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="id000387">
                <pb pagenum="18" xlink:href="015/01/037.jpg"/>
              tates, &
                <expan abbr="diuiderent̃">diuiderentur</expan>
              ſingulę
                <expan abbr="ſecundũ">ſecundum</expan>
              numerum
                <expan abbr="illarũ">illarum</expan>
              , ſi quatuor in
                <lb/>
              quatuor partes æquales, ſi quinque in quinque, ſi decem in decem, ea ra
                <lb/>
              tione ut ultima
                <expan abbr="diuideret̃">diuideretur</expan>
              , ubi eſt finis primæ partis, penultima ubi
                <lb/>
              eſt finis ſecundæ partis, ante penultima ubi eſt finis tertiæ, & ſic de
                <lb/>
              alijs. </s>
              <s id="id000388">Vocabo ergo primas
                <expan abbr="quãtitates">quantitates</expan>
              propoſitas a b c d e f g h quan­
                <lb/>
              titates primi ordinis, ſed quantitates æquales quæ
                <expan abbr="conſtãt">conſtant</expan>
              ex quan
                <lb/>
              titatis. </s>
              <s id="id000389">primi ordinis, & ſupplementis, appellabo quantitates ſecun
                <lb/>
              di ordinis: ex quo patet quòd prima
                <expan abbr="quãtitas">quantitas</expan>
              erit ex utro que ordine,
                <lb/>
              quia non eſt diuiſa, reliquæ omnes differunt, quantitates uerò quas
                <lb/>
              adiunxi nominabo
                <expan abbr="ſupplemẽta">ſupplementa</expan>
              , & ſunt una minus
                <expan abbr="quã">quam</expan>
              quantitates
                <lb/>
              ordinum: ut ſi
                <expan abbr="quãtitates">quantitates</expan>
              ordinum ſint octo, erunt ſupplementa ſe­
                <lb/>
              ptem, & ſi quantitates
                <expan abbr="ordinũ">ordinum</expan>
              , eſſent ſeptem eſſent
                <expan abbr="ſupplemẽta">ſupplementa</expan>
              ſex,
                <lb/>
              quia inter ſupplementa
                <expan abbr="nõ">non</expan>
                <expan abbr="adnumerat̃">adnumeratur</expan>
              quantitas indiuiſa. </s>
              <s id="id000390">Erunt er
                <lb/>
              go ſupplementa i k l m n o p, quæ tanto erunt maiora quanto quan
                <lb/>
              titates primi ordinis ſunt minores, & contrà tanto maiora, quanto
                <lb/>
                <expan abbr="quãtitates">quantitates</expan>
              primi ordinis ſunt maiores. </s>
              <s id="id000391">quantitates
                <expan abbr="aũt">aut</expan>
              ſecundi ordi
                <lb/>
              nis
                <expan abbr="appellabunt̃">appellabuntur</expan>
              a, b i, ck, dl, em, fn, go, & hp. </s>
              <s id="id000392">Hæc uolui pluribus
                <lb/>
              agere, ut dilucidior eſſet propoſitio. </s>
              <s id="id000393">quæ licet
                <expan abbr="nõ">non</expan>
              ſit difficilis, eſt
                <expan abbr="tamẽ">tamen</expan>
                <lb/>
              confuſa ualde propter multitudinem
                <expan abbr="quantitatũ">quantitatum</expan>
              & ordinum. </s>
              <s id="id000394">Dico
                <lb/>
              ergo q̊d aggregatum
                <expan abbr="quadratorũ">quadratorum</expan>
              quantitatum ſecundi ordinis pri
                <lb/>
              mo quadrato bis repetito, ſeu uno addito
                <expan abbr="cũ">cum</expan>
              eo quod fit ex minima
                <lb/>
              in aggregatum quantitatum primi ordinis eſt
                <expan abbr="triplũ">triplum</expan>
              aggregato ex
                <lb/>
              quadratis omnibus
                <expan abbr="quantitatũ">quantitatum</expan>
                <expan abbr="eiuſdẽ">eiuſdem</expan>
              primi ordinis, & utres exem
                <lb/>
              plo facilius innoteſcat, ſint
                <expan abbr="quãtitates">quantitates</expan>
              primi ordinis 8. 7. 6. 5. 4. 3. 2. 1.
                <lb/>
              quorum quadrata ſint 64. 49. 36. 25. 16. & 9.4 & 1. quæ iuncta
                <expan abbr="faciũt">faciunt</expan>
                <lb/>
              204, dico quod ſi ſumamus quadrata omnium
                <expan abbr="quãtitatum">quantitatum</expan>
              ſecundi
                <lb/>
              ordinis, quæ ſunt octies 64, & eis addiderimus unum
                <expan abbr="quadratũ">quadratum</expan>
              ex
                <lb/>
              his, ut fiant nouies 64, & erunt 556, ſimul iuncta & eis addamus, q̊d
                <lb/>
              fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti­
                <lb/>
              tatum omnium primi ordinis, & eſt tale
                <expan abbr="productũ">productum</expan>
              36, ut fiat totum
                <lb/>
              612, quod tale 612 eſt triplum 204, aggregati
                <expan abbr="quadratorũ">quadratorum</expan>
              primi or­
                <lb/>
              dinis unius demonſtratio hęc eſt. </s>
              <s id="id000395">Quia ex quarta ſecundi Element.
                <lb/>
              Euclidis ſingula quadrata
                <expan abbr="quantitatũ">quantitatum</expan>
                <expan abbr="diuiſarũ">diuiſarum</expan>
              ſecundi ordinis con
                <lb/>
              ſtant ex quatuor partibus quarum duę ſunt quadrata partium, reli­
                <lb/>
              quæ duæ ſunt producta ex partibus
                <expan abbr="inuicẽ">inuicem</expan>
              bis, & quia h fuit æqua­
                <lb/>
              lis 1, & p ęqualis b, quia ſupplementa
                <expan abbr="fuerũtęqualia">fuerunt ęqualia</expan>
              mutuò quanti
                <lb/>
              tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis
                <lb/>
              f, e
                <expan abbr="aũt">aut</expan>
              ęqualis m. </s>
              <s id="id000396">
                <expan abbr="Sequit̃">Sequitur</expan>
              ergo quod ſumptis duabus quantitatibus
                <lb/>
              ſecundi ordinis habentibus
                <expan abbr="ſupplemẽta">ſupplementa</expan>
              mutuò æqualia ipſis quan
                <lb/>
              titatibus quod quadrata partium
                <expan abbr="erũt">erunt</expan>
              dupla quadratis primarum
                <lb/>
              quantitatum: ueluti capio b i ſecundam & h p ultimam,
                <expan abbr="quarũ">quarum</expan>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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