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="id002492">
                <pb pagenum="141" xlink:href="015/01/160.jpg"/>
              a b, & latus eius detracto dimidio reſidui erit b c linea, quare diui­
                <lb/>
              ſio nota, & eſt ut dicamus : uolo diuidere datam lineam, ut quantita­
                <lb/>
              tes adiectæ ſub mutua proportione ad unam tertiam cum parti­
                <lb/>
              bus obtineant inter ſe proportionem datam.</s>
            </p>
            <p type="margin">
              <s id="id002493">
                <margin.target id="marg487"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              1.
                <emph type="italics"/>
              ſecun
                <lb/>
              di
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lement.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id002494">Propoſitio centeſima quadrageſima nona.</s>
            </p>
            <p type="main">
              <s id="id002495">Datam lineam ſic diuidere, ut proportio quadratorum ad du­
                <lb/>
              plum unius partis in alteram ſit, ut lineę datæ ad lineam datam.</s>
            </p>
            <p type="main">
              <s id="id002496">Sit data a b quam uolo diuidere, ut proponitur ſub proportio­
                <lb/>
                <arrow.to.target n="marg488"/>
                <lb/>
              ne c d ad e, diuido a b bifariam in f, & abſcindo
                <lb/>
                <figure id="id.015.01.160.1.jpg" xlink:href="015/01/160/1.jpg" number="160"/>
                <lb/>
              g d æqualem d e, & inter c g
                <expan abbr="reſiduũ">reſiduum</expan>
              & c e inter­
                <lb/>
              pono proportione, & ut h ad c g ita a f medietatis a b ad fk. </s>
              <s id="id002497">Omnia
                <lb/>
              iſta ſunt notiſsima ex primo & ſexto Elemento­
                <lb/>
                <figure id="id.015.01.160.2.jpg" xlink:href="015/01/160/2.jpg" number="161"/>
                <lb/>
                <expan abbr="rũ">rum</expan>
              Euclidis. </s>
              <s id="id002498">Si ergo abſcindantur fk ex fa, dico
                <lb/>
              quod proportio quadratorum l k & k a ad du­
                <lb/>
              plum rectanguli a k in k b eſt ut c d ad d e. </s>
              <s id="id002499">Quia. n. </s>
              <s id="id002500">c e ad c g dupli­
                <lb/>
              cata eſt ei quę eſt h ad c g, duplicata eſt
                <expan abbr="etiã">etiam</expan>
              ei quæ eſt f a ad fk, qua­
                <lb/>
              re ut quadrati a f ad fk, ita c e ad c g, igitur diſiungendo c g ad g e ut
                <lb/>
              reſidui quadrati k f ad reſiduum quadrati a f, quare c g ad g d ut
                <lb/>
              quadrati k f ad dimidium reſidui quadrati a f, igitur coniunctim c d
                <lb/>
              ad d g ut quadrati k f & dimidij reſidui quadrati a f ad ipſum dimi­
                <lb/>
              dium reſidui. </s>
              <s id="id002501">At uerò cum g d ſit æqualis d e, erit c d ad d e ut qua­
                <lb/>
              drati k f cum dimidio reſidui ſæpius dicti ad ipſum dimidium reſi­
                <lb/>
              dui. </s>
              <s id="id002502">Igitur etiam ut dupli quadrati k f cum reſiduo ad
                <expan abbr="reſiduũ">reſiduum</expan>
              , ſunt
                <lb/>
              enim omnia duplicata. </s>
              <s id="id002503">At
                <expan abbr="duplũ">duplum</expan>
              quadrati k f
                <expan abbr="cũ">cum</expan>
              reſiduo eſt æqua­
                <lb/>
              le quadratis a f & f k, igitur quadratorum a f & f k ad differentiam
                <lb/>
              eo rum proportio eſt ut c d ad d e, igitur dupli quadratorum a f &
                <lb/>
              f k ad duplum differentiæ quadratorum a f & fk ut c d ad d e. </s>
              <s id="id002504">Ve­
                <lb/>
                <arrow.to.target n="marg489"/>
                <lb/>
              rum duplum quadratorum a f & f k æquatur quadratis b k & k a.
                <lb/>
                <arrow.to.target n="marg490"/>
                <lb/>
              Et duplum differentiæ quadratorum a f & fk eſt ęquale duplo pro
                <lb/>
              ducti b k in k a, igitur proportio quadratorum k b & k a ad
                <expan abbr="duplũ">duplum</expan>
                <lb/>
              producti k b in k a eſt ueluti c d ad d e, quod eſt propoſitum.</s>
            </p>
            <p type="margin">
              <s id="id002505">
                <margin.target id="marg488"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="margin">
              <s id="id002506">
                <margin.target id="marg489"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              9.
                <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="id002507">
                <margin.target id="marg490"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              5.
                <emph type="italics"/>
              ſecun
                <lb/>
              di
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id002508">Propoſitio centeſima quinquageſima.</s>
            </p>
            <p type="main">
              <s id="id002509">Propoſitis duabus lineis
                <expan abbr="lineã">lineam</expan>
              communem
                <lb/>
                <figure id="id.015.01.160.3.jpg" xlink:href="015/01/160/3.jpg" number="162"/>
                <lb/>
              utrique adiungere, ut ſit maioris ad additam pro­
                <lb/>
              portio, uelut quadratorum minoris & adiectæ
                <lb/>
              ad duplum unius in alteram.</s>
            </p>
            <p type="main">
              <s id="id002510">Hæc eſt quaſi conuerſa
                <expan abbr="præcedẽtis">præcedentis</expan>
              . </s>
              <s id="id002511">Sit a ma­
                <lb/>
                <arrow.to.target n="marg491"/>
                <lb/>
              ior, & b c minor, & fiat b d dupla b c, ſuper
                <expan abbr="quã">quam</expan>
                <lb/>
              erigatur b f æqualis a; & ſit rectangulum d f &
                <lb/>
              deſcribatur quadratum b c quod ſit b g reſiduę
                <lb/>
              ſuperficiei ad d f latus ſit h, dico h eſſe lineam quæſitam. </s>
              <s id="id002512">Superficies </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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