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

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            <s xml:id="echoid-s1062" xml:space="preserve">
              <pb o="40" file="0060" n="60" rhead="GEOMETRIÆ"/>
            ſeuntis, vt ipſius, BDFO, quod ſemper eſt trapezium, & </s>
            <s xml:id="echoid-s1063" xml:space="preserve">ipſarum, V
              <lb/>
            BO, TDF, ſiue eiſdem æquidiſtantium inter eaſdem ductarum, eſſe ea-
              <lb/>
            rundem lineas, vel latera homologa, vnde patet communes ſectiones
              <lb/>
            planiper latera fruſti conici ducti, & </s>
            <s xml:id="echoid-s1064" xml:space="preserve">eiuſdem baſium oppoſitarum, ſiue
              <lb/>
            eiſdem æquidiſtantium inter eas productarum figurarum, eſſe earundem
              <lb/>
            lineas, vel latera homologa; </s>
            <s xml:id="echoid-s1065" xml:space="preserve">lineas, inquam, cum ſunt intra figuras,
              <lb/>
            nec ſumuntur in plano tangente: </s>
            <s xml:id="echoid-s1066" xml:space="preserve">latera, cum ſunt in earum circuitu,
              <lb/>
            cum nempè ſunt in eodem plano tangente, in eo præcisè, quod eſt pla-
              <lb/>
            num contactus fruſti conici (contactus ſcilicet cius plani, quod per ver-
              <lb/>
            ticem ducitur) quod ſemper erit trapezium, vel trapezia, vt patere po-
              <lb/>
            teſt in trapezijs, BDCR, IEFO, quæ eſſent planum contactus fruſti
              <lb/>
            conici, ſiidem fruſtum tangeretur à plano trianguli, ADF.</s>
            <s xml:id="echoid-s1067" xml:space="preserve"/>
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        <div xml:id="echoid-div128" type="section" level="1" n="89">
          <head xml:id="echoid-head100" xml:space="preserve">THEOREMA XIX. PROPOS. XXII.</head>
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            <s xml:id="echoid-s1068" xml:space="preserve">SI duæ figuræ planę ſimiles, non exiſtentes in eodem pla-
              <lb/>
            no, fuerint inæquales, & </s>
            <s xml:id="echoid-s1069" xml:space="preserve">ſimiliter poſitæ; </s>
            <s xml:id="echoid-s1070" xml:space="preserve">erunt cuiu-
              <lb/>
            ſdam fruſticonici oppoſitæ baſes.</s>
            <s xml:id="echoid-s1071" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1072" xml:space="preserve">Vtamuradhuc figura Propoſ. </s>
            <s xml:id="echoid-s1073" xml:space="preserve">19. </s>
            <s xml:id="echoid-s1074" xml:space="preserve">& </s>
            <s xml:id="echoid-s1075" xml:space="preserve">ſint duæ figuræ planæ quæ-
              <lb/>
            cumque ſimiles, inæquales, & </s>
            <s xml:id="echoid-s1076" xml:space="preserve">ſimiliter poſitę, non tamen exiſten-
              <lb/>
            tesin eodem plano, ipſæ, VBO, TDF. </s>
            <s xml:id="echoid-s1077" xml:space="preserve">Dico, quod erunt am-
              <lb/>
            bæ cuiuſdam fruſti conici oppoſitę baſes. </s>
            <s xml:id="echoid-s1078" xml:space="preserve">Quoniam ergo figure, V
              <lb/>
            BO, TDF, ſunt ſimiliter poſitæ, & </s>
            <s xml:id="echoid-s1079" xml:space="preserve">non in eodem plano, erunt in
              <lb/>
              <note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">D.Def.0.
                <lb/>
              huius.</note>
            planis ęquidiſtantibus, & </s>
            <s xml:id="echoid-s1080" xml:space="preserve">quia ſunt ſimiles ſint earum incldentes, & </s>
            <s xml:id="echoid-s1081" xml:space="preserve">
              <lb/>
            oppoſitarum tangentium, quæ ſunt earundem homologarum regu-
              <lb/>
            læ, ipſæ, KN, HP; </s>
            <s xml:id="echoid-s1082" xml:space="preserve">KN, ipſius, VBO, &</s>
            <s xml:id="echoid-s1083" xml:space="preserve">, HP, ipſius, TDF,
              <lb/>
            & </s>
            <s xml:id="echoid-s1084" xml:space="preserve">prædictæ tangentes figuræ, VBO, ſint ipſæ, VK, XN, & </s>
            <s xml:id="echoid-s1085" xml:space="preserve">fi-
              <lb/>
            guræ, TDF, ipſæ, TH, SP, erunt ergo ipſæ, KN, HP, æqui-
              <lb/>
            diſtantes, & </s>
            <s xml:id="echoid-s1086" xml:space="preserve">quia ad tangentes, quæ ſunt regulæ homologarum, illę
              <lb/>
              <note position="left" xlink:label="note-0060-02" xlink:href="note-0060-02a" xml:space="preserve">Conuerla
                <lb/>
              10. Vnde-
                <lb/>
              cimi El</note>
            efficiunt ad eandem partem angulos æquales, erit angulus, KNX,
              <lb/>
            æqualis angulo, HPS, & </s>
            <s xml:id="echoid-s1087" xml:space="preserve">quia, KN, eſt parallela ipſi, HP, erit
              <lb/>
            etiam, XN, parallela ipſi, SP. </s>
            <s xml:id="echoid-s1088" xml:space="preserve">Eodem pacto oſtendemus, VK,
              <lb/>
            eſſe parallelam ipſi, TH; </s>
            <s xml:id="echoid-s1089" xml:space="preserve">ducantur in figuris, VBO, TDF, duæ
              <lb/>
            earum homologæ regulis dictis tang entibus, quæ ſint ipſæ, BR, I
              <lb/>
            O, DC, EF, ſint autem totæ, BO, DF, productæ, ſi opus ſit, vt
              <lb/>
            ſecent ipſas, KN, HP, quas diuident ſimiliter ad eandem partem,
              <lb/>
            vt in punctis, M, G, & </s>
            <s xml:id="echoid-s1090" xml:space="preserve">quia figuræ propoſitæ ſunt inæquales, ſit
              <lb/>
            maior ipſa, TDF, igitur etiam maior erit, DC, ipſa, BR, vel, E
              <lb/>
            F, ipſa, IO, ſi, n. </s>
            <s xml:id="echoid-s1091" xml:space="preserve">eſſent eiſdem æquales, etiam reliquæ homologæ
              <lb/>
            his parallelæ eſſent ęquales, cum omnes ſint proportionales (ſunt.</s>
            <s xml:id="echoid-s1092" xml:space="preserve">n.</s>
            <s xml:id="echoid-s1093" xml:space="preserve"/>
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