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

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[531.] COROLLARIVM XXVII.
Page: 383 (363)
[532.] SCHOLIV M.
Page: 384 (364)
[533.] Finis quarti Libri.
Page: 384 (364)
[534.] GEOMETRIÆ CAVALERII. LIBER QVINTVS. In quo de Hyperbola, Oppoſitis Sectionib us, ac ſolidis ab eiſdem genitis, babetur contemplatio. THEOREMA I. PROPOS. I.
Page: 385 (365)
[535.] THEOREMA II. PROPOS. II.
Page: 387 (367)
[536.] THEOREMA III. PROPOS. III.
Page: 388 (368)
[537.] THEOREMA IV. PROPOS. IV.
Page: 390 (370)
[538.] THEOREMA V. PROPOS. V.
Page: 391 (371)
[539.] PROBLEMA I. PROPOS. VI.
Page: 393 (373)
[540.] THEOREMA VI. PROPOS. VII.
Page: 394 (374)
[541.] THEOREMA VII. PROPOS. VIII.
Page: 395 (375)
[542.] THEOREMA VIII. PROPOS. IX.
Page: 396 (376)
[543.] THEOREMA IX. PROPOS. X.
Page: 398 (378)
[544.] THEOREMA X. PROPOS. XI.
Page: 399 (379)
[545.] THEOREMA XI. PROPOS. XII.
Page: 401 (381)
[546.] THEOREMA XII. PROPOS. XIII.
Page: 403 (383)
[547.] THEOREMA XIII, PROPOS. XIV.
Page: 404 (384)
[548.] SCHOLIVM.
Page: 406 (386)
[549.] THEOREMA XIV. PROPOS. XV.
Page: 406 (386)
[550.] THEOREMA XV. PROPOS. XVI.
Page: 407 (387)
[551.] COROLLARIVM.
Page: 407 (387)
[552.] THEOREMA XVI. PROPOS. XVII.
Page: 407 (387)
[553.] THE OREMA XVII. PROPOS. XVIII.
Page: 408 (388)
[554.] THEOREMA XVIII. PROPOS. XIX.
Page: 409 (389)
[555.] COROLLARIVM.
Page: 409 (389)
[556.] SCHOLIVM.
Page: 409 (389)
[557.] THEOREMA XIX. PROPOS. XX.
Page: 410 (390)
[558.] THEOREMA XX. PROPOS. XXI.
Page: 412 (392)
[559.] A@@ter ſupradictam rationem explicare.
Page: 414 (394)
[560.] COROLLARIVM:
Page: 415 (395)
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            ptis omnibus quadratis, HQ, æquatur autem rectangulum, EM
              <lb/>
            N, cum rectangulo ſub, EM, & </s>
            <s xml:id="echoid-s9472" xml:space="preserve">iub compoſita ex {1/3}. </s>
            <s xml:id="echoid-s9473" xml:space="preserve">EM, & </s>
            <s xml:id="echoid-s9474" xml:space="preserve">{1/2}. </s>
            <s xml:id="echoid-s9475" xml:space="preserve">N
              <lb/>
            O, rectangulo ſub, EM, & </s>
            <s xml:id="echoid-s9476" xml:space="preserve">ſub compoſita ex {1/3}. </s>
            <s xml:id="echoid-s9477" xml:space="preserve">EM. </s>
            <s xml:id="echoid-s9478" xml:space="preserve">integra, MN,
              <lb/>
            & </s>
            <s xml:id="echoid-s9479" xml:space="preserve">{1/2}. </s>
            <s xml:id="echoid-s9480" xml:space="preserve">NO, ergo omnia quadrata, SF, ad omnia quadrata fruſti, D
              <lb/>
            HGF, demptis omnibus quadratis, HQ, erunt vt rectangulum,
              <lb/>
            OEN, ad rectangulum ſub, EM, & </s>
            <s xml:id="echoid-s9481" xml:space="preserve">ſub compoſita ex {1/3}. </s>
            <s xml:id="echoid-s9482" xml:space="preserve">EM, in-
              <lb/>
            tegra, MN, & </s>
            <s xml:id="echoid-s9483" xml:space="preserve">{1/2}. </s>
            <s xml:id="echoid-s9484" xml:space="preserve">NO.</s>
            <s xml:id="echoid-s9485" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s9486" xml:space="preserve">Omnia verò quadrata trianguli, DMF, ad eadem erunt, vt {1/3}.
              <lb/>
            </s>
            <s xml:id="echoid-s9487" xml:space="preserve">rectanguli, OEN, ad rectangulum ſub, EM, & </s>
            <s xml:id="echoid-s9488" xml:space="preserve">ſub compoſita ex
              <lb/>
            {1/3}. </s>
            <s xml:id="echoid-s9489" xml:space="preserve">EM, integra, MN, & </s>
            <s xml:id="echoid-s9490" xml:space="preserve">{1/2}. </s>
            <s xml:id="echoid-s9491" xml:space="preserve">NO, .</s>
            <s xml:id="echoid-s9492" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9493" xml:space="preserve">vt totum rectangulum ſub, O
              <lb/>
            EN, ad rectangulum ſub, EM, & </s>
            <s xml:id="echoid-s9494" xml:space="preserve">ſub compoſita ex, EM, tripla,
              <lb/>
            MN, &</s>
            <s xml:id="echoid-s9495" xml:space="preserve">, NX, .</s>
            <s xml:id="echoid-s9496" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9497" xml:space="preserve">ſub, EM, & </s>
            <s xml:id="echoid-s9498" xml:space="preserve">ſub compoſita ex, EX, & </s>
            <s xml:id="echoid-s9499" xml:space="preserve">dupla, MN,
              <lb/>
            quæ oſtendenda erant.</s>
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          <head xml:id="echoid-head562" xml:space="preserve">THEOREMA V. PROPOS. V.</head>
          <p>
            <s xml:id="echoid-s9501" xml:space="preserve">IN eadem figura, regula eadem retenta, oſtendemus om-
              <lb/>
            mnia quadrata, AF, demptis omnibus quadratis hyper-
              <lb/>
            bolæ, DNF, ad omnia quadrata, SF, demptis omnibus
              <lb/>
            quadratis fruſti, HDFG, eſſe vt parallelepipedum ſub cõ-
              <lb/>
            poſita ex ipſa, XE, EN, & </s>
            <s xml:id="echoid-s9502" xml:space="preserve">ſub quadrato, NE, ad parallele-
              <lb/>
            pipedum ſub compoſita ex eadem, XE, & </s>
            <s xml:id="echoid-s9503" xml:space="preserve">cum, EN, NM,
              <lb/>
            & </s>
            <s xml:id="echoid-s9504" xml:space="preserve">ſub quadrato, ME.</s>
            <s xml:id="echoid-s9505" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s9506" xml:space="preserve">Quia enim omnia quadrata, AF, ad omnia quadrata hyperbo-
              <lb/>
              <note position="right" xlink:label="note-0391-01" xlink:href="note-0391-01a" xml:space="preserve">Is huius.</note>
            læ, DNF, ſunt vt, OE, ad compoſitam ex {1/2}. </s>
            <s xml:id="echoid-s9507" xml:space="preserve">ON, & </s>
            <s xml:id="echoid-s9508" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9509" xml:space="preserve">NE, ideò
              <lb/>
            per conuerſionem rationis, & </s>
            <s xml:id="echoid-s9510" xml:space="preserve">conuertendo omnia quadrata, AF,
              <lb/>
            demptis omnibus quadratis hyperbolæ, DNF, ad omnia quadra-
              <lb/>
            ta, AF, erunt vt compoſita ex {1/2}. </s>
            <s xml:id="echoid-s9511" xml:space="preserve">ON, & </s>
            <s xml:id="echoid-s9512" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s9513" xml:space="preserve">NE, ad, OE, .</s>
            <s xml:id="echoid-s9514" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9515" xml:space="preserve">ſum-
              <lb/>
            pta, NE, communialtitudine, vt rectangulum ſub compoſita ex
              <lb/>
            {1/2}. </s>
            <s xml:id="echoid-s9516" xml:space="preserve">ON, & </s>
            <s xml:id="echoid-s9517" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s9518" xml:space="preserve">NE, & </s>
            <s xml:id="echoid-s9519" xml:space="preserve">ſub, NE, ad rectangulum, OEN. </s>
            <s xml:id="echoid-s9520" xml:space="preserve">Quoniam
              <lb/>
            verò omnia quadrata, AF, demptis omnibus quadratis hyperbo-
              <lb/>
            læ, DNF, ad omnia quadrata, SF, demptis omnibus quadratis
              <lb/>
            fruſti, DHGF, habent rationem compoſitam ex ea, quam habent
              <lb/>
            omnia quadrata, AF, demptis omnibus quadratis hyperbolæ, D
              <lb/>
              <note position="right" xlink:label="note-0391-02" xlink:href="note-0391-02a" xml:space="preserve">Defin. 12.
                <lb/>
              1. I.</note>
            NF, ad omnia quadrata, AF, .</s>
            <s xml:id="echoid-s9521" xml:space="preserve">i. </s>
            <s xml:id="echoid-s9522" xml:space="preserve">ex ea, quam habet rectangulum
              <lb/>
            ſub compoſita ex {1/2}. </s>
            <s xml:id="echoid-s9523" xml:space="preserve">ON, & </s>
            <s xml:id="echoid-s9524" xml:space="preserve">{2/3}. </s>
            <s xml:id="echoid-s9525" xml:space="preserve">NE, & </s>
            <s xml:id="echoid-s9526" xml:space="preserve">ſub, N E, ad rectangu-
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            lum, NEO; </s>
            <s xml:id="echoid-s9527" xml:space="preserve">& </s>
            <s xml:id="echoid-s9528" xml:space="preserve">ex ratione, quam habent omnia quadrata,
              <lb/>
            AF, ad omnia quadrata, SF, ideſt ex ea, quam habet, NE,
              <lb/>
            ad, EM, & </s>
            <s xml:id="echoid-s9529" xml:space="preserve">tandem ex ea, quam habent omnia quadrata, SF,
              <lb/>
              <note position="right" xlink:label="note-0391-03" xlink:href="note-0391-03a" xml:space="preserve">10, 1.2.</note>
            ad omnia quadrata, SF, demptis omnibus quadratis fruſti, </s>
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