Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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              <pb o="19" file="0049" n="49" rhead="DE IIS QVAE VEH. IN AQVA."/>
            eam proportionem babebit, quam a f ad a e. </s>
            <s xml:id="echoid-s1084" xml:space="preserve">Sed & </s>
            <s xml:id="echoid-s1085" xml:space="preserve">eandem habet
              <lb/>
            a s ad a r. </s>
            <s xml:id="echoid-s1086" xml:space="preserve">quare a s ipſi a x eſt æqualis, pars toti, quod fieri non
              <lb/>
              <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">9. quinti</note>
            poteſt. </s>
            <s xml:id="echoid-s1087" xml:space="preserve">Idem abſurdum ſequetur, ſi ponamus punctum t cadere ul-
              <lb/>
            tra lineam a c. </s>
            <s xml:id="echoid-s1088" xml:space="preserve">neceſſarium igitur est, ut in ipſam a c cadat. </s>
            <s xml:id="echoid-s1089" xml:space="preserve">quod
              <lb/>
            demonſtrandum propoſuimus.</s>
            <s xml:id="echoid-s1090" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div82" type="section" level="1" n="33">
          <head xml:id="echoid-head38" xml:space="preserve">LEMMA III.</head>
          <p style="it">
            <s xml:id="echoid-s1091" xml:space="preserve">Sit parabole, cuius diameter a b: </s>
            <s xml:id="echoid-s1092" xml:space="preserve">atque eam cŏtingen
              <lb/>
            tes rectæ lineæ a c, b d; </s>
            <s xml:id="echoid-s1093" xml:space="preserve">a c quidem in puncto c, b d ue
              <lb/>
            ro in b: </s>
            <s xml:id="echoid-s1094" xml:space="preserve">& </s>
            <s xml:id="echoid-s1095" xml:space="preserve">per c ductis duabus lineis; </s>
            <s xml:id="echoid-s1096" xml:space="preserve">quarum alter a c e
              <lb/>
            diametro æquidiſtet, alter a c f æquidiſtet ipſi b d: </s>
            <s xml:id="echoid-s1097" xml:space="preserve">ſuma
              <lb/>
            tur quod uis punctum g in diametro: </s>
            <s xml:id="echoid-s1098" xml:space="preserve">fiatque ut f b, ad
              <lb/>
            b g, ita b g ad b h: </s>
            <s xml:id="echoid-s1099" xml:space="preserve">& </s>
            <s xml:id="echoid-s1100" xml:space="preserve">per g h ducantur g k l, h e m,
              <lb/>
            æquidiſtantes b d: </s>
            <s xml:id="echoid-s1101" xml:space="preserve">per m uero ducatur m n o ipſi a c
              <lb/>
            æquidistans, quæ diametrum ſecet in o: </s>
            <s xml:id="echoid-s1102" xml:space="preserve">& </s>
            <s xml:id="echoid-s1103" xml:space="preserve">per n ducta
              <lb/>
            n p uſque ad diametrum, ipſi b d æquidistet. </s>
            <s xml:id="echoid-s1104" xml:space="preserve">Dico h o
              <lb/>
            ipſius g b duplam eſſe.</s>
            <s xml:id="echoid-s1105" xml:space="preserve"/>
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            <s xml:id="echoid-s1106" xml:space="preserve">V_EL_ igitur linea m n o ſccat diametrum in g, uel in alijs pun-
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            ctis: </s>
            <s xml:id="echoid-s1107" xml:space="preserve">& </s>
            <s xml:id="echoid-s1108" xml:space="preserve">ſi quidem ſecat in g, unum at que idem punctum duabus li-
              <lb/>
            teris go notabitur. </s>
            <s xml:id="echoid-s1109" xml:space="preserve">Itaque quoniam f c, p n, h e m ſibiipſis æqui
              <lb/>
            distant: </s>
            <s xml:id="echoid-s1110" xml:space="preserve">& </s>
            <s xml:id="echoid-s1111" xml:space="preserve">ipſi a c æquidiſtat m n o: </s>
            <s xml:id="echoid-s1112" xml:space="preserve">fient triangula a f c, o p n,
              <lb/>
            o h m inter ſe ſimilia. </s>
            <s xml:id="echoid-s1113" xml:space="preserve">quare erit o h ad h m, ut a f ad fc: </s>
            <s xml:id="echoid-s1114" xml:space="preserve">& </s>
            <s xml:id="echoid-s1115" xml:space="preserve">per-
              <lb/>
              <note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">4. ſexti.</note>
            mut ando o h ad a f, ut h m ad fc. </s>
            <s xml:id="echoid-s1116" xml:space="preserve">est autem quadratum h m ad
              <lb/>
            quadratum g l, ut linea h b ad lineam b g, ex uigeſima primi libri
              <lb/>
            conicorum: </s>
            <s xml:id="echoid-s1117" xml:space="preserve">& </s>
            <s xml:id="echoid-s1118" xml:space="preserve">quadratum g l ad quadratum fc, ut linea g b ad
              <lb/>
            ipſam b f: </s>
            <s xml:id="echoid-s1119" xml:space="preserve">ſuntq; </s>
            <s xml:id="echoid-s1120" xml:space="preserve">h b, b g, b f lineæ deinceps proportionales. </s>
            <s xml:id="echoid-s1121" xml:space="preserve">er-
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              <note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">22. ſexti.
                <lb/>
              cor. 20. ſe
                <lb/>
              xti.</note>
            go & </s>
            <s xml:id="echoid-s1122" xml:space="preserve">quadrata h m, g l, f c, & </s>
            <s xml:id="echoid-s1123" xml:space="preserve">ipſorum latera proportionalia
              <lb/>
            erunt. </s>
            <s xml:id="echoid-s1124" xml:space="preserve">atque idcirco ut quadratum h m ad quadratum g l, ita </s>
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