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

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            <s xml:id="echoid-s1683" xml:space="preserve">
              <pb o="65" file="0085" n="85" rhead="LIBER I."/>
            ptis regulis, OS, fp, omnes ęquidiſtare: </s>
            <s xml:id="echoid-s1684" xml:space="preserve">Et quidem ſi plana ſecent
              <lb/>
            figuras, oOS, sfp, hoc manifeſtum eſt, etenim productę lineę ip-
              <lb/>
            ſis baſibus, OS, sp, erunt parallelæ, & </s>
            <s xml:id="echoid-s1685" xml:space="preserve">latera homologa ſimilium
              <lb/>
            figurarum ex traiectis planis in ſolidis productarum. </s>
            <s xml:id="echoid-s1686" xml:space="preserve">Siverò plana
              <lb/>
            parallela ſecent duas figuras ipſis, oOS, sfp, continuatas, vti ſa-
              <lb/>
            ciunt plana figurarum, HMPL, VZdY, quæ etiam ſecant plana
              <lb/>
            figurarum, oOS, sfp, producta in rectis, KN, ug, oſtendemus
              <lb/>
            ipſas, KN, ug, eſſe regulas homologarum ſimilium figurarum, L
              <lb/>
            HMP, VZdY, iunctis, PK, du. </s>
            <s xml:id="echoid-s1687" xml:space="preserve">Quia enim, Oo, fs, ſunt ip-
              <lb/>
            ſorum ſimilium ſolidorum latera homologa, producta, ac terminata
              <lb/>
            ad baſium plana, & </s>
            <s xml:id="echoid-s1688" xml:space="preserve">oppoſitorum tangentium, in punctis, O, B; </s>
            <s xml:id="echoid-s1689" xml:space="preserve">f,
              <lb/>
              <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">Elicitur
                <lb/>
              ex Corol.
                <lb/>
              Lem. 6.</note>
            Q, ideò, BO, Qf, ſunt ſimiliter ad eandem partem ſectæ in, o, s,
              <lb/>
            & </s>
            <s xml:id="echoid-s1690" xml:space="preserve">nedum, Oo, fs, ſed etiam, oB, sQ, ſunt vt eorum altitudines
              <lb/>
            ſumptæ reſpectu dictarum baſium, ſed ſic etiam ſunt ipſæ, oS, sp,
              <lb/>
            latera homologa, ergo, Bo, ad, oS, eſt vt, Qs, ad, sp, & </s>
            <s xml:id="echoid-s1691" xml:space="preserve">angu-
              <lb/>
            los æquales, BoS, Qsp, complectuntur latera proportionalia, er
              <lb/>
              <note position="right" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">6. Sex. El.</note>
            go triangula, BoS, Qsp, ſunt ſimilia, cum verò ſint in planis trian-
              <lb/>
            gulorum, oOS, sfp, ſunt etiam ſimilibus figuris, LPSo, Ydps,
              <lb/>
            ęquè ad eandem partem inclinata, quibus communia ſunt homolo-
              <lb/>
              <note position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">Ex Lem.
                <lb/>
              1.</note>
            galatera, oS, sp, ergo anguli, KoL, usY, interſe, necnon, PS
              <lb/>
              <note position="right" xlink:label="note-0085-04" xlink:href="note-0085-04a" xml:space="preserve">Corollar.
                <lb/>
              Lem. 6.</note>
            I, dpX, æquales erunt; </s>
            <s xml:id="echoid-s1692" xml:space="preserve">cum verò, BS, Qp, ſint vt dictæ altitudi-
              <lb/>
            nes, & </s>
            <s xml:id="echoid-s1693" xml:space="preserve">ſic etiam, IS, Xp, necnon, PS, dp, (etenim, BS, Qp, in,
              <lb/>
              <note position="right" xlink:label="note-0085-05" xlink:href="note-0085-05a" xml:space="preserve">Corol. 26
                <lb/>
              huius.</note>
            I, X, &</s>
            <s xml:id="echoid-s1694" xml:space="preserve">, ES, lp, ſimiliter ſecantur, & </s>
            <s xml:id="echoid-s1695" xml:space="preserve">ad eandem partem, in pun-
              <lb/>
            ctis, P, d,) erit, IS, ad, SP, vt, Xp, ad, pd, & </s>
            <s xml:id="echoid-s1696" xml:space="preserve">circumſtant an-
              <lb/>
              <note position="right" xlink:label="note-0085-06" xlink:href="note-0085-06a" xml:space="preserve">6. Sex. El.</note>
            gulos æquales, ISP, Xpd, ergo triangula, ISP, Xpd, ſunt ſimi-
              <lb/>
            lia. </s>
            <s xml:id="echoid-s1697" xml:space="preserve">Eodem modo oſtendemus ſimilia eſſe triangula, LoK, Y su.
              <lb/>
            </s>
            <s xml:id="echoid-s1698" xml:space="preserve">Vlterius, quia eſt, Ko, ad, oS, vt, us, ad, sp, &</s>
            <s xml:id="echoid-s1699" xml:space="preserve">, oS, ad, SI,
              <lb/>
            vt, sp, ad, pX, & </s>
            <s xml:id="echoid-s1700" xml:space="preserve">anguli, KoS, usp, necnon, oSI, spX, ſunt
              <lb/>
            æquales, ideò trapezia, KoSI, uspX, erunt ſimilia, ſed etiam fi-
              <lb/>
            guræ, LPSo, Ydps, ſunt ſimiles, eſt autem, KL, ad, Lo, vt,
              <lb/>
            uY, ad, Ys, &</s>
            <s xml:id="echoid-s1701" xml:space="preserve">, oL, ad, LP, vt, sY, ad, Yd, ergo, KL, ad, L
              <lb/>
            P, erit vt, uY, ad, Yd, eodem modo autem oſtendemus, LP, PI,
              <lb/>
            IK, KL, binas eſte in eadem proportione cum ipſis, Yd, dX, Xu. </s>
            <s xml:id="echoid-s1702" xml:space="preserve">
              <lb/>
            uY. </s>
            <s xml:id="echoid-s1703" xml:space="preserve">Manifeſtum eſt autem ſi iungeremus, AO, Tf, AS, Tp, quod
              <lb/>
            fierent ſimiles pyramides triangulatæ ipſæ, AOoS, Tfsp, ſimili-
              <lb/>
            bus n. </s>
            <s xml:id="echoid-s1704" xml:space="preserve">triangulis comprehenderentur, vt meditanti compertum fiet,
              <lb/>
              <note position="right" xlink:label="note-0085-07" xlink:href="note-0085-07a" xml:space="preserve">Ex Lem.
                <lb/>
              4.</note>
            ideò plana, AoO, Tsf, ideſt triangula ſimilia, LKo, Yus, ſunt
              <lb/>
              <note position="right" xlink:label="note-0085-08" xlink:href="note-0085-08a" xml:space="preserve">Ex Lem.
                <lb/>
              1.</note>
            æquè ad eandem partem ipſis ſimilibus figuris, LPSo, Ydps, in-
              <lb/>
            clinata, cum quibus coincidunt in lateribus homologis, Lo, Ys,
              <lb/>
              <note position="right" xlink:label="note-0085-09" xlink:href="note-0085-09a" xml:space="preserve">Ex Lem.
                <lb/>
              2.</note>
            ergo anguli, KLP, uYd, erunt æquales, quibus circumſtant latera
              <lb/>
            proportionalia, vt probatum eſt, ergo triangula, KLP, uYd, ſi-
              <lb/>
            milia erunt, & </s>
            <s xml:id="echoid-s1705" xml:space="preserve">erit, KP, ad, PL, vt, ud, ad, dY, eſt verò, </s>
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