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

Table of contents

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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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            <s xml:id="echoid-s9244" xml:space="preserve">Sitigitur hyperbola, DBF, in baſi, DF, cuius axis, vel diameter,
              <lb/>
            EB, & </s>
            <s xml:id="echoid-s9245" xml:space="preserve">tranſuerſum latus, BO, bifariam diuiſum in, N, deſcribatur
              <lb/>
            vero paralielogrammũ, AF, in eadem altitudine, & </s>
            <s xml:id="echoid-s9246" xml:space="preserve">baſi cum hy-
              <lb/>
            perbola, DBF, & </s>
            <s xml:id="echoid-s9247" xml:space="preserve">nunc circa axim, vel diametrum, BE, circa quam
              <lb/>
              <figure xlink:label="fig-0386-01" xlink:href="fig-0386-01a" number="264">
                <image file="0386-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0386-01"/>
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            tit etiam triangulum, BDF. </s>
            <s xml:id="echoid-s9248" xml:space="preserve">Dico ergo
              <lb/>
            omnia quadrata hyperbolæ, DBF, regu-
              <lb/>
            la, DF, ad omnia quadrata, AF, eſſe vt
              <lb/>
            compoſitam ex, NB, & </s>
            <s xml:id="echoid-s9249" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9250" xml:space="preserve">BE, ad, OE;
              <lb/>
            </s>
            <s xml:id="echoid-s9251" xml:space="preserve">ad omnia verò quadrata trianguli, DBF,
              <lb/>
            eſſe vt compoſitam ex ſexquialtera, OB,
              <lb/>
            & </s>
            <s xml:id="echoid-s9252" xml:space="preserve">ipſa, BE, ad, OE, ſumatur in, BE, vt-
              <lb/>
            cunq; </s>
            <s xml:id="echoid-s9253" xml:space="preserve">punctum, M, & </s>
            <s xml:id="echoid-s9254" xml:space="preserve">per, M, ducatur,
              <lb/>
            MG, parallela ipſi, DF, ſecans curuam
              <lb/>
            hyperbolæ in, H. </s>
            <s xml:id="echoid-s9255" xml:space="preserve">Eſt ergo quadratum,
              <lb/>
              <note position="left" xlink:label="note-0386-01" xlink:href="note-0386-01a" xml:space="preserve">39. l. 3. &
                <lb/>
              Scho. 40.</note>
            EF, vel quadratum, GM, ad quadratum,
              <lb/>
            MH, vt rectangulum, OEB, ad rectangu-
              <lb/>
            lum, OMB, eſt autem, BF, parallelogrã@
              <lb/>
            mum in eadem altitudine, & </s>
            <s xml:id="echoid-s9256" xml:space="preserve">baſi cum ſemihyperbola, BEF, & </s>
            <s xml:id="echoid-s9257" xml:space="preserve">
              <lb/>
            punctum, M, vtcunq; </s>
            <s xml:id="echoid-s9258" xml:space="preserve">ſumptum, per quod acta eſtipſi, DF, pa-
              <lb/>
            rallela, MG, regula, DF, repertumque eſt, vt quadratum, GM, ad
              <lb/>
            quadratum, MH, ita eſſe rectangulum, OEB, ad rectangulum, O
              <lb/>
              <note position="left" xlink:label="note-0386-02" xlink:href="note-0386-02a" xml:space="preserve">Coroll. 3.
                <lb/>
              26. l. 2.</note>
            MB, ergo horum quatuor ordinum magnitudines erunt propor-
              <lb/>
            tionales .</s>
            <s xml:id="echoid-s9259" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s9260" xml:space="preserve">omnia quadrata, BF, magnitudines primi ordinis col-
              <lb/>
            lectæ iuxta primam .</s>
            <s xml:id="echoid-s9261" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s9262" xml:space="preserve">iuxta quadratum, GM, ad omnia quadra-
              <lb/>
            ta ſernihy perbolæ, BEF, magnitudines ſecundi ordinis collectas
              <lb/>
            iuxta ſecundam .</s>
            <s xml:id="echoid-s9263" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s9264" xml:space="preserve">iuxta quadratum, MH, erunt vt rectangula ſub
              <lb/>
            maximis abſciſſarum, BE, magnitudines tertij ordinis collectę iux-
              <lb/>
            ta tertjam .</s>
            <s xml:id="echoid-s9265" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s9266" xml:space="preserve">iuxta rectangulum ſub, OE, EB, ad rectangula ſub
              <lb/>
            omnibus abſciſsis, EB, adiuncta, BO, & </s>
            <s xml:id="echoid-s9267" xml:space="preserve">ſub omnibus abſciſsis, EB,
              <lb/>
            magnitudines quarti ordinis collectas iuxta primam, .</s>
            <s xml:id="echoid-s9268" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s9269" xml:space="preserve">iuxta re-
              <lb/>
            ctangulum, OMB; </s>
            <s xml:id="echoid-s9270" xml:space="preserve">verum rectangula ſub maximis abſciſſarum,
              <lb/>
            EB, adiuncta, BO, & </s>
            <s xml:id="echoid-s9271" xml:space="preserve">ſub maximis abiciſſarum, EB, ad rectangula
              <lb/>
            ſub omnibus abſeiſsis, EB, adiuncta, BO, & </s>
            <s xml:id="echoid-s9272" xml:space="preserve">ſub omnibus abſciſsis,
              <lb/>
            EB, recti, vel eiuſdem obliqui tranſitus, ſunt vt, OE, ad compoſi-
              <lb/>
            tam ex, NB, & </s>
            <s xml:id="echoid-s9273" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9274" xml:space="preserve">BE, ergo, conuertendo, omnia quadrara ſemi-
              <lb/>
              <note position="left" xlink:label="note-0386-03" xlink:href="note-0386-03a" xml:space="preserve">Corol. 30.
                <lb/>
              l. 2.</note>
            hyperbolæ, BEF, ad omnia quadrata, BF, vel eorum quadrupla .</s>
            <s xml:id="echoid-s9275" xml:space="preserve">i.
              <lb/>
            </s>
            <s xml:id="echoid-s9276" xml:space="preserve">omnia quadrata hyperbolæ, DBF, ad omnia quadrata, AF, etiam
              <lb/>
            ſi, AF, non eſſet circa axim, vel diametrum, BE, ſed tantum in ea-
              <lb/>
            dem altitudine cum hyperbola, DBF, erunt, vt compoſita ex {1/2}. </s>
            <s xml:id="echoid-s9277" xml:space="preserve">O
              <lb/>
            B, & </s>
            <s xml:id="echoid-s9278" xml:space="preserve">{1/3}. </s>
            <s xml:id="echoid-s9279" xml:space="preserve">BE, ad, OE.</s>
            <s xml:id="echoid-s9280" xml:space="preserve"/>
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          <note position="left" xml:space="preserve">24. l. 2.</note>
          <p>
            <s xml:id="echoid-s9281" xml:space="preserve">Quoniam verò omnia quadrata, AF, ſunt tripla omnium </s>
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