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

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        <div xml:id="echoid-div1186" type="section" level="1" n="709">
          <p>
            <s xml:id="echoid-s13351" xml:space="preserve">
              <pb o="516" file="0536" n="536" rhead="GEOMETRIÆ"/>
            Elem. </s>
            <s xml:id="echoid-s13352" xml:space="preserve">à prima illius def. </s>
            <s xml:id="echoid-s13353" xml:space="preserve">dependentes, animaduertet ſuam ſortiri
              <lb/>
            veritatem ſiue ſecundum hanc, ſiue ſecundum adductam defini-
              <lb/>
            tionem intelligantur; </s>
            <s xml:id="echoid-s13354" xml:space="preserve">conſimilem autem demonſtrationum ſeriẽ
              <lb/>
            exſuperioribus definitionibus emanantem, inferius & </s>
            <s xml:id="echoid-s13355" xml:space="preserve">ipſæ ſubiun-
              <lb/>
            gam.</s>
            <s xml:id="echoid-s13356" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1188" type="section" level="1" n="710">
          <head xml:id="echoid-head743" xml:space="preserve">THEOREMA XII. PROPOS. XII.</head>
          <p>
            <s xml:id="echoid-s13357" xml:space="preserve">PRopoſito quocunq; </s>
            <s xml:id="echoid-s13358" xml:space="preserve">ſolido rectangulo iuxta datas re-
              <lb/>
            gulas, ac ſub duabus quibuſdam ſuperficiebus con-
              <lb/>
            tento; </s>
            <s xml:id="echoid-s13359" xml:space="preserve">indefinita numero ſolida rectangula pariter dari
              <lb/>
            poſſunt, iuxta eaſdem regulas, quorum vnumquodq; </s>
            <s xml:id="echoid-s13360" xml:space="preserve">pro-
              <lb/>
            poſito ſolido æquale erit, ac ſub eiſdem ſuperficiebus
              <lb/>
            continebitur.</s>
            <s xml:id="echoid-s13361" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13362" xml:space="preserve">Sit propoſitum quodcunq; </s>
            <s xml:id="echoid-s13363" xml:space="preserve">ſolidum rectangulum, POIS, ſub
              <lb/>
            duabus ſuperficiebus, QSIB, OBIH, contentum, cuius regulæ
              <lb/>
            ſint, HI, IS. </s>
            <s xml:id="echoid-s13364" xml:space="preserve">Dico indefinita numero ſolida rectangula regulis
              <lb/>
            eiſdem pariter dari poſſe, quorum vnumquodq; </s>
            <s xml:id="echoid-s13365" xml:space="preserve">ipſi, POIS, æqua-
              <lb/>
            le erit, ac ſub eiſdem ſuperficiebus, QSIB, OBIH, continebitur.
              <lb/>
            </s>
            <s xml:id="echoid-s13366" xml:space="preserve">Igitur rectangulum ſolidum, POIS, ſuperficiebus cylindraceis cõ-
              <lb/>
            prehendetur, illæ ergo ſuperficies indefinitè hincinde produci in-
              <lb/>
            telligantur, in quibus latera fignata per plana parallela, in ſolido
              <lb/>
            parallelogramma rectangula gignentia, vni regulæ, vt ipſi, HI, æ-
              <lb/>
            quidiſtant, tales autem ſunt ſuperficies, PS, SH, HB, BP, ſicut
              <lb/>
              <note position="left" xlink:label="note-0536-01" xlink:href="note-0536-01a" xml:space="preserve">11. huius.</note>
            etiam, PH, HS, SB, BP, quarum eſt pariter regula, SI, cum enim,
              <lb/>
            RI, PB, fuerint parallelogramma rectangula, tam iuxta regulam,
              <lb/>
            HI, quam iuxta, SI, poſſunt in ipſis rectę lineæ vni cuidam paral-
              <lb/>
            lelæ deſignari: </s>
            <s xml:id="echoid-s13367" xml:space="preserve">Producatur autem ipſæ, PS, SH, HB, BP, hinc
              <lb/>
            inde inderinitè, intelligaturq; </s>
            <s xml:id="echoid-s13368" xml:space="preserve">ſimiliter in quacunq; </s>
            <s xml:id="echoid-s13369" xml:space="preserve">productarum
              <lb/>
            ſuperficierum, vt in, OI, producta, exiſtere figura quæcunque, ΔΚ.
              <lb/>
            </s>
            <s xml:id="echoid-s13370" xml:space="preserve">Μλ, homologa, iuxta regulam, RI, ipſi, OHIB, in eadem ſuperfi-
              <lb/>
            cie exi@tenti, deinde per illus ambitum, ΔΚΜλ, feratur quædam
              <lb/>
            recta linea indeficitè hinc inde producta, temper ipſi, SI, æquidi-
              <lb/>
            ſtanter, donec omnem illius percurrerit ambitum, gignens ſuper-
              <lb/>
            ficies cylindraceas, CΔΚΝ, NM, GMλD, DΔ, abſcindenſq; </s>
            <s xml:id="echoid-s13371" xml:space="preserve">a fu-
              <lb/>
            perficie, QR, indefinitè producta ſuperficiem cylindraceam, DCN
              <lb/>
            G. </s>
            <s xml:id="echoid-s13372" xml:space="preserve">Eſto igitur, quod vnum parallelorum planorum in ſolido, PI,
              <lb/>
            rectangula plana gignentium, vt, quod genuit, XV, indefinitè pro-
              <lb/>
            ductum, ita vt fecet ſolidum, CM, in eo produxerit figuram, E℟,
              <lb/>
            quoniam ergo, EF, eſt parallela ipſi, & </s>
            <s xml:id="echoid-s13373" xml:space="preserve">℟, nam eſt portio, EY, quę,
              <lb/>
            eſt parallela ipſi, & </s>
            <s xml:id="echoid-s13374" xml:space="preserve">V, ſimiliter, E&</s>
            <s xml:id="echoid-s13375" xml:space="preserve">, eſt parallela ipſi, F℟, erit, </s>
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