Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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              <pb o="32" file="0075" n="75" rhead="DE IIS QVAE VEH. IN AQVA."/>
            ad ſectionem e f g ex parte e linea l m, eidem a c baſi æquidi-
              <lb/>
            stans. </s>
            <s xml:id="echoid-s1860" xml:space="preserve">Sit autem ſectionis a b c, linea b n iuxta quam poſſunt, quæ
              <lb/>
            à ſectione ducuntur: </s>
            <s xml:id="echoid-s1861" xml:space="preserve">& </s>
            <s xml:id="echoid-s1862" xml:space="preserve">ſectionis e f c ſit ipſa f o. </s>
            <s xml:id="echoid-s1863" xml:space="preserve">quoniam igi-
              <lb/>
            tur triangula c d b, c f g ſimilia ſunt, erit ut b c ad c f, ita d c
              <lb/>
              <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">4. ſexti.</note>
            ad c g; </s>
            <s xml:id="echoid-s1864" xml:space="preserve">& </s>
            <s xml:id="echoid-s1865" xml:space="preserve">b d ad f g. </s>
            <s xml:id="echoid-s1866" xml:space="preserve">rurſus quoniam triangula c k b, c l f etiã
              <lb/>
            inter ſe ſunt ſimilia, ut b c ad c f, boc eſt ut b d ad f g, ita erit k c
              <lb/>
            ad c l; </s>
            <s xml:id="echoid-s1867" xml:space="preserve">& </s>
            <s xml:id="echoid-s1868" xml:space="preserve">b K ad f l. </s>
            <s xml:id="echoid-s1869" xml:space="preserve">quare K c ad c l, & </s>
            <s xml:id="echoid-s1870" xml:space="preserve">b k ad f l ſunt ut d c
              <lb/>
            ad c g: </s>
            <s xml:id="echoid-s1871" xml:space="preserve">hoc eſt ut earum duplæ a c ad c e. </s>
            <s xml:id="echoid-s1872" xml:space="preserve">ſed ut b d ad f g, ita d c
              <lb/>
              <note position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">15. quin-
                <lb/>
              ti.</note>
            ad c g; </s>
            <s xml:id="echoid-s1873" xml:space="preserve">hoc ẽ a d ad e g: </s>
            <s xml:id="echoid-s1874" xml:space="preserve">& </s>
            <s xml:id="echoid-s1875" xml:space="preserve">permutãdo ut b d ad a d, ita f g ad e g.
              <lb/>
            </s>
            <s xml:id="echoid-s1876" xml:space="preserve">quadratum autem a d æquale eſt rectangulo d b n ex undecima pri
              <lb/>
            mi conicorum. </s>
            <s xml:id="echoid-s1877" xml:space="preserve">ergo tres lineæ b d, a d, b n inter ſe ſunt proportio
              <lb/>
              <note position="right" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">17. ſexti.</note>
            nales. </s>
            <s xml:id="echoid-s1878" xml:space="preserve">eadem quoque ratione cum quadratum e g æquale ſit rectan
              <lb/>
            gulo g f o, tres aliæ lineæ f g, e g, f o, deinceps proportionales
              <lb/>
            erũt. </s>
            <s xml:id="echoid-s1879" xml:space="preserve">& </s>
            <s xml:id="echoid-s1880" xml:space="preserve">ut b d ad, a d, ita f g ad e g. </s>
            <s xml:id="echoid-s1881" xml:space="preserve">quare ut a d ad b n, ita e g
              <lb/>
            ad f o. </s>
            <s xml:id="echoid-s1882" xml:space="preserve">ex æquali igitur, ut d b ad b n, ita g f ad f o: </s>
            <s xml:id="echoid-s1883" xml:space="preserve">& </s>
            <s xml:id="echoid-s1884" xml:space="preserve">permu-
              <lb/>
            tando ut d b ad g f, ita b n ad f o. </s>
            <s xml:id="echoid-s1885" xml:space="preserve">ut autem d b ad g f, ita b k
              <lb/>
            ad f l. </s>
            <s xml:id="echoid-s1886" xml:space="preserve">ergo b k ad f l, ut b n ad f o: </s>
            <s xml:id="echoid-s1887" xml:space="preserve">& </s>
            <s xml:id="echoid-s1888" xml:space="preserve">permutando, ut b k ad
              <lb/>
            bn, ita f l ad f o. </s>
            <s xml:id="echoid-s1889" xml:space="preserve">Rurſus quoniá quadratú h K æquale eſt rectan
              <lb/>
              <note position="right" xlink:label="note-0075-04" xlink:href="note-0075-04a" xml:space="preserve">11. primi
                <lb/>
              conicorũ</note>
            gulo k b n: </s>
            <s xml:id="echoid-s1890" xml:space="preserve">& </s>
            <s xml:id="echoid-s1891" xml:space="preserve">quadratum m l rectangulo l f o æquale: </s>
            <s xml:id="echoid-s1892" xml:space="preserve">erunt tres
              <lb/>
            lineæ b k, k h, b n proportionales: </s>
            <s xml:id="echoid-s1893" xml:space="preserve">itémq; </s>
            <s xml:id="echoid-s1894" xml:space="preserve">proportionales inter ſe
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            f l, l m, f o. </s>
            <s xml:id="echoid-s1895" xml:space="preserve">quare ut linea b K ad lineam b n, ita quadratum b K
              <lb/>
              <note position="right" xlink:label="note-0075-05" xlink:href="note-0075-05a" xml:space="preserve">cor. 20. ſe
                <lb/>
              xti.</note>
            ad quadratum h k: </s>
            <s xml:id="echoid-s1896" xml:space="preserve">& </s>
            <s xml:id="echoid-s1897" xml:space="preserve">ut linea f l ad ipſam f o, ita quadratú f l
              <lb/>
            ad quadratum l m. </s>
            <s xml:id="echoid-s1898" xml:space="preserve">Itaque quoniam, ut b K ad b n, ita eſt f l ad
              <lb/>
            f o; </s>
            <s xml:id="echoid-s1899" xml:space="preserve">erit ut quadratum b K ad quadratum k h, ita quadratum f l
              <lb/>
            ad l m quadratum. </s>
            <s xml:id="echoid-s1900" xml:space="preserve">ergo ut linea b k, ad lineam K h, ita linea f l
              <lb/>
              <note position="right" xlink:label="note-0075-06" xlink:href="note-0075-06a" xml:space="preserve">22. ſexti</note>
            ad ipsã lm: </s>
            <s xml:id="echoid-s1901" xml:space="preserve">& </s>
            <s xml:id="echoid-s1902" xml:space="preserve">permutãdo ut b k ad f l, ita k h ad lm. </s>
            <s xml:id="echoid-s1903" xml:space="preserve">ſed b k ad
              <lb/>
            f l erat ut k c ad c l. </s>
            <s xml:id="echoid-s1904" xml:space="preserve">ergo k h ad lm, ut K c ad c l. </s>
            <s xml:id="echoid-s1905" xml:space="preserve">quare ex eo
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            dem lemmate patet lineam h c, & </s>
            <s xml:id="echoid-s1906" xml:space="preserve">per m punctum tranſire. </s>
            <s xml:id="echoid-s1907" xml:space="preserve">ut igi-
              <lb/>
            tur K c ad c l: </s>
            <s xml:id="echoid-s1908" xml:space="preserve">hoc eſt ut a c ad c e, ita h c ad c m; </s>
            <s xml:id="echoid-s1909" xml:space="preserve">hoc eſt ad eam
              <lb/>
            ipſius partem, quæ inter c, & </s>
            <s xml:id="echoid-s1910" xml:space="preserve">e g c ſectionem interyeitur. </s>
            <s xml:id="echoid-s1911" xml:space="preserve">ſimiliter
              <lb/>
            demonſtrabimus idem contingere in alijs lineis, quæ à puncto c ad
              <lb/>
            a b c ſectionem perducuntur. </s>
            <s xml:id="echoid-s1912" xml:space="preserve">At uero b c ad e f eandern propor-
              <lb/>
            tionem habere, liquido apparet; </s>
            <s xml:id="echoid-s1913" xml:space="preserve">nam b c ad c f, eſt ut d c ad c g;
              <lb/>
            </s>
            <s xml:id="echoid-s1914" xml:space="preserve">uidelicet ut earum duplæ, a c ad c e.</s>
            <s xml:id="echoid-s1915" xml:space="preserve"/>
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