Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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[Figure 151]
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        <div xml:id="echoid-div139" type="section" level="1" n="44">
          <p style="it">
            <s xml:id="echoid-s1936" xml:space="preserve">
              <pb o="25" file="0077" n="77" rhead="DE IIS QVAE VEH. IN AQVA."/>
            itêmq; </s>
            <s xml:id="echoid-s1937" xml:space="preserve">quadratum c q æquale rectangulo q u y, hoc eſt ſectionum
              <lb/>
            h s c, m u c lineas s x, u y, eas eſſe, iuxta quas poſſunt, quæ à ſectio-
              <lb/>
            ne ad diametrum ducuntur. </s>
            <s xml:id="echoid-s1938" xml:space="preserve">ſed cú triangula c p r, c q t ſimilia ſint,
              <lb/>
            habebit c r ad c p eandem proportionem, quam c t ad c q: </s>
            <s xml:id="echoid-s1939" xml:space="preserve">& </s>
            <s xml:id="echoid-s1940" xml:space="preserve">id-
              <lb/>
              <note position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">22. fexti</note>
            circo quadratum c r ad quadratum c p eandem habebit, quam
              <lb/>
            quadratum c t ad quadratum c q. </s>
            <s xml:id="echoid-s1941" xml:space="preserve">ergo & </s>
            <s xml:id="echoid-s1942" xml:space="preserve">linea b n, ad lineam
              <lb/>
            ſ x ita erit, ut linea fo ad ipſam u y. </s>
            <s xml:id="echoid-s1943" xml:space="preserve">erat autem b c ad c m, ut a c
              <lb/>
            ad c e. </s>
            <s xml:id="echoid-s1944" xml:space="preserve">quare & </s>
            <s xml:id="echoid-s1945" xml:space="preserve">earum dimidiæ c p ad c q, ut a d ad e g: </s>
            <s xml:id="echoid-s1946" xml:space="preserve">& </s>
            <s xml:id="echoid-s1947" xml:space="preserve">
              <lb/>
            permutando c p ad a d, ut c q ad e g. </s>
            <s xml:id="echoid-s1948" xml:space="preserve">Sed oſtenſum est a d ad b n
              <lb/>
            ita eſſe, ut e g ad f o: </s>
            <s xml:id="echoid-s1949" xml:space="preserve">& </s>
            <s xml:id="echoid-s1950" xml:space="preserve">b n ad s x, ut f o ad u y. </s>
            <s xml:id="echoid-s1951" xml:space="preserve">ergo ex
              <lb/>
            æquali c p ad ſ x erit, ut c q ad u y. </s>
            <s xml:id="echoid-s1952" xml:space="preserve">Quòd cum quadratú c p æqua
              <lb/>
            le ſit rectangulo p s x & </s>
            <s xml:id="echoid-s1953" xml:space="preserve">quadratum c q rectangulo q u y, erunt
              <lb/>
            tres lineæ ſ p, p c, ſ x proportionales; </s>
            <s xml:id="echoid-s1954" xml:space="preserve">itemq; </s>
            <s xml:id="echoid-s1955" xml:space="preserve">proportionales ip-
              <lb/>
            ſæ u q, q c, u y. </s>
            <s xml:id="echoid-s1956" xml:space="preserve">quare & </s>
            <s xml:id="echoid-s1957" xml:space="preserve">ſ p ad p c, ut u q ad q c: </s>
            <s xml:id="echoid-s1958" xml:space="preserve">& </s>
            <s xml:id="echoid-s1959" xml:space="preserve">ut p c ad
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            c h, ita q c ad c m. </s>
            <s xml:id="echoid-s1960" xml:space="preserve">ex æquali igitur ut portionis h ſ c diameter ſ p
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            ad eius baſim c h, ita portionis m u s diameter u q ad baſim c m.
              <lb/>
            </s>
            <s xml:id="echoid-s1961" xml:space="preserve">& </s>
            <s xml:id="echoid-s1962" xml:space="preserve">anguli, quos diametri cum baſibus continent, ſunt æquales, quòd
              <lb/>
            lineæ ſ p, u q ſibi ipſis æquidiſtent, ergo & </s>
            <s xml:id="echoid-s1963" xml:space="preserve">portiones h ſ c, m u c
              <lb/>
            inter ſe ſimiles erunt. </s>
            <s xml:id="echoid-s1964" xml:space="preserve">id quod demonstrandum proponebatur.</s>
            <s xml:id="echoid-s1965" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div141" type="section" level="1" n="45">
          <head xml:id="echoid-head50" xml:space="preserve">LEMMA IIII.</head>
          <p style="it">
            <s xml:id="echoid-s1966" xml:space="preserve">Sint duæ lineæ a b, c d, quæ ſecentur in punctis e f,
              <lb/>
            ita ut quam proportionem habet a e ad e b, habeat c f
              <lb/>
            ad f d: </s>
            <s xml:id="echoid-s1967" xml:space="preserve">rurſus ſecentur in aliis duobus punctis g h; </s>
            <s xml:id="echoid-s1968" xml:space="preserve">& </s>
            <s xml:id="echoid-s1969" xml:space="preserve">
              <lb/>
            habeat c h ad h d eandem proportionem, quam a g ad
              <lb/>
            g b. </s>
            <s xml:id="echoid-s1970" xml:space="preserve">Dico c f ad f h ita eſſe, ut a e ad e g.</s>
            <s xml:id="echoid-s1971" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1972" xml:space="preserve">
              <emph style="sc">Q_voniam_</emph>
            enim ut a e ad e b, ita c f ad f d, erit componen
              <lb/>
            do ut a b ad e b, ita c d ad f d. </s>
            <s xml:id="echoid-s1973" xml:space="preserve">Rurſus cum ſit ut a g ad g b, ita
              <lb/>
            c h ad h d; </s>
            <s xml:id="echoid-s1974" xml:space="preserve">componendo, conuertendoq; </s>
            <s xml:id="echoid-s1975" xml:space="preserve">ut g b ad a b, ita erit h d
              <lb/>
            ad c d. </s>
            <s xml:id="echoid-s1976" xml:space="preserve">ergo ex æquali, conuertendoq; </s>
            <s xml:id="echoid-s1977" xml:space="preserve">ut e b ad g b, ita f d ad h d:</s>
            <s xml:id="echoid-s1978" xml:space="preserve"/>
          </p>
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