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

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        <div xml:id="echoid-div1078" type="section" level="1" n="646">
          <p style="it">
            <s xml:id="echoid-s11817" xml:space="preserve">
              <pb o="462" file="0482" n="482" rhead="GEOMETRIÆ"/>
            & </s>
            <s xml:id="echoid-s11818" xml:space="preserve">ſecunda, & </s>
            <s xml:id="echoid-s11819" xml:space="preserve">ſub compoſito ex prima, ſecunda, & </s>
            <s xml:id="echoid-s11820" xml:space="preserve">tertia parte,
              <lb/>
            vna cum tertia parte quadrati tertiæ partis, & </s>
            <s xml:id="echoid-s11821" xml:space="preserve">ſic trilinea deinceps
              <lb/>
            ſequentia eſſe, vt hæc rectangula deinceps ſequentia cum tertia parte
              <lb/>
            dictorum quadratorum, eodem enim modo ſupra adhibito hoc oſtende-
              <lb/>
            tur. </s>
            <s xml:id="echoid-s11822" xml:space="preserve">Quotieſcunq; </s>
            <s xml:id="echoid-s11823" xml:space="preserve">autem tangens ſit æqualis radio circuli ſpiralium
              <lb/>
            alicuius numeri veluti fuit, EZ, æqualis ipſi, AB, & </s>
            <s xml:id="echoid-s11824" xml:space="preserve">diuidatur in tot
              <lb/>
            partes æquales, in quot radius talis circuli diuiditur à circumferen-
              <lb/>
            tijs infertorum circulorum, tunc nedum in parabola dicta ſpatia ſe
              <lb/>
            habent, vt dictum eſt, ſed etiam ſunt æqualia ſpatijs dictorum circu-
              <lb/>
            lorum, primum nempè primo, ſecundum ſecundo, & </s>
            <s xml:id="echoid-s11825" xml:space="preserve">ſic deinceps, à
              <lb/>
            puncto contactus parabolæ dictorum ſpatiorum enumeratione facta,
              <lb/>
            quod eſt admirabile, hęc autem ex ſupradictis manifeſta ſunt.</s>
            <s xml:id="echoid-s11826" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1079" type="section" level="1" n="647">
          <head xml:id="echoid-head677" xml:space="preserve">THEOREMA XX. PROPOS. XX.</head>
          <p>
            <s xml:id="echoid-s11827" xml:space="preserve">SI parabolam tetigerit recta linea, quæ diuidatur in
              <lb/>
            quotcunq; </s>
            <s xml:id="echoid-s11828" xml:space="preserve">partes æquales, per puncta autem diuiſio-
              <lb/>
            num, & </s>
            <s xml:id="echoid-s11829" xml:space="preserve">extremum ducantur rectæ lineæ diametro para-
              <lb/>
            bolæ, æquidiſtautes, quouſq; </s>
            <s xml:id="echoid-s11830" xml:space="preserve">in eiuſdem curuam incidant,
              <lb/>
            iungantur autem puncta incidentiæ cum puncto cõtactus.
              <lb/>
            </s>
            <s xml:id="echoid-s11831" xml:space="preserve">Spatium ſub prima iungente, & </s>
            <s xml:id="echoid-s11832" xml:space="preserve">ſubtenſa ab eadem curua
              <lb/>
            erit ſeptima pars ſpatij ſub prima, & </s>
            <s xml:id="echoid-s11833" xml:space="preserve">ſecunda iungente, & </s>
            <s xml:id="echoid-s11834" xml:space="preserve">
              <lb/>
            ab ijs appræhenſa curua compræhenſi. </s>
            <s xml:id="echoid-s11835" xml:space="preserve">Hoc verò ad ſpa-
              <lb/>
            tium ſub ſecunda, & </s>
            <s xml:id="echoid-s11836" xml:space="preserve">tertia iungente, & </s>
            <s xml:id="echoid-s11837" xml:space="preserve">appræhenſa curua,
              <lb/>
            erit vt 7. </s>
            <s xml:id="echoid-s11838" xml:space="preserve">ad 19 Hoc autem ad ſpatium ſub tertia, & </s>
            <s xml:id="echoid-s11839" xml:space="preserve">quar-
              <lb/>
            ta iungente & </s>
            <s xml:id="echoid-s11840" xml:space="preserve">ab ijs incluſa curua, vt 19. </s>
            <s xml:id="echoid-s11841" xml:space="preserve">ad 37. </s>
            <s xml:id="echoid-s11842" xml:space="preserve">& </s>
            <s xml:id="echoid-s11843" xml:space="preserve">ſic de-
              <lb/>
            i@ceps, prout indicat appoſita numerorum ſeries.</s>
            <s xml:id="echoid-s11844" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11845" xml:space="preserve">Sit tangens parabolam, AHF, ipſa, AE, diuiſa in quotcumq;
              <lb/>
            </s>
            <s xml:id="echoid-s11846" xml:space="preserve">partes æquales, AB, BC, CD, DE, ductis autem à punctis, B, C,
              <lb/>
            D, E, diametro parallelis, quouſq; </s>
            <s xml:id="echoid-s11847" xml:space="preserve">incidant curuæ, AHF, ipſæ, B
              <lb/>
            N, CM, DH, EF, iungantur puncta incidèntiæ, quæ ſint, F, H, M,
              <lb/>
            N, cum puncto, A, &</s>
            <s xml:id="echoid-s11848" xml:space="preserve">, AN, dicatur prima iungens, AM, tecunda,
              <lb/>
            AH, tertia, & </s>
            <s xml:id="echoid-s11849" xml:space="preserve">ſic deinceps. </s>
            <s xml:id="echoid-s11850" xml:space="preserve">Dico ipatium ſub, AN, & </s>
            <s xml:id="echoid-s11851" xml:space="preserve">ab ea ſub-
              <lb/>
            tenſa curua, eſſe ad ſpatium ſub, NA, AM, & </s>
            <s xml:id="echoid-s11852" xml:space="preserve">curua, MN, ideſt
              <lb/>
            ad trilineum, AMN, vt 1. </s>
            <s xml:id="echoid-s11853" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s11854" xml:space="preserve">hoc verò ad trilineum, AHM, vt
              <lb/>
            7. </s>
            <s xml:id="echoid-s11855" xml:space="preserve">ad 19. </s>
            <s xml:id="echoid-s11856" xml:space="preserve">& </s>
            <s xml:id="echoid-s11857" xml:space="preserve">ſic deinceps, prout indicat oppoſita numerorum ſeries
              <lb/>
              <note position="left" xlink:label="note-0482-01" xlink:href="note-0482-01a" xml:space="preserve">Ex Coro.
                <lb/>
              antec.</note>
            ſe habere trilinea deinceps ſubſequentia. </s>
            <s xml:id="echoid-s11858" xml:space="preserve">Eſt enim ſpatium, AIN,
              <lb/>
            ad trilineum, AMN, vt {1/3}. </s>
            <s xml:id="echoid-s11859" xml:space="preserve">quadrati, AB, ad rectangulum, CAB,</s>
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