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="id002512">
                <pb pagenum="142" xlink:href="015/01/161.jpg"/>
              enim d f cum fiat ex a in duplum b c, dupla erit ſuperficiei a in b c, ſu
                <lb/>
              perficies f d, tota æquatur quadratis h & b c, igitur quadrata h & b
                <lb/>
              c dupla ſunt ſuperficiei a in b c, quod uerò fit ex a in duplum b c ſe
                <lb/>
              habet ad id quod fit ex h in duplum b c, ut a ad h, cum per eandem
                <lb/>
              lineam ducantur, igitur quod fit ex a in duplum b c, & ſunt quadra­
                <lb/>
              ta h & b c, ſe habent ad duplum h in b c, ut a ad h, quod fuit de­
                <lb/>
              monſtrandum.</s>
            </p>
            <p type="margin">
              <s id="id002513">
                <margin.target id="marg491"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              m.</s>
            </p>
            <p type="main">
              <s id="id002514">Propoſitio centeſima quinquageſima prima.</s>
            </p>
            <p type="main">
              <s id="id002515">Proportio differentiæ quadratorum partium, cuiuſuis lineæ ad
                <lb/>
              quadratum differentiæ
                <expan abbr="illarũ">illarum</expan>
              eſt uelut totius lineę ad differentiam.
                <lb/>
                <arrow.to.target n="marg492"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002516">
                <margin.target id="marg492"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id002517">Sit a b diuiſa in puncto c, & fiat c d æqualis
                <lb/>
              c b, manifeſtum eſt quod differentia partium
                <lb/>
                <figure id="id.015.01.161.1.jpg" xlink:href="015/01/161/1.jpg" number="163"/>
                <lb/>
              eſt a d, dico proportionem differentiæ quadra
                <lb/>
              torum a c & c b ad quadratum a d differentiæ partium eſſe ut a b ad </s>
            </p>
            <p type="main">
              <s id="id002518">
                <arrow.to.target n="marg493"/>
                <lb/>
              a d. </s>
              <s id="id002519">Quoniam differentia quadratorum a c & c b eſt, quod fit ex a d
                <lb/>
              in d c bis cum quadrato a d, & ideò quod fit ex a d in d b cum qua­
                <lb/>
              drato a d, & ideò quod fit ex tota a b in a d. </s>
              <s id="id002520">Igitur differentia qua­
                <lb/>
                <arrow.to.target n="marg494"/>
                <lb/>
              drato a c & c b eſt quod fit ex a b in a d, quare cum quadratum a d
                <lb/>
              fiat ex a d in a d, erit proportio a b ad a d, uelut differentiæ quadra­
                <lb/>
                <arrow.to.target n="marg495"/>
                <lb/>
              torum a c & b c ad quadratum a d differentiæ partium. </s>
              <s id="id002521">Quod fuit
                <lb/>
              propoſitum.</s>
            </p>
            <p type="margin">
              <s id="id002522">
                <margin.target id="marg493"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              4.
                <emph type="italics"/>
              ſecun
                <lb/>
              di
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002523">
                <margin.target id="marg494"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              3.
                <emph type="italics"/>
              ſecun
                <lb/>
              di
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002524">
                <margin.target id="marg495"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              1.
                <emph type="italics"/>
              ſexti
                <emph.end type="italics"/>
                <lb/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id002525">Propoſitio centeſima quinquageſima ſecunda.</s>
            </p>
            <p type="main">
              <s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue­
                <lb/>
              ritque proportio aggregati ex maiore & dimidio ad ipſam maiorem
                <lb/>
              uelut ex minore, & aliqua linea ad ipſam minorem, & rurſus aggre­
                <lb/>
              gati ex minore dimidio ad ipſam minorem, uelut aggregati ex ma­
                <lb/>
              iore & alia addita ad ipſam maiorem, erit proportio dimidij ad par
                <lb/>
              tem unam inæqualem, uelut alterius partis inæqualis ad ſuam ad­
                <lb/>
              ditam mutuò, & etiam proportio additarum inuicem, uelut pro­
                <lb/>
              portio partium inæqualium duplicata, & rurſus ipſum dimidium
                <lb/>
              lineæ aſſumptæ medium erit proportione inter additas. </s>
              <s id="id002527">Demum
                <lb/>
              proportio dimidij cum ad dita maiore ad dimidium cum addita mi
                <lb/>
              nore, uelut maioris partis ad minorem.
                <lb/>
                <arrow.to.target n="marg496"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002528">
                <margin.target id="marg496"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id002529">Sit propoſita a b diuiſa per
                <lb/>
                <figure id="id.015.01.161.2.jpg" xlink:href="015/01/161/2.jpg" number="164"/>
                <lb/>
              æqualia in c per inæqualia in
                <lb/>
              d, & ſit ut addantur a g & b f,
                <lb/>
              ita ut proportio c a, & a d ad a d ſit ueluti f d ad d b, & c b & b d ad
                <lb/>
              b d, uelut g d ad d a, & hæc eſt quarta
                <expan abbr="ſecũdi">ſecundi</expan>
              Archimedis de ſphęra,
                <lb/>
              & Cylindro: quia ergo a c & a d ad a d, ut f d ad d b erit a c ad a d,
                <lb/>
              fb ad b d. </s>
              <s id="id002530">Et ſimiliter quia eſt c b & b d ad b d, uelut g d ad d a erit </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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