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

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              <pb o="91" file="0111" n="111" rhead="LIBER I."/>
            tantum ſui parte, velin vno puncto tantum, prędictas figuras, tan-
              <lb/>
            gant in punctis tantum, P, R, & </s>
            <s xml:id="echoid-s2248" xml:space="preserve">ab ipſis ducantur parallelę regulis,
              <lb/>
            dO, gY, ipſæ, PN, RT, occurrentes incidentibus, LO, uY, in
              <lb/>
            punctis, N, T, dico, LN, uY, ſimiliter ad eandem partem ſecari
              <lb/>
            in, N, T, ſi.</s>
            <s xml:id="echoid-s2249" xml:space="preserve">n. </s>
            <s xml:id="echoid-s2250" xml:space="preserve">hoc non ſit, diuidatur, LO, in, M, ſimiliter ad ean-
              <lb/>
            dem partem, acdiuiditur, uY, in, T, & </s>
            <s xml:id="echoid-s2251" xml:space="preserve">per, M, extendatur, MI,
              <lb/>
            parallela, dO, incidentes ambitui figuræ in, I, & </s>
            <s xml:id="echoid-s2252" xml:space="preserve">rurſus ſecetur, u
              <lb/>
            Y, in, V, ſimiliter ad eandem partem, vt ſecatur, LO, in, N, quia
              <lb/>
            ergo, N, eſt intra puncta, M, O, etiam, V, erit inter puncta, T,
              <lb/>
            Y; </s>
            <s xml:id="echoid-s2253" xml:space="preserve">ducatur tandem, VS, parallela, gY, incidens ambitui figurę in,
              <lb/>
            S. </s>
            <s xml:id="echoid-s2254" xml:space="preserve">Quia igitur, MI, non incidit in punctum contactus rectæ, H o,
              <lb/>
            cum figura, erit, MI, maior, NP, eadem ratione oſtendemus, SV,
              <lb/>
            fore maiorem ipſa, RT, eſt enim, RT, minima earum, quæ abin-
              <lb/>
            cidente, uY, ad ambitum figuræ duci poſſunt æquidiſtanter ipſi, g
              <lb/>
            Y. </s>
            <s xml:id="echoid-s2255" xml:space="preserve">Cum verò, IM, RT, ſimiliter diuidant, & </s>
            <s xml:id="echoid-s2256" xml:space="preserve">ad eandem partem
              <lb/>
            ipſas incidentes, LO, uY, erit, IM, ad, RT, vt, LO, ad, uY,
              <lb/>
              <note position="right" xlink:label="note-0111-01" xlink:href="note-0111-01a" xml:space="preserve">Defin. 10.
                <lb/>
              huius.</note>
            ideſt vt, PN, ad, SV, ergo, permutando, IM, ad, PN, erit vt,
              <lb/>
            RT, ad, SV, eſt autem, IM, maior, PN, ergo etiam, RT, erit
              <lb/>
            maior, SV, ſed etiam minor, quod eſt abſurdum, ergo falſum eſt ip-
              <lb/>
            ſas, PN, RT, non ſecare ſimiliter ad eandem partem ipſas, LO,
              <lb/>
            uY, ſic igitur eaſdem diuidunt, eritque, PN, ad, RT, hoc eſt, H
              <lb/>
            L, ad, Xu, vt, LO, ad, uY, idem oſtendemus etiam ſi contactus
              <lb/>
            eſſet in parte linearum, Ho, XZ, ſeu in totis eiſdem lineis, vt conſi-
              <lb/>
            deranti facilè innoteſcet. </s>
            <s xml:id="echoid-s2257" xml:space="preserve">Eadem autem methodo probabimus etiam,
              <lb/>
            DL, pu, eſſe vt ipſas, LO, uY, ergo reſiduæ, DH, pX, hoc eſt,
              <lb/>
            AC, FK, erunt vt, LO, uY, ideſt vt, E4, ℟ &</s>
            <s xml:id="echoid-s2258" xml:space="preserve">, ſed, AC, FK,
              <lb/>
            ſimiliter ſunt diuiſę ab homologis, sl, 7 4, productis, in punctis, B,
              <lb/>
              <note position="right" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">Defin. 10.
                <lb/>
              huius.</note>
            G, ergo, AB, ad, FG, ideſt, DE, ad, p℟, erit vt, AC, ad, F
              <lb/>
            K, ideſt vt, E 4, ad, ℟ &</s>
            <s xml:id="echoid-s2259" xml:space="preserve">. Extendantur, NP, TR, quę diuidunt,
              <lb/>
            LO, uY, ſimiliter ad eandem partem, ſecentque ipſos, E4, ℟ &</s>
            <s xml:id="echoid-s2260" xml:space="preserve">,
              <lb/>
            in punctis, 2, 3, incidat autem, NQ, in, Q, punctum contactus
              <lb/>
            lineæ, Ad, cum figura, oſtendemus, vt factum eſt circa ipſas, NP,
              <lb/>
            TR, etiam, T8, incidere in punctum contactus rectę, p g, cum fi-
              <lb/>
            gura, quod ſit ipſum, 8, quoniam ergo probatum eſt, DE, ad, p
              <lb/>
            ℟, eſſe vt, E4, ad, ℟ &</s>
            <s xml:id="echoid-s2261" xml:space="preserve">, erit etiam, Q2, ad, 83, vt, E4, ad,
              <lb/>
            ℟ &</s>
            <s xml:id="echoid-s2262" xml:space="preserve">. Similiter probabimus, 2P, ad, 3R, eſſe vt, E4, ad, ℟ &</s>
            <s xml:id="echoid-s2263" xml:space="preserve">,
              <lb/>
            & </s>
            <s xml:id="echoid-s2264" xml:space="preserve">diuidunt ipſas, E4, ℟ &</s>
            <s xml:id="echoid-s2265" xml:space="preserve">, ſimiliter ad eandem partem, à quibus
              <lb/>
            viciſſim ſecantur ad eundem angulum ex eadem parte, cum, E4, ℟
              <lb/>
            &</s>
            <s xml:id="echoid-s2266" xml:space="preserve">, ſint parallelæ ipſis, LO, uY, ergo, E4, ℟ &</s>
            <s xml:id="echoid-s2267" xml:space="preserve">, erunt incidentes
              <lb/>
            ſimilium figurarum, PlQs, R487, & </s>
            <s xml:id="echoid-s2268" xml:space="preserve">oppoſitarum tangentium,
              <lb/>
              <note position="right" xlink:label="note-0111-03" xlink:href="note-0111-03a" xml:space="preserve">24, huius.</note>
            DH, do, pX, gZ, quod etiam veriſicaretur de ipſis homologis, ls,
              <lb/>
            47, ſi fuiſſent ad oppoſitas tangentes terminatę in punctis, E, 4, </s>
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