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>
            <pb pagenum="14" xlink:href="015/01/033.jpg"/>
            <p type="margin">
              <s id="id000309">
                <margin.target id="marg46"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              18.
                <emph type="italics"/>
              diff.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id000310">Propoſitio quinta decima.</s>
            </p>
            <p type="main">
              <s id="id000311">Si fuerint quatuor quantitas proportio confuſa aggregati pri­
                <lb/>
              mæ & tertiæ ad aggregatum ſecundæ, & quartæ erit ut monadis
                <lb/>
              addito prouentu, qui fit diuiſa differentia differentiarum primæ &
                <lb/>
              ſecundæ, atque quartæ & tertiæ per aggregatum tertiæ, & quartæ ad
                <lb/>
              ipſam monadem.</s>
            </p>
            <figure id="id.015.01.033.1.jpg" xlink:href="015/01/033/1.jpg" number="24"/>
            <p type="main">
              <s id="id000312">Sint quatuor quantitates a b, c, d, e f, &
                <lb/>
                <arrow.to.target n="marg47"/>
                <lb/>
              ſit a b maior cin a h, & e f maior d in f g, &
                <lb/>
              differentia f g & a h ſit a k: dico proportio­
                <lb/>
              nem a b, & d confuſam ad c & e f, eſſe ut mo
                <lb/>
              nadis addito prouentu, uel detracto a k diuiſæ per aggregatum c.
                <lb/>
              & e f ad ipſam monadem, & manifeſtum eſt, quòd poteſt continge­
                <lb/>
              re pluribus modis: Primus ut a b ſit maior c & e f minor d, & tunc
                <lb/>
              differentiæ coniungentur, & prouentus, addetur monadi. </s>
              <s id="id000313">Idem fa­
                <lb/>
              ciendum erit ſi a b ſit maior c, & e f ſit minor d, ſed exceſſus ſuperet
                <lb/>
              defectum. </s>
              <s id="id000314">At ſi uel a b ſit minor c, & e f maior d, uel ita minor, ut c
                <lb/>
              exceſſus ſupra b a ſit maior defectu, detrahemus prouentum à mo­
                <lb/>
              nade. </s>
              <s id="id000315">Alia cautio eſt quòd ſi fuerint utrinque exceſſus, aut defectus,
                <lb/>
              minuemus minorem de maiore: ſi autem unus ſit exceſſus alter de­
                <lb/>
              fectus, iungemus illos, & poſt diuidemus. </s>
              <s id="id000316">uno ergo demonſtrato
                <lb/>
              ut pote primo intelligentur reliqui. </s>
              <s id="id000317">Quia ergo b h eſt æqualis c &
                <lb/>
              e g æqualis d & h k æqualis g f, erit ex communi animi ſententia ag
                <lb/>
              gregatum ex d & k b æquale aggregato ex c & e f, igitur per dicta
                <lb/>
              proportio aggregati ad aggregatum eſt unum. </s>
              <s id="id000318">at uerò diuiſa k a
                <lb/>
              per c & e f fit quantum diuiſa eadem per b k, & d, ſed diuiſa k a per b
                <lb/>
              k, & d iunctas, exit proportio a k ad aggregatum b k & d: igitur di­
                <lb/>
              uiſa a k per aggregatum e f & c, exibit eadem proportio, igitur a b
                <lb/>
              & d ad aggregatum c & e f eſt coniuncta ex monade & proportio­
                <lb/>
              ne a k ad aggregatum c & e f, quod erat demonſtrandum.</s>
            </p>
            <p type="margin">
              <s id="id000319">
                <margin.target id="marg47"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <figure id="id.015.01.033.2.jpg" xlink:href="015/01/033/2.jpg" number="25"/>
            <p type="main">
              <s id="id000320">Ex hoc patet quod proportionum confuſio
                <lb/>
                <arrow.to.target n="marg48"/>
                <lb/>
              fit iunctis denominatoribus numeratoris: mul­
                <lb/>
              tiplicatio multiplicatis: additio multiplicatis
                <lb/>
              decuſſatim in numeratores ad productum ex
                <lb/>
              denominatoribus, ut in exemplis.</s>
            </p>
            <p type="margin">
              <s id="id000321">
                <margin.target id="marg48"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id000322">Propoſitio ſexta decima.</s>
            </p>
            <p type="main">
              <s id="id000323">Omnium quatuor quantitatum propoſita
                <lb/>
              prima, quæ non minorem habet proportionem
                <lb/>
              ad ſuam correſpondentem, quàm alia ad aliam
                <lb/>
                <figure id="id.015.01.033.3.jpg" xlink:href="015/01/033/3.jpg" number="26"/>
                <lb/>
              erit proportio confuſa illarum, ut pro­
                <lb/>
              ducti ex aggregato primæ & tertiæ in </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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