Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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ARCHIMEDIS
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        <div type="section" level="1" n="37">
          <p style="it">
            <s xml:space="preserve">
              <pb file="0062" n="62" rhead="ARCHIMEDIS"/>
            quadratum e ψ ad quadr. </s>
            <s xml:space="preserve">itum ψ b.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="5">
            <note position="right" xlink:label="note-0061-08" xlink:href="note-0061-08a" xml:space="preserve">E</note>
            <note position="right" xlink:label="note-0061-09" xlink:href="note-0061-09a" xml:space="preserve">4. ſexti.</note>
            <note position="right" xlink:label="note-0061-10" xlink:href="note-0061-10a" xml:space="preserve">8. quinti.</note>
            <note position="right" xlink:label="note-0061-11" xlink:href="note-0061-11a" xml:space="preserve">13. quin-
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            ti.</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Sed quam proportionem habet qua-_
              <lb/>
              <anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/>
              <anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/>
            _dratum p i ad quadratum i y, eandem li_
              <lb/>
            _nea k r habet ad lineam i y.</s>
            <s xml:space="preserve">]_ Est enim ex
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            undecima primi conicorum quadratum p i æqua
              <lb/>
            le rectangulo contento linea i o, & </s>
            <s xml:space="preserve">ea, iuxta quam poſſunt quæ à
              <lb/>
            ſectione ad diametrum ducuntur, uidelicet duplaipſius k r. </s>
            <s xml:space="preserve">atque
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            est i y dupla i o, extrigeſimatertia eiuſdem: </s>
            <s xml:space="preserve">quare ex decimaſext a
              <lb/>
            ſexti elementorum, rectangulum, quod fit ex k r, & </s>
            <s xml:space="preserve">i y æ quale eſt
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            rectangulo contento linea i o & </s>
            <s xml:space="preserve">ea, iuxta quam poſſunt: </s>
            <s xml:space="preserve">hoc eſt qua
              <lb/>
            drato p i. </s>
            <s xml:space="preserve">Sed ut rectangulnm ex k r, & </s>
            <s xml:space="preserve">i y ad quadratum i y, ita
              <lb/>
              <anchor type="note" xlink:label="note-0062-02a" xlink:href="note-0062-02"/>
            linea κ r ad ipſam i y. </s>
            <s xml:space="preserve">ergo linea κ r ad i y eandem proportionem
              <lb/>
            habebit, quam rectangulum ex κ r & </s>
            <s xml:space="preserve">i y, hoc eſt quadratum p i ad
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            quadratum i y.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="6">
            <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a">
              <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0062-01"/>
            </figure>
            <note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">F</note>
            <note position="left" xlink:label="note-0062-02" xlink:href="note-0062-02a" xml:space="preserve">lem. 22.
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            decimi.</note>
          </div>
          <p>
            <s xml:space="preserve">Et quam proportionem habet quadratũ e ψ ad quadra
              <lb/>
              <anchor type="note" xlink:label="note-0062-03a" xlink:href="note-0062-03"/>
            tum ψ b, eandem habet dimidium lineæ K r ad lineã ψ b.</s>
            <s xml:space="preserve">]</s>
          </p>
          <div type="float" level="2" n="7">
            <note position="left" xlink:label="note-0062-03" xlink:href="note-0062-03a" xml:space="preserve">G</note>
          </div>
          <p style="it">
            <s xml:space="preserve">Nam cum quadratum e ψ poſitum ſit æquale dimidio rectanguli
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            contenti linea κ r, & </s>
            <s xml:space="preserve">ψ b; </s>
            <s xml:space="preserve">hoc est ei, quod dimidia ipſius κ r
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            & </s>
            <s xml:space="preserve">linea ψ b continetur: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ut rectangulum ex dimidia κ r, & </s>
            <s xml:space="preserve">ψ b
              <lb/>
              <anchor type="note" xlink:label="note-0062-04a" xlink:href="note-0062-04"/>
            ad quadratum ψ b, ita ſit dimidia κ r ad line am ψ b: </s>
            <s xml:space="preserve">habebit dimi-
              <lb/>
            dia κ r ad ψ b proportionem eandem, quam quadratum e ψ ad qua-
              <lb/>
            dratum ψ b.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="8">
            <note position="left" xlink:label="note-0062-04" xlink:href="note-0062-04a" xml:space="preserve">lem. 22.
              <lb/>
            decimi</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Etidcirco i y minor eſt, quàm dupla ψ b.</s>
            <s xml:space="preserve">]_ Quam enim pro
              <lb/>
              <anchor type="note" xlink:label="note-0062-05a" xlink:href="note-0062-05"/>
            portionem habet dimidium κ r ad ψ b, habeat κ r ad aliam lineam.
              <lb/>
            </s>
            <s xml:space="preserve">erit ea maior, quàm i y; </s>
            <s xml:space="preserve">nempe ad quam κ r minorem proportioné
              <lb/>
              <anchor type="note" xlink:label="note-0062-06a" xlink:href="note-0062-06"/>
            habet: </s>
            <s xml:space="preserve">at que erit dupla ψ b. </s>
            <s xml:space="preserve">ergo i y minor eſt, quam dupla ψ b.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="9">
            <note position="left" xlink:label="note-0062-05" xlink:href="note-0062-05a" xml:space="preserve">H</note>
            <note position="left" xlink:label="note-0062-06" xlink:href="note-0062-06a" xml:space="preserve">10. quinti.</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Et i ω maior, quam ψ r.</s>
            <s xml:space="preserve">]_ Cum enim o ω poſita ſit æ qualis b r
              <lb/>
              <anchor type="note" xlink:label="note-0062-07a" xlink:href="note-0062-07"/>
            ſi ex b r dematur ψ b, & </s>
            <s xml:space="preserve">ex o ω dematur o i, quæ minor eſt ψ b: </s>
            <s xml:space="preserve">erit
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            reliqua i ω maior reliqua ψ r.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="10">
            <note position="left" xlink:label="note-0062-07" xlink:href="note-0062-07a" xml:space="preserve">K</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Atqueideo f q æqualis eſt ipſi p m.</s>
            <s xml:space="preserve">]_ Ex decimaquarta
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              <anchor type="note" xlink:label="note-0062-08a" xlink:href="note-0062-08"/>
            quinti elementorum, nam linea o n ipſi b d eſt æ qualis.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="11">
            <note position="left" xlink:label="note-0062-08" xlink:href="note-0062-08a" xml:space="preserve">L</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Demonſtrata eſt autem p h maior, quàm f.</s>
            <s xml:space="preserve">]_ Etenim de-
              <lb/>
              <anchor type="note" xlink:label="note-0062-09a" xlink:href="note-0062-09"/>
            monstrata est i ω maior, quàm f; </s>
            <s xml:space="preserve">atque est p h æqualis ipſi i ω.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="12">
            <note position="left" xlink:label="note-0062-09" xlink:href="note-0062-09a" xml:space="preserve">M</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Eodem modo demonſtrabitur t h perpendicularis ad_
              <lb/>
              <anchor type="note" xlink:label="note-0062-10a" xlink:href="note-0062-10"/>
            </s>
          </p>
        </div>
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