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

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            <s xml:id="echoid-s815" xml:space="preserve">
              <pb o="29" file="0049" n="49" rhead="LIBER I."/>
            milium. </s>
            <s xml:id="echoid-s816" xml:space="preserve">Ducantur plana oppoſita tangentia cylindrici, AM, re-
              <lb/>
              <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">Coroll. 1.
                <lb/>
              huius.</note>
            ſpectu plani, BF, in eo ducti, vnius quorum, & </s>
            <s xml:id="echoid-s817" xml:space="preserve">planorum figura-
              <lb/>
            rum, YNO, TDF, productorum, communes ſectiones ſint, XS,
              <lb/>
            MG, alterius autem, & </s>
            <s xml:id="echoid-s818" xml:space="preserve">eorundem planorum ſint rectæ, YP, TQ,
              <lb/>
            indefinitè ambæ productæ, ſumpto autem in, YP, vtcumque pun-
              <lb/>
            cto, P, ducatur per, P, ipſi, CF, æquidiſtans, PQ, & </s>
            <s xml:id="echoid-s819" xml:space="preserve">ab eodem
              <lb/>
            in plano per, YP, XS, tranſeunte vſquead, XS, ducatur vtcum-
              <lb/>
            queipſa, PS, per ipſas autem, QP, PS, intelligatur extenſum pla-
              <lb/>
            num, quod ſecetaliud tangens planum in, SG, & </s>
            <s xml:id="echoid-s820" xml:space="preserve">planum per, T
              <lb/>
              <figure xlink:label="fig-0049-01" xlink:href="fig-0049-01a" number="22">
                <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/05TCTFNR/figures/0049-01"/>
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            Q, MG, ductum in, QG, producan-
              <lb/>
            tur autem ipſæ, NO, DF, verſus, P
              <lb/>
            S, QG, quibus occurrant in, V, R,
              <lb/>
            & </s>
            <s xml:id="echoid-s821" xml:space="preserve">iungatur, VR, erunt igitur, VR,
              <lb/>
            PQ, communes ſectiones æquidiſtan-
              <lb/>
            tium planorum, YQ, NR, & </s>
            <s xml:id="echoid-s822" xml:space="preserve">plani,
              <lb/>
            PR, & </s>
            <s xml:id="echoid-s823" xml:space="preserve">ideò erunt parallelæ, vt & </s>
            <s xml:id="echoid-s824" xml:space="preserve">ip-
              <lb/>
            ſæ, PV, QR, &</s>
            <s xml:id="echoid-s825" xml:space="preserve">, PR, erit paralle-
              <lb/>
            logrammum: </s>
            <s xml:id="echoid-s826" xml:space="preserve">Similiter, vt in Prop. </s>
            <s xml:id="echoid-s827" xml:space="preserve">11.
              <lb/>
            </s>
            <s xml:id="echoid-s828" xml:space="preserve">oſtendemus eſſe parallelogramma ip-
              <lb/>
            ſa, VG, PG, NF, OR, NR, & </s>
            <s xml:id="echoid-s829" xml:space="preserve">an-
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            gulum, PSX, æqualem eſſe angulo,
              <lb/>
            QGM, & </s>
            <s xml:id="echoid-s830" xml:space="preserve">tandem, PS, QG, eſſe in-
              <lb/>
              <note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">B. Def. 10.
                <lb/>
              huius.</note>
            cidentes ſimilium figurarum, YNO,
              <lb/>
            TDF, & </s>
            <s xml:id="echoid-s831" xml:space="preserve">oppofitarum tangentium, YP, XS, TQ, MG, & </s>
            <s xml:id="echoid-s832" xml:space="preserve">tan-
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            gentes eſſe homologarum earundem regulas, & </s>
            <s xml:id="echoid-s833" xml:space="preserve">quia eiſdem æqui-
              <lb/>
            diſtant ipſæ, NO, DF, & </s>
            <s xml:id="echoid-s834" xml:space="preserve">productæ ſimiliter, & </s>
            <s xml:id="echoid-s835" xml:space="preserve">ad eandem par-
              <lb/>
            tem ipſas incidentes, PS, QG, diuidunt; </s>
            <s xml:id="echoid-s836" xml:space="preserve">nam, PV, æquatur ipſi,
              <lb/>
            QR, &</s>
            <s xml:id="echoid-s837" xml:space="preserve">, VS, ipſi, RG, ideò ipſæ, NO, DF, erunt lineæ homo-
              <lb/>
            logæ figurarum, YNO, TDF, ſimilium, quæ in plures homolo-
              <lb/>
            gas ſecari contingere poteſt, prout ſe habet ambitus ſuperficiei cy-
              <lb/>
            lindraceæ huius cylindrici, AM, ſunt lineæ homologæ inquam, ſi
              <lb/>
              <note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">C. Def. 10.
                <lb/>
              huius.</note>
            ſint intra ambitum figurarum, quarum ſunt homologæ, ſunt verò
              <lb/>
            latera homologa, ſi ſint in earundem ambitu, veluti contingeret ſi
              <lb/>
            planum per latera ductum eſſet planum contactus vnius oppoſito-
              <lb/>
            rum tangentium, veluti ſi cylindricus fuiſſet, cuius oppoſitæ baſes
              <lb/>
            ſunt, ABC, TDF, excluſis reſiduis figuris, quæ ab ipſis, BC, D
              <lb/>
            F, abſcinduntur, tunc enim eodem modo facta fuiſſet demonſtra-
              <lb/>
            tio, vt conſideranti facilè patebit; </s>
            <s xml:id="echoid-s838" xml:space="preserve">idem oſtendemus in recta, BC,
              <lb/>
            & </s>
            <s xml:id="echoid-s839" xml:space="preserve">in quibuſuis alijs, quæ ſunt communes ſectiones planorum baſi-
              <lb/>
            bus æquidiſtantium, & </s>
            <s xml:id="echoid-s840" xml:space="preserve">parallelogrammi, BF, probantes ſcilicet
              <lb/>
            eaſdem eſſe lineas, vellatera homologa figurarum in cylindrico per
              <lb/>
            baſibus æquidiſtantia plana productarum, quod oſtendere opus erat.</s>
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