Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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              <pb o="35" file="0081" n="81" rhead="DE IIS QVAE VEH. IN AQVA."/>
            proportionem, quam c e ad e a. </s>
            <s xml:id="echoid-s2055" xml:space="preserve">ſimiliter demonſtrabitur eandem
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            babere n o ad o f: </s>
            <s xml:id="echoid-s2056" xml:space="preserve">& </s>
            <s xml:id="echoid-s2057" xml:space="preserve">reliquas eiuſmodi, at uero b K ad K e eam
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            habere proportionem, quam habet c e ad e a, ex eadem quinta. </s>
            <s xml:id="echoid-s2058" xml:space="preserve">Ar-
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            chimedis perſpicue apparet. </s>
            <s xml:id="echoid-s2059" xml:space="preserve">at que illud eſt, quod demonſtr andum
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            propoſuimus.</s>
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        <div xml:id="echoid-div146" type="section" level="1" n="47">
          <head xml:id="echoid-head52" xml:space="preserve">LEMMA VI.</head>
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            <s xml:id="echoid-s2061" xml:space="preserve">Itaque maneant eadem, quæ ſupra: </s>
            <s xml:id="echoid-s2062" xml:space="preserve">& </s>
            <s xml:id="echoid-s2063" xml:space="preserve">itidem deſcri-
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            batur alia portio ſimilis contenta recta linea & </s>
            <s xml:id="echoid-s2064" xml:space="preserve">rectan-
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            guli coni ſectione d r c; </s>
            <s xml:id="echoid-s2065" xml:space="preserve">cuius diameter r s, ut ſecet li-
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            neam f g in t: </s>
            <s xml:id="echoid-s2066" xml:space="preserve">producaturque s r ad lineam c h in u;
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            </s>
            <s xml:id="echoid-s2067" xml:space="preserve">cuiſectio a b c occurrat in x, & </s>
            <s xml:id="echoid-s2068" xml:space="preserve">e f c in y. </s>
            <s xml:id="echoid-s2069" xml:space="preserve">Dico b m
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            ad m d proportionem habere compoſitam ex propor-
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            tione, quam babet e a ad a c; </s>
            <s xml:id="echoid-s2070" xml:space="preserve">& </s>
            <s xml:id="echoid-s2071" xml:space="preserve">ex ea, quam c d ba-
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            bet ad de.</s>
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            <s xml:id="echoid-s2073" xml:space="preserve">
              <emph style="sc">S_imiliter_</emph>
            enim ut ſupra, demonſtrabimus lineam c h con-
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            tingere ſectioné d r c in c puncto: </s>
            <s xml:id="echoid-s2074" xml:space="preserve">& </s>
            <s xml:id="echoid-s2075" xml:space="preserve">l m ad m d, itêmq; </s>
            <s xml:id="echoid-s2076" xml:space="preserve">n f ad f t;
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            </s>
            <s xml:id="echoid-s2077" xml:space="preserve">& </s>
            <s xml:id="echoid-s2078" xml:space="preserve">u y ad y r ita eſſe, ut c d ad d e. </s>
            <s xml:id="echoid-s2079" xml:space="preserve">Quoniam igitur lb ad b m eſt,
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            ut c e ad e a; </s>
            <s xml:id="echoid-s2080" xml:space="preserve">erit componendo, conuertendôq; </s>
            <s xml:id="echoid-s2081" xml:space="preserve">bm ad lm, ut e a ad
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            a c: </s>
            <s xml:id="echoid-s2082" xml:space="preserve">& </s>
            <s xml:id="echoid-s2083" xml:space="preserve">ut lm ad m d, ita c d ad d e. </s>
            <s xml:id="echoid-s2084" xml:space="preserve">proportio autem b m ad m d
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            compoſita eſt ex proportione, quam habet b m ad l m, & </s>
            <s xml:id="echoid-s2085" xml:space="preserve">ex propor
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            tione, quam l m habet ad m d. </s>
            <s xml:id="echoid-s2086" xml:space="preserve">ergo proportio b m ad m d etiam com
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            poſita erit ex proportione, quam habet e a, ad a c; </s>
            <s xml:id="echoid-s2087" xml:space="preserve">& </s>
            <s xml:id="echoid-s2088" xml:space="preserve">ex ea, quam
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            c d habet ad d e. </s>
            <s xml:id="echoid-s2089" xml:space="preserve">Eadem ratione demonſtrabitur o f ad f t; </s>
            <s xml:id="echoid-s2090" xml:space="preserve">itêmq; </s>
            <s xml:id="echoid-s2091" xml:space="preserve">
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            x y ad y r proportionem habere ex eiſdem proportionibus compo-
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            ſitam: </s>
            <s xml:id="echoid-s2092" xml:space="preserve">& </s>
            <s xml:id="echoid-s2093" xml:space="preserve">ita in aijs. </s>
            <s xml:id="echoid-s2094" xml:space="preserve">quod demonſtrare oportebat.</s>
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            <s xml:id="echoid-s2096" xml:space="preserve">Ex quibus apparet lineas ſic ductas, quæ inter ſectio
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            nes a b c, d r c interiiciuntur à ſectione e f c in eandem
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            proportionem diuidi.</s>
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