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="id000408">
                <pb pagenum="21" xlink:href="015/01/040.jpg"/>
              cum ſuis duplicatis ſequentibus, & in c, & in d, & in reliquis pa­
                <lb/>
              riter conduplicatis ſuis ſequentibus ex altera, quod fit ex h in b ſe­
                <lb/>
              mel, in c ter, in d quinquies, in e ſepties, in f nouies, in g undecies,
                <lb/>
              in h tredecies, detractis ergo rurſus quod fit ex h in b ſemel, & ex
                <lb/>
              h in c d e f g h bis relinquetur, quod fit ex h in c, & duplo ſequen­
                <lb/>
              tium, & d & duplo ſequentium, & e & aliarum pariter: & ex alia
                <lb/>
              parte, quod fit ex h in c ſemel, & in d ter, & in e quinquies, in f ſe­
                <lb/>
              pties, in g nouies, in h undecies. </s>
              <s id="id000409">Ab his rurſus detractis, quòd fit
                <lb/>
              ex h in c ſemel, & in ſequentes bis, relinquetur h in d ſemel cum ſuis
                <lb/>
              ſequentibus bis, & in e ſemel cum ſuis ſequentibus & in f, & in g &
                <lb/>
              in h pariter, & ex alia parte, quod fit ex h in d ſemel, in e ter, f quin­
                <lb/>
              quies, g ſepties, h nouies, ab his rurſus detraho, quod fit ex h in d
                <lb/>
              ſemel, & in ſequentes bis, relinquetur ex una parte, quod fit ex h
                <lb/>
              in e f g h cum duplo ſequentium ex alia, quod fit ex h in e ſe­
                <lb/>
              mel, f ter, g quinquies, h ſepties, & ſimiliter ab his detractis, quod
                <lb/>
              fit ex h in e ſemel, & bis in ſequentes, relinquetur ex una par­
                <lb/>
              te; quod fit ex h in f ſemel, & in g h bis, & in g ſemel, & in h bis,
                <lb/>
              & in h ſemel, & ex alia, quod fit ex h in f ſemel, in g ter, in h quin­
                <lb/>
              quies. </s>
              <s id="id000410">Iterum detractis, quod fit ex h in f ſemel, & in g h bis com­
                <lb/>
              muniter relinquetur, quod fit ex h in g ſemel, & in h bis, & in h ſe­
                <lb/>
              mel, & ex alia parte quod fit ex h in g ſemel, & ex h in h ter. </s>
              <s id="id000411">Sed
                <lb/>
              iſta, quæ relicta ſunt iam, ſunt manifeſtè æqualia, ergo etiam pri­
                <lb/>
              ma aggregata ab initio fuere æqualia, ergo & æqualia illis qua­
                <lb/>
              drata a b c d e f g h his, quæ fiunt, ex h in eaſdem quantita­
                <lb/>
              tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o,
                <lb/>
              h in p, ſed iam his quadratis a b c d e f g h demonſtrata ſunt eſſe du­
                <lb/>
              pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra­
                <lb/>
              ti a, ergo quadrata omnium quantitatum ſecundi ordinis cum
                <lb/>
              quadrato a rurſus repetito, & producto h in aggregatum quanti­
                <lb/>
              tatum primi ordinis ſunt tripla quadratis quantitatum primi ordi­
                <lb/>
              nis pariter acceptis, quod fuit propoſitum, & fuit Archimedis in li
                <lb/>
              bro de lineis ſpiralibus, & ego adieci hic propter modum demon
                <lb/>
              ſtrandi, qui eſt elegantiſsimus, & procedit ex principijs arithmeti­
                <lb/>
              cis, & diuerſis à communibus, & ideo non reuoluitur, ut ſolent re­
                <lb/>
              liquæ quæſtiones.</s>
            </p>
            <p type="margin">
              <s id="id000412">
                <margin.target id="marg67"/>
              I
                <emph type="italics"/>
              n
                <emph.end type="italics"/>
              5. E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
                <lb/>
              P
                <emph type="italics"/>
              rop.
                <emph.end type="italics"/>
              12.</s>
            </p>
            <p type="margin">
              <s id="id000413">
                <margin.target id="marg68"/>
              L
                <emph type="italics"/>
              ib.
                <emph.end type="italics"/>
              6. E
                <emph type="italics"/>
              le.
                <emph.end type="italics"/>
                <lb/>
              P
                <emph type="italics"/>
              rop.
                <emph.end type="italics"/>
              17.</s>
            </p>
            <p type="main">
              <s id="id000414">Propoſitio uigeſima.</s>
            </p>
            <p type="main">
              <s id="id000415">Cùm fuerint quatuor quantitates, fueritque ſecunda æqualis ter­
                <lb/>
              tiæ, aut primæ æqualis quartæ, erit proportio primæ ad quartam,
                <lb/>
              aut tertiæ ad ſecundam producta ex proportionibus primæ ad ſe­
                <lb/>
              cundam, & tertiæ ad quartam.
                <lb/>
                <arrow.to.target n="marg69"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000416">
                <margin.target id="marg69"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id000417">Cùm enim quantitates hæ non fuerint ęquales,
                <expan abbr="cõſtat">conſtat</expan>
              per </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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