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

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              <pb o="35" file="0055" n="55" rhead="LIBERI."/>
            ad verticem, A, duci poſſunt, iacent autem omnes illæ in plano
              <lb/>
            trianguli, cuius baſis eſt linea contactus vertex reſpectu eius, pun-
              <lb/>
            ctus, A, igitur, contactus plani per, AB, DF, ductifit vel in vna,
              <lb/>
            vel pluribus rectis lineis, vel in plano, quod eſt triangulum, ſiue
              <lb/>
            plura triangula, non ſecabit autem alicubi tale planum ipſum coni-
              <lb/>
            cum, tunc enim aliquis punctus talis plani per, AB, DF, tranſeun-
              <lb/>
            tis eſſet intra ſuperficiem conicularem, ſit is punctus, I, iuncta igi-
              <lb/>
            tur, AI, & </s>
            <s xml:id="echoid-s948" xml:space="preserve">producta verſus baſim incidet intra baſim, vt facilè o-
              <lb/>
            ſtendi poteſt, & </s>
            <s xml:id="echoid-s949" xml:space="preserve">quia eſt, AX, in plano per, AB, DF, ducto, & </s>
            <s xml:id="echoid-s950" xml:space="preserve">
              <lb/>
            punctus, X, eſt etiam in plano baſis, erit in communi fectione, ideſt
              <lb/>
            in linea, DF, igitur aliquis punctus rectæ, DF, erit intra baſim,
              <lb/>
            igitur illam ſecabit, quod eſt abſurdum, ergo falſum eſt planum per,
              <lb/>
            A, DF, ductum ſecare alicubi ipſum conicum, igitur illum tanget
              <lb/>
            in his, quæ dicta ſunt, quod oſtendere oportebat.</s>
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          </p>
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        <div xml:id="echoid-div117" type="section" level="1" n="82">
          <head xml:id="echoid-head93" xml:space="preserve">COROLLARIVM.</head>
          <p style="it">
            <s xml:id="echoid-s952" xml:space="preserve">_E_X hoc habetur, ſi conicus ſecetur plano baſi æquidiſtante, commu-
              <lb/>
            nem ſectionem huius, & </s>
            <s xml:id="echoid-s953" xml:space="preserve">plani per verticem, & </s>
            <s xml:id="echoid-s954" xml:space="preserve">tangentem baſim
              <lb/>
            ducti, tangere figuram à plano æquidiſtante baſi in conico productam, ſi
              <lb/>
            enim eam ſecaret, etiam tangens planum ſecaret conicum, quod eſt ab-
              <lb/>
            ſurdum.</s>
            <s xml:id="echoid-s955" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div118" type="section" level="1" n="83">
          <head xml:id="echoid-head94" xml:space="preserve">THEOREMA XVI. PROPOS. XIX.</head>
          <p>
            <s xml:id="echoid-s956" xml:space="preserve">SI conicus planoſecetur baſi æquidiſtante, concepta in
              <lb/>
            eo figura erit ſimilis baſi, & </s>
            <s xml:id="echoid-s957" xml:space="preserve">eidem ſimiliter poſita.</s>
            <s xml:id="echoid-s958" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s959" xml:space="preserve">Sit conicus, cuius vertex, A, baſis, TDF, ſecetur autem plano
              <lb/>
            baſi æquidiſtante, quod in eo producat figura, VBO. </s>
            <s xml:id="echoid-s960" xml:space="preserve">Dico hanc
              <lb/>
            eſſe ſimilem baſi, & </s>
            <s xml:id="echoid-s961" xml:space="preserve">eidem ſimiliter poſitam. </s>
            <s xml:id="echoid-s962" xml:space="preserve">Ducantur ipſius ba-
              <lb/>
            ſis duæ vtcumque oppoſitæ tangentes, quæ ſint, TH, SP, indefi-
              <lb/>
              <note position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">Coroll. 1.
                <lb/>
              huius.</note>
            nitè productæ, deinde per verticem, & </s>
            <s xml:id="echoid-s963" xml:space="preserve">quamlibet dictarum tangen-
              <lb/>
            tium extendatur planum, erunt ergo hęc plana tangentia conicum,
              <lb/>
            ADF, ſecent autem figuræ, VBO, productum planum in rectis,
              <lb/>
              <note position="right" xlink:label="note-0055-02" xlink:href="note-0055-02a" xml:space="preserve">Penãtes. 1</note>
            VK, XN, quæ erunt ipſius figuræ, VBO, oppoſitæ tangentes,
              <lb/>
              <note position="right" xlink:label="note-0055-03" xlink:href="note-0055-03a" xml:space="preserve">Corol. an-
                <lb/>
              teced.</note>
            ſumatur deinde in altera ipſarum, TH, SP, vtin, TH, vtcumq;
              <lb/>
            </s>
            <s xml:id="echoid-s964" xml:space="preserve">punctum, vt, H, à quo verſus reliquam tangentem eiuſdem figurę,
              <lb/>
            TDF, in eiuſdem plano ducatur vtcumque, HP, in, SP, termi-
              <lb/>
            nata, deinde intelligatur extenſum planum per, A, &</s>
            <s xml:id="echoid-s965" xml:space="preserve">, HP, tran-
              <lb/>
            ſiens ita, vtſecet plana conicum tangentia in rectis, AH, AP, &</s>
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