Cardano, Geronimo, Opvs novvm de proportionibvs nvmerorvm, motvvm, pondervm, sonorvm, aliarvmqv'e rervm mensurandarum, non solùm geometrico more stabilitum, sed etiam uarijs experimentis & observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, & in V libros digestum. Praeterea Artis Magnae, sive de regvlis algebraicis, liber vnvs abstrvsissimvs & inexhaustus planetotius Ariothmeticae thesaurus ... Item De Aliza Regvla Liber, hoc est, algebraicae logisticae suae, numeros recondita numerandi subtilitate, secundum Geometricas quantitates inquirentis ...

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s id="id002530">
                <pb pagenum="143" xlink:href="015/01/162.jpg"/>
              c b ad b d, uelut g a ad a d, & hoc eſt primum. </s>
              <s id="id002531">Quia ergo c a eſt æ­
                <lb/>
              qualis c b, erit c a ad b d, uelut g a ad a d, & iam fuit a d ad c a, ut b d
                <lb/>
              ad f b, per conuerſam igitur a d ad b d, ut g a ad a d, & ut b d ad fb,
                <lb/>
              interpoſitis ergo a d & d b inter a g & b f cum compoſita ſit pro­
                <lb/>
              portio a g ad b f ex proportione a g ad a d, & ad d b, & d b
                <lb/>
              ad b f, & proportio a d ad d b, ſit æqualis proportioni
                <lb/>
                <figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg" number="165"/>
                <lb/>
              a g ad a d, & d b ad b f, igitur proportio a g ad b f. </s>
              <s id="id002532">Per de­
                <lb/>
              monſtrata ab Alchindo eſt duplicata proportioni a d ad
                <lb/>
              d b quod eſt ſecundum. </s>
              <s id="id002533">Rurſus quia ex primo demon­
                <lb/>
              ſtrato, uel eius conuerſo proportio a d ad a c eſt uelut b d
                <lb/>
              ad b f, & d b ad a c, ut a d ad a g, proportiones ergo
                <lb/>
                <figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg" number="166"/>
                <lb/>
              a d & d b ad a c componunt proportionem produ­
                <lb/>
              cti a d in d b, quod ſit h ad quadratum a c quod ſit
                <lb/>
              k, & ſimiliter proportio b d ad b f & a d ad a g com­
                <lb/>
              ponunt proportionem producti ex b d in a d, quod
                <lb/>
              ſit l ad productum b f in a g, quod ſit m, per demonſtrata ab Eucli­
                <lb/>
              de in ſexto Elementorum, igitur proportio h ad k ut l ad m, ſed h & </s>
            </p>
            <p type="main">
              <s id="id002534">
                <arrow.to.target n="marg497"/>
                <lb/>
              l ſunt æquales, quia producuntur ex eiſdem, igitur per demonſtra­
                <lb/>
              ta in quinto Elementorum Euclidis, k eſt æquale m, ergo a c eſt me­
                <lb/>
              dia pro portione inter b f & g a, quod eſt tertium. </s>
              <s id="id002535">Quia uerò ex pri­
                <lb/>
              mo demonſtrato eſt fb ad b d, ut a c ad a d, & c b ad idem b d, ut g a
                <lb/>
              ad idem a d erit coniungendo fb & b c ad b d, ut coniun­
                <lb/>
                <figure id="id.015.01.162.3.jpg" xlink:href="015/01/162/3.jpg" number="167"/>
                <lb/>
              gendo g a & a c ad a d, ſed fb & b c componunt f c & g a,
                <lb/>
              & a c componunt g c, igitur ut f c ad b d, ita g c ad a d, er­
                <lb/>
              go permutando g c ad f c, ut a d ad b d, quod eſt quartum.</s>
            </p>
            <p type="margin">
              <s id="id002536">
                <margin.target id="marg497"/>
              I
                <emph type="italics"/>
              n
                <emph.end type="italics"/>
              P
                <emph type="italics"/>
              rop.
                <emph.end type="italics"/>
              23
                <lb/>
              P
                <emph type="italics"/>
              ropoſ.
                <emph.end type="italics"/>
              9.</s>
            </p>
            <p type="main">
              <s id="id002537">Cum ergo punctum d fuerit datum, licet inuenire a g & b f, faci­
                <lb/>
              lè, ut Archimedes præſupponit proportionem g d ad d f datam &
                <lb/>
              quærit eam, quæ eſt a d ad d b, & peruenitur ad res numero triplo
                <lb/>
              quadrati dimidij lineæ aſſumptæ æquales cubo & numero, qui ſit
                <lb/>
              ex duplo cubi dimidij in 1 m: ipſa proportione, & quod produci­
                <lb/>
              tur diuiſo per 1 p: ipſa proportione. </s>
              <s id="id002538">Veluti poſita a b 10, & propor­
                <lb/>
              tione quam uolo g d ad d f ſexcupla, duco 5 dimidium 10 in ſe fit 25,
                <lb/>
              & triplico, fit 75 numerus rerum. </s>
              <s id="id002539">Inde duco 5 idem dimidium ad
                <lb/>
              cubum fit 125, duplico fit 250, duco in 5, qui eſt 1 m: proportione fit
                <lb/>
              1250, diuido per 7, qui eſt 1 p: proportione exit 178 4/7 numerus, qui
                <lb/>
              cum cubo æquatur 75 rebus. </s>
              <s id="id002540">Cum ergo conſtituta fuerit diuiſio in
                <lb/>
              c, non recipit proportionem g d ad f d quam uolueris, ſed ſequitur
                <lb/>
              una ſola ad
                <expan abbr="illã">illam</expan>
              , & eſt mirabile, quoniam lineę uidentur ſumi liberè.
                <lb/>
              </s>
              <s id="id002541">Sed non eſt ita. </s>
              <s id="id002542">Et
                <expan abbr="etiã">etiam</expan>
              quia Archimedes
                <expan abbr="uidet̃">uidetur</expan>
              aſſumere
                <expan abbr="aliã">aliam</expan>
              lineam,
                <lb/>
              ſed non inueſtigat eam, imò oſtendit eam ex aſſumptis. </s>
              <s id="id002543">At Eutoci­
                <lb/>
              us oſtendit ambas,
                <expan abbr="unã">unam</expan>
              ex propria inuentione, aliam ex Diocle, ſed </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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