Apollonius <Pergaeus>, Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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        <div xml:id="echoid-div63" type="section" level="1" n="42">
          <head xml:id="echoid-head67" xml:space="preserve">LEMMA I.</head>
          <p style="it">
            <s xml:id="echoid-s1083" xml:space="preserve">Si quatuor quantitates eandem proportionem habuerint, antecedentes,
              <lb/>
            vel cońſequentes ad terminorum ſummas, vel differentias in eadem ra-
              <lb/>
            tione erunt; </s>
            <s xml:id="echoid-s1084" xml:space="preserve">& </s>
            <s xml:id="echoid-s1085" xml:space="preserve">è contra.</s>
            <s xml:id="echoid-s1086" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1087" xml:space="preserve">HAbeat A B ad B C eandem proportionem, quàm D E ad E H: </s>
            <s xml:id="echoid-s1088" xml:space="preserve">ſequitur pri-
              <lb/>
            mò, quod A C ad C B ſit, vt D H ad H E; </s>
            <s xml:id="echoid-s1089" xml:space="preserve">& </s>
            <s xml:id="echoid-s1090" xml:space="preserve">huiuſmodi argumentatio
              <lb/>
            vocatur in elementis compoſitio terminorum proportionis: </s>
            <s xml:id="echoid-s1091" xml:space="preserve">itaque ſummæ antece-
              <lb/>
            dentium, & </s>
            <s xml:id="echoid-s1092" xml:space="preserve">conſequentium ad eaſdem conſequentes ſunt etiam proportionales:
              <lb/>
            </s>
            <s xml:id="echoid-s1093" xml:space="preserve">ſi vero ex eadem hypotbeſi concludai
              <unsure/>
            ur, quod A C ad A B, ſit vt D H ad D E,
              <lb/>
            vt nimirum ſummæ terminorum proportionis ad antecedentes ſint proportiona-
              <lb/>
            les: </s>
            <s xml:id="echoid-s1094" xml:space="preserve">quod quidem manifeſtum eſt, nam poſita fuit A B ad B C, vt D E ad E H; </s>
            <s xml:id="echoid-s1095" xml:space="preserve">
              <lb/>
            erit inuertendo C B ad B A, vt H E ad E D, & </s>
            <s xml:id="echoid-s1096" xml:space="preserve">componendo C A ad A B erit
              <lb/>
            vt H D ad D E: </s>
            <s xml:id="echoid-s1097" xml:space="preserve">modo huiuſmodi argumentandi forma innominata eſt; </s>
            <s xml:id="echoid-s1098" xml:space="preserve">poteſt
              <lb/>
            autem breuitatis gratia appellari, Per comparationem ſummæ terminorum ad
              <lb/>
            antecedentes.</s>
            <s xml:id="echoid-s1099" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1100" xml:space="preserve">Secundò concludi poteſt, quod A B ad A
              <lb/>
            C ſit vt D E ad D H; </s>
            <s xml:id="echoid-s1101" xml:space="preserve">quia, vt in prima
              <lb/>
              <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a" number="20">
                <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0051-01"/>
              </figure>
            parte dictum eſt, A C ad A B erat vt D H
              <lb/>
            ad D E, ergo inuertendo A B ad A C erit
              <lb/>
            vt D E ad D H: </s>
            <s xml:id="echoid-s1102" xml:space="preserve">hæc argumentandi forma
              <lb/>
            vocari poteſt, Per comparationem antece-
              <lb/>
            dentium ad terminorum ſummas.</s>
            <s xml:id="echoid-s1103" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1104" xml:space="preserve">Tertiò concludi poteſt: </s>
            <s xml:id="echoid-s1105" xml:space="preserve">quod B C ad C A, ſit vt E H ad H D; </s>
            <s xml:id="echoid-s1106" xml:space="preserve">nam componen-
              <lb/>
            do A C ad C B, erat vt D H ad H E, quare inuertendo B C ad C A erit vt E
              <lb/>
            H ad H D, & </s>
            <s xml:id="echoid-s1107" xml:space="preserve">hæc argumentatio fieri dicetur comparando conſequentes ad ter-
              <lb/>
            minorum ſummas.</s>
            <s xml:id="echoid-s1108" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1109" xml:space="preserve">Deindè ſint eædem quatuor proportiona-
              <lb/>
            les in ſecunda figura, nimirum totum A B
              <lb/>
              <figure xlink:label="fig-0051-02" xlink:href="fig-0051-02a" number="21">
                <image file="0051-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0051-02"/>
              </figure>
            ad ſegmentum eius B C ſit vt totum D E
              <lb/>
            ad portionem eius E H; </s>
            <s xml:id="echoid-s1110" xml:space="preserve">tunc reſiduum A C
              <lb/>
            ad C B erit, vt reſiduum D H ad H E; </s>
            <s xml:id="echoid-s1111" xml:space="preserve">hæc
              <lb/>
            argumentatio ſieri dicitur in elementis, di-
              <lb/>
            uidendo terminos proportionis, eſtque comparatio differentiarum terminorum ad
              <lb/>
            conſequentes.</s>
            <s xml:id="echoid-s1112" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1113" xml:space="preserve">At ſi concludatur ex eadem hypotbeſi quod A B ad A C ſit vt D E ad D H;
              <lb/>
            </s>
            <s xml:id="echoid-s1114" xml:space="preserve">hæc argumentatio in elementis fieri dicitur per conuerſionem rationis eſtque
              <lb/>
            comparatio antecedentium ad differentias terminorum.</s>
            <s xml:id="echoid-s1115" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1116" xml:space="preserve">Poſtea ex eadem hypotbeſi ſequitur quod A C ad A B ſit vt D H ad D E: </s>
            <s xml:id="echoid-s1117" xml:space="preserve">quia
              <lb/>
            per conuerſionem rationis, ſeu referendo antecedentes ad differentias terminorum
              <lb/>
            eſt A B ad A C, vt D E ad D H; </s>
            <s xml:id="echoid-s1118" xml:space="preserve">ergo inuertendo A C ad A B erit vt D H ad
              <lb/>
            D E, & </s>
            <s xml:id="echoid-s1119" xml:space="preserve">hæc argumentatio innominata fiet comparando differentias terminorum
              <lb/>
            ad antecedentes.</s>
            <s xml:id="echoid-s1120" xml:space="preserve"/>
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