Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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DE IIS QVAE VEH. IN AQVA.
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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:space="preserve">ſimiliter demonſtrabitur eandem
              <lb/>
            babere n o ad o f: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">reliquas eiuſmodi, at uero b K ad K e eam
              <lb/>
            habere proportionem, quam habet c e ad e a, ex eadem quinta. </s>
            <s xml:space="preserve">Ar-
              <lb/>
            chimedis perſpicue apparet. </s>
            <s xml:space="preserve">at que illud eſt, quod demonſtr andum
              <lb/>
            propoſuimus.</s>
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          <div type="float" level="2" n="1">
            <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a">
              <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0079-01"/>
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            <note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">4. ſexti:</note>
            <note position="right" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">11. quinti</note>
            <note position="right" xlink:label="note-0079-03" xlink:href="note-0079-03a" xml:space="preserve">14. quinti</note>
            <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a">
              <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0080-01"/>
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            <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">2. ſexti.</note>
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        <div type="section" level="1" n="47">
          <head xml:space="preserve">LEMMA VI.</head>
          <p style="it">
            <s xml:space="preserve">Itaque maneant eadem, quæ ſupra: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">itidem deſcri-
              <lb/>
            batur alia portio ſimilis contenta recta linea & </s>
            <s xml:space="preserve">rectan-
              <lb/>
            guli coni ſectione d r c; </s>
            <s xml:space="preserve">cuius diameter r s, ut ſecet li-
              <lb/>
            neam f g in t: </s>
            <s xml:space="preserve">producaturque s r ad lineam c h in u;
              <lb/>
            </s>
            <s xml:space="preserve">cuiſectio a b c occurrat in x, & </s>
            <s xml:space="preserve">e f c in y. </s>
            <s xml:space="preserve">Dico b m
              <lb/>
            ad m d proportionem habere compoſitam ex propor-
              <lb/>
            tione, quam babet e a ad a c; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ex ea, quam c d ba-
              <lb/>
            bet ad de.</s>
            <s xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:space="preserve">
              <emph style="sc">S_imiliter_</emph>
            enim ut ſupra, demonſtrabimus lineam c h con-
              <lb/>
            tingere ſectioné d r c in c puncto: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">l m ad m d, itêmq; </s>
            <s xml:space="preserve">n f ad f t;
              <lb/>
            </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">u y ad y r ita eſſe, ut c d ad d e. </s>
            <s xml:space="preserve">Quoniam igitur lb ad b m eſt,
              <lb/>
            ut c e ad e a; </s>
            <s xml:space="preserve">erit componendo, conuertendôq; </s>
            <s xml:space="preserve">bm ad lm, ut e a ad
              <lb/>
            a c: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ut lm ad m d, ita c d ad d e. </s>
            <s xml:space="preserve">proportio autem b m ad m d
              <lb/>
            compoſita eſt ex proportione, quam habet b m ad l m, & </s>
            <s xml:space="preserve">ex propor
              <lb/>
            tione, quam l m habet ad m d. </s>
            <s xml:space="preserve">ergo proportio b m ad m d etiam com
              <lb/>
            poſita erit ex proportione, quam habet e a, ad a c; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ex ea, quam
              <lb/>
            c d habet ad d e. </s>
            <s xml:space="preserve">Eadem ratione demonſtrabitur o f ad f t; </s>
            <s xml:space="preserve">itêmq; </s>
            <s xml:space="preserve">
              <lb/>
            x y ad y r proportionem habere ex eiſdem proportionibus compo-
              <lb/>
            ſitam: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ita in aijs. </s>
            <s xml:space="preserve">quod demonſtrare oportebat.</s>
            <s xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:space="preserve">Ex quibus apparet lineas ſic ductas, quæ inter ſectio
              <lb/>
            nes a b c, d r c interiiciuntur à ſectione e f c in eandem
              <lb/>
            proportionem diuidi.</s>
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