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

Table of contents

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[51.] PROBLEMA I. PROPOS. 1.
[52.] COROLLARIVM.
[53.] PROBLEMA II. PROPOS. II.
[54.] PROBLEMA III. PROPOS. III.
[55.] SCHOLIVM.
[56.] THEOREMA I. PROPOS. IV.
[57.] COROLLARIVM I.
[58.] COROLLARIVM II.
[59.] THEOREMA II. PROPOS. V.
[60.] THEOREMA III. PROPOS. VI.
[61.] COROLLARIVM.
[62.] THEOREMA IV. PROPOS. VII.
[63.] THEOREMA V. PROPOS. VIII.
[64.] COROLLARIV M.
[65.] THEOREMA VI. PROPOS. IX.
[66.] COROLLARIVM.
[67.] THEOREMA VII. PROPOS. X.
[68.] THEOREMA VIII. PROPOS. XI.
[69.] COROLLARIV M.
[70.] LEMMA PRO ANTECED. PROP.
[71.] THEOREMA IX. PROPOS. XII.
[72.] COROLLARIV M.
[73.] THEOREMA X. PROPOS. XIII.
[74.] THEOREMA XI. PROPOS. XIV.
[75.] THEOREMA XII. PROPOS. XV.
[76.] SCHOLIVM.
[77.] THEOREMA XIII. PROPOS. XVI.
[78.] COROLLARIVM.
[79.] THEOREMA XIV. PROPOS. XVII.
[80.] COROLLARIVM.
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              <pb o="37" file="0057" n="57" rhead="LIBERI."/>
            fit eas, quæ inter taliter incidentes, & </s>
            <s xml:id="echoid-s986" xml:space="preserve">perimetrum figurarum con-
              <lb/>
            tinentur, eodem ordine ſumptas, eſſe vt ipſas, HP, KN, inciden-
              <lb/>
              <note position="right" xlink:label="note-0057-01" xlink:href="note-0057-01a" xml:space="preserve">A. Def. 10.</note>
            tes, ſunt igitur figuræ planę, BVO, DTF, inter ſe ſimiles, & </s>
            <s xml:id="echoid-s987" xml:space="preserve">ho-
              <lb/>
            mologarum earundem regulæ ipſæ tangentes, dictæ figuræ ſunt in
              <lb/>
            planis æquidiſtantibus, quarum incidentes fibi inuicem ęquidiſtant,
              <lb/>
            & </s>
            <s xml:id="echoid-s988" xml:space="preserve">homologæ earundem figurarum ſunt ad eandem partem inciden-
              <lb/>
            tium, & </s>
            <s xml:id="echoid-s989" xml:space="preserve">ipſarum incidentium partes homologæ pariter ad eandem
              <lb/>
            partem conſtitutæ, igitur figuræ, VBO, TDF, nedum erunt ſimi-
              <lb/>
            les, ſed etiam ſimiliter poſitæ, quod oſtendendum erat.</s>
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        <div xml:id="echoid-div120" type="section" level="1" n="84">
          <head xml:id="echoid-head95" xml:space="preserve">COROLLARIVMI.</head>
          <p style="it">
            <s xml:id="echoid-s991" xml:space="preserve">_E_T quia oſtenſum eſt ipſas tangentes, SP, XN, eſſe bomologárum
              <lb/>
            earundem ſimilium figurarum regulas, & </s>
            <s xml:id="echoid-s992" xml:space="preserve">ductæ ſunt vtcumque,
              <lb/>
            patet ſi duxerimus alias duas eiuſdem baſis oppoſitas tangentes, quæ cum
              <lb/>
            primò ductis angulos efficient æquales, & </s>
            <s xml:id="echoid-s993" xml:space="preserve">per ipſas, & </s>
            <s xml:id="echoid-s994" xml:space="preserve">verticem, A,
              <lb/>
            extenderimus duo plana (quorum & </s>
            <s xml:id="echoid-s995" xml:space="preserve">plani figuræ, BVO, producti com-
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            munes ſectiones erunt aliæ duæ figuræ, BVO, oppoſitæ tangentes) quod
              <lb/>
            eodem modo oſtendemus has ſecundas tangentes eſſe homologarum earun-
              <lb/>
            dem ſimilium figurarum regulas, & </s>
            <s xml:id="echoid-s996" xml:space="preserve">intra ipſas contineri earundem
              <lb/>
            quoq; </s>
            <s xml:id="echoid-s997" xml:space="preserve">incidentes, ſacient autem ſecunda tangentes cum primis angu-
              <lb/>
            los æquales, prima. </s>
            <s xml:id="echoid-s998" xml:space="preserve">n. </s>
            <s xml:id="echoid-s999" xml:space="preserve">ex. </s>
            <s xml:id="echoid-s1000" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s1001" xml:space="preserve">tangens figuræ, BVO, quæ eſt, XN, eſt
              <lb/>
            parallela ipſi, SP, primæ tangenti figuræ, DTF, & </s>
            <s xml:id="echoid-s1002" xml:space="preserve">ſecundatangens
              <lb/>
            figuræ, BVO, eſt pariter parallela ſecundæ tangenti figuræ, DTF, nam
              <lb/>
            tum primæ, tum ſecundæ tangentes ſunt communes ſectiones æquidiſtan-
              <lb/>
            tium planorum, ipſarum nempè figurarum, BVO, DTF, productorum
              <lb/>
            planorum, & </s>
            <s xml:id="echoid-s1003" xml:space="preserve">ideò ſunt parallelæ, & </s>
            <s xml:id="echoid-s1004" xml:space="preserve">angulos continent æquales, vnde
              <lb/>
              <note position="right" xlink:label="note-0057-02" xlink:href="note-0057-02a" xml:space="preserve">_10. Vnde-_
                <lb/>
              _cimi El._</note>
            in figuris, quæ à planis baſi conici parallelis producuntur, ſi babeamus
              <lb/>
            bomologas cum àuabus quibuſdam regulis, eaſdem etiam babebimus cum
              <lb/>
            duabus quibaſuis alijs angulos æquales cum prædictis ad eandem partem
              <lb/>
            continentibus.</s>
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        <div xml:id="echoid-div122" type="section" level="1" n="85">
          <head xml:id="echoid-head96" xml:space="preserve">COROLLARIVM II.</head>
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            <s xml:id="echoid-s1006" xml:space="preserve">_P_Atet in ſuper ex bac, & </s>
            <s xml:id="echoid-s1007" xml:space="preserve">11. </s>
            <s xml:id="echoid-s1008" xml:space="preserve">ac 12. </s>
            <s xml:id="echoid-s1009" xml:space="preserve">huius ſimilium planarum figu-
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            rarum, quæex ſectione planorum baſi cylindrici, vel conici æqui-
              <lb/>
            diſtantium in illis producuntur, vel ſunt oppoſitæ baſes cylindrici, aut
              <lb/>
            fruſti conici, poſſibile eſſe inuenire incidentes, quæ ſint & </s>
            <s xml:id="echoid-s1010" xml:space="preserve">ductarum vt-
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            cumq; </s>
            <s xml:id="echoid-s1011" xml:space="preserve">oppoſitarum earundem tangentium incidentes, & </s>
            <s xml:id="echoid-s1012" xml:space="preserve">quia punctum,
              <lb/>
            H, ſumptum eſt vtcumque, & </s>
            <s xml:id="echoid-s1013" xml:space="preserve">ab ipſo ducta quælibet incidens, HP, pa-
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            tet, quod, ducta vtcumque in dictis figuris incidente earum </s>
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