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="id000396">
                <pb pagenum="19" xlink:href="015/01/038.jpg"/>
              drata partium ſunt quadrata b & i, & h & p, ſed b eſt æqualis p, & h
                <lb/>
              æqualis i. </s>
              <s id="id000397">Ergo quatuor quadrata b i & h p ſunt dupla quadratis b
                <lb/>
              & h, & ita
                <expan abbr="concludã">concludam</expan>
              de omnibus ubi duæ quantitates duabus com
                <lb/>
              parantur: ſed in e m quia eſt ſola una quantitas, iſtud eſt etiam cla­
                <lb/>
              rius, quia quadrata e & m ſunt dupla quadrato e ſoli eo, quod & m
                <lb/>
                <arrow.to.target n="marg67"/>
                <lb/>
              ſunt æquales. </s>
              <s id="id000398">Igitur per demonſtrata ab Euclide erit proportio o­
                <lb/>
              mnium quadratorum b i, c k, d l, e m, f n, g o, h p, ad quadrata b c d e
                <lb/>
              f g h, pariter accepta proportio dupla. </s>
              <s id="id000399">at uerò addito quadrato a
                <lb/>
              quadratis b c d e f g h, & erunt quadrata omnium quantitatum, &
                <lb/>
              quadratis b i, c k, d l, e m, f n, g o, h p, duplo quadrati a ſcilicet ſemel,
                <lb/>
              quia a eſt ex ſecundo ordine quantitatum, & ſemel, quia hoc fuit aſ­
                <lb/>
              ſumptum in Problemate. </s>
              <s id="id000400">Sequitur ut quadrata omnia
                <expan abbr="quãtitatum">quantitatum</expan>
                <lb/>
              ſecundi ordinis, pro ut ſunt diuiſa in partes addito quadrato a, ſint
                <lb/>
              dupla quadratis primarum quantítatum, ſimul pariter acceptis. </s>
              <s id="id000401">Re
                <lb/>
              liquum eſt modo ut oſtendamus dupla
                <expan abbr="illorũ">illorum</expan>
              productorum, cum
                <lb/>
              eo quod fit ex minima quantitate, ſcilicet h in aggregatum ipſarum
                <lb/>
              quantitatum primi ordinis eſſe æquale quadratis,
                <expan abbr="quantitatũ">quantitatum</expan>
              eiuſ­
                <lb/>
              dem primi ordinis pariter acceptis. </s>
              <s id="id000402">Conſtat igitur, quod duplum i
                <lb/>
              in b eſt æquale duplo h in ipſum b, quia h & i ſunt æquales, & du­
                <lb/>
              plum k in ipſum c, eſt æquale quadruplo h in idem c, quia k eſt du­
                <lb/>
              pla h, & ſimiliter duplum l in ipſum d eſt æquale ſexcuplo, h in d,
                <lb/>
              quia l eſt tripla h, & ita procedendo erunt illa dupla producta æ­
                <lb/>
              qualia productis ex h in ipſas quantitates toties ſumptis quantus
                <lb/>
              eſt numerus, qui prouenit duplicato numero, ſecundum
                <expan abbr="quẽ">quem</expan>
              h con
                <lb/>
              tinetur in illo ſupplemento, exemplum uolo duplum producti lin
                <lb/>
              d bis, ſcio quòd ſupplementum l continet h ter, duplicabo tria & fi­
                <lb/>
              ent ſex,
                <expan abbr="igit̃">igitur</expan>
                <expan abbr="duplũ">duplum</expan>
              lin d æquale eſt ſexcuplo h in ipſum d. </s>
              <s id="id000403">Quo con­
                <lb/>
              ſtituto, cum ſuppoſitum ſit producta illa duplicata cum producto h
                <lb/>
              in aggregatum primarum
                <expan abbr="quãtitatum">quantitatum</expan>
              eſſe æqualia quadratis ipſa­
                <lb/>
              rum quantitatum, igitur addemus
                <expan abbr="productũ">productum</expan>
              ex h in ſingulas quan­
                <lb/>
              titates productis illis prioribus, & fiet productum h in a ſemel, in b
                <lb/>
              ter, in c quinquies, in d ſepties, in e nouies, in f undecies, in g trede­
                <lb/>
              cies, & in h quindecies æquale duplo producti uniuſcuiuſque quan­
                <lb/>
              titatis in ſuum ſupplementum cum producto h in
                <expan abbr="aggregatũ">aggregatum</expan>
              ipſa­
                <lb/>
              rum quantitatum, at quadratum a eſt ęquale producto ex h in eam,
                <lb/>
              quę talem habet proportionem ad ipſum a,
                <expan abbr="qualẽ">qualem</expan>
              habet a ad ipſum
                <lb/>
                <arrow.to.target n="marg68"/>
                <lb/>
              h per demonſtrata ab Euclide, & pariter de quadrato b, quod eſt ę­
                <lb/>
              quale ei quod fit ex h in eam quæ toties continet b, quotiens b con
                <lb/>
              tinet h, & ita quadratum c æquale eſt ei, quod continetur ſub h, &
                <lb/>
              habente proportionem ad b eandem, quam b ad h, & ſimiliter de
                <lb/>
              quadrato c & omnibus reliquis, uſque ad h ipſum. </s>
              <s id="id000404">Gratia ergo exem</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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