 (335)
[480.] THEOREMA XL. PROPOS. XLII.
Page: 356 (336)
[481.] THEOREMA XLI. PROPOS. XLIII.
Page: 357 (337)
[482.] THEOREMA XLII. PROPOS. XLIV.
Page: 358 (338)
[483.] THEOREMA XLIII. PROP. XLV.
Page: 358 (338)
[484.] THEOREMA XLIV. PROP. XLVI.
Page: 359 (339)
[485.] THEOREMA XLV. PROP. XLVII.
Page: 360 (340)
[486.] THEOREMA XLVI. PROPOS. XLVIII.
Page: 361 (341)
[487.] THEOREMA XLVII. PROPOS. XLIX.
Page: 362 (342)
[488.] THEOREMA XLVIII. PROPOS. L.
Page: 364 (344)
[489.] THEOREMA XLIX. PROPOS. LI.
Page: 365 (345)
[490.] SCHOLIVM.
Page: 365 (345)
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            EP, PH, vt quadratum, GP, ad rectangulum, GPZ, ergo, exæ.
              <lb/>
            </s>
            <s xml:id="echoid-s8437" xml:space="preserve">quali, omnia quadrata, DG, ad rectangul a ſub trilineis, DPG, D
              <lb/>
            EG, erunt vt quadratum, PG, ad {1/6}. </s>
            <s xml:id="echoid-s8438" xml:space="preserve">quadrati, PG, .</s>
            <s xml:id="echoid-s8439" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8440" xml:space="preserve">erunt eorum
              <lb/>
            fexcupla: </s>
            <s xml:id="echoid-s8441" xml:space="preserve">Quoniam ergo omnia quadrata, DG, ad rectangula, ſub,
              <lb/>
            DG, &</s>
            <s xml:id="echoid-s8442" xml:space="preserve">?? </s>
            <s xml:id="echoid-s8443" xml:space="preserve">trilineo, DGP, ſunt vt, DG, ad, DGP, .</s>
            <s xml:id="echoid-s8444" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8445" xml:space="preserve">vt, ZP, ad
              <lb/>
            compofitam ex {1/2}. </s>
            <s xml:id="echoid-s8446" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8447" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8448" xml:space="preserve">PG, ſunt autem omnia quadrata, D
              <lb/>
              <note position="right" xlink:label="note-0361-01" xlink:href="note-0361-01a" xml:space="preserve">5. huius@</note>
            G, ſexcupla rectangulorum ſub trilineis, DPG, DEG, .</s>
            <s xml:id="echoid-s8449" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8450" xml:space="preserve">ad ea,
              <lb/>
            vt, ZP, ad {1/6}, ZP, ergo omnia quadrata, DG, ad omnia quadrata,
              <lb/>
            D GP, erunt vt, ZG, ad reſiduum, dempto {1/6}. </s>
            <s xml:id="echoid-s8451" xml:space="preserve">ZP, à compoſita ex
              <lb/>
            {1/2}. </s>
            <s xml:id="echoid-s8452" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8453" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8454" xml:space="preserve">PG, quia verò ſi ab {1/2}. </s>
            <s xml:id="echoid-s8455" xml:space="preserve">ZP, dematur, {1/6}. </s>
            <s xml:id="echoid-s8456" xml:space="preserve">ZP, remanent
              <lb/>
            1. </s>
            <s xml:id="echoid-s8457" xml:space="preserve">ZP, ideò omnia quadrata, DG, ad omnia quadiata, DPG, erunt
              <lb/>
            vt, ZP, ad compoſitam ex {1/3}. </s>
            <s xml:id="echoid-s8458" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8459" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8460" xml:space="preserve">PG, vt dictum eſt.</s>
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            <s xml:id="echoid-s8462" xml:space="preserve">Quia verò nunc oſtenſum eſt omnia quadrata, DG, ad omnia
              <lb/>
            quadrata, DPG, eſſe vt, ZP, ad compoſitam ex {1/3}. </s>
            <s xml:id="echoid-s8463" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8464" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8465" xml:space="preserve">PG,
              <lb/>
            omnia autem quadrata, DG, ad rectangula ſub trilineis, DPG, D
              <lb/>
            E G, ſunt vt, ZP, ad {1/6}. </s>
            <s xml:id="echoid-s8466" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8467" xml:space="preserve">ad eadem bis ſumpta, vt, ZP, ad {1/3}.
              <lb/>
            </s>
            <s xml:id="echoid-s8468" xml:space="preserve">Z P, ideò omnia quadrata, DG, ad omnia quadrata, DPG, & </s>
            <s xml:id="echoid-s8469" xml:space="preserve">ad
              <lb/>
            rectangula bis ſub, DPG, DEG, erunt vt, ZP, ad compoſitam ex
              <lb/>
            {2/3}. </s>
            <s xml:id="echoid-s8470" xml:space="preserve">ZP. </s>
            <s xml:id="echoid-s8471" xml:space="preserve">& </s>
            <s xml:id="echoid-s8472" xml:space="preserve">{1/6}: </s>
            <s xml:id="echoid-s8473" xml:space="preserve">PG, ergo omnia quadrata. </s>
            <s xml:id="echoid-s8474" xml:space="preserve">DG, ad reſiduum, demptis
              <lb/>
            omnibus quadratis, DPG, & </s>
            <s xml:id="echoid-s8475" xml:space="preserve">rectangulis bis ſub, DPG, DEG, .</s>
            <s xml:id="echoid-s8476" xml:space="preserve">i. </s>
            <s xml:id="echoid-s8477" xml:space="preserve">
              <lb/>
            ad omnia quadrata trilinei, DEG, erunt vt, ZP, ad reſiduum, dem-
              <lb/>
            ptis {2/3}. </s>
            <s xml:id="echoid-s8478" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8479" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8480" xml:space="preserve">PG, ab eadem, ZP, quæ nobis oſtendenda erat.</s>
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        <div xml:id="echoid-div821" type="section" level="1" n="486">
          <head xml:id="echoid-head506" xml:space="preserve">THEOREMA XLVI. PROPOS. XLVIII.</head>
          <p>
            <s xml:id="echoid-s8482" xml:space="preserve">IN ſupradictæ Propoſ. </s>
            <s xml:id="echoid-s8483" xml:space="preserve">figura, ducta, AX, parallela baſi,
              <lb/>
            Z G, quæ tanget parabolam in, A, cui occurrat, GE,
              <lb/>
            producta, in puncto, X, oſtendemus omnia quadrata trili-
              <lb/>
            nei, DPG, ad omnia quadrata ſemiparabolæ, AQG, ha-
              <lb/>
            bere rationem compoſitam ex ea, quam habet compoſita ex
              <lb/>
            {1/3}. </s>
            <s xml:id="echoid-s8484" xml:space="preserve">ZP, & </s>
            <s xml:id="echoid-s8485" xml:space="preserve">{1/6}. </s>
            <s xml:id="echoid-s8486" xml:space="preserve">PG, ad, ZP, & </s>
            <s xml:id="echoid-s8487" xml:space="preserve">ex ratione parallelepipedi ſub,
              <lb/>
            D P, & </s>
            <s xml:id="echoid-s8488" xml:space="preserve">quadrato, PG, ad dimidium parallelepipedi ſub,
              <lb/>
            A Q, & </s>
            <s xml:id="echoid-s8489" xml:space="preserve">quadrato, QG; </s>
            <s xml:id="echoid-s8490" xml:space="preserve">Omnia vero quadrata trilinei, AX
              <lb/>
            G. </s>
            <s xml:id="echoid-s8491" xml:space="preserve">ad omnia quadrata trilinei, DEG, habere rationem cõ-
              <lb/>
            poſitam ex ea, quam habet parallelepipedi ſub, AQ, & </s>
            <s xml:id="echoid-s8492" xml:space="preserve">qua-
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            drato, QG, ſexta pars, ad parallelepipedum ſub, DP, & </s>
            <s xml:id="echoid-s8493" xml:space="preserve">
              <lb/>
            quadrato, PG, & </s>
            <s xml:id="echoid-s8494" xml:space="preserve">ex ea, quam habet, ZP, ad reſidunm, dem-
              <lb/>
            ptis ab eadem, ZP, {2/3}. </s>
            <s xml:id="echoid-s8495" xml:space="preserve">ZP, cum {1/6}. </s>
            <s xml:id="echoid-s8496" xml:space="preserve">PG.</s>
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          <p>
            <s xml:id="echoid-s8498" xml:space="preserve">Omnia .</s>
            <s xml:id="echoid-s8499" xml:space="preserve">n. </s>
            <s xml:id="echoid-s8500" xml:space="preserve">quadrata trilinei, DPG, ad omnia quadrata </s>
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