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="id000404">
                <pb pagenum="20" xlink:href="015/01/039.jpg"/>
              pli quadratum a, erit æquale producto ex h in omnes quantitates ſe­
                <lb/>
              cundas, quia quotus eſt numerus quantitatum, totus eſt numerus
                <lb/>
              ſecundum quem a continet h, & ſimiliter quotus eſt numerus quan
                <lb/>
              títatum incipiendo à b, & quotus eſt numerus quantitatum incipi­
                <lb/>
              endo à c, toties b uel c
                <expan abbr="continẽt">continent</expan>
              h, & ita de alijs, quadrata ergo om­
                <lb/>
              nium quantitatum ſimul iuncta ſunt æqualia productis ex h in ſin­
                <lb/>
              gulas illarum toties ſumptis, quoties illæ
                <expan abbr="cõtinent">continent</expan>
              h, ſeu quotus eſt
                <lb/>
              numerus illius quantitatis, incipiendo ab h, &
                <expan abbr="numerãdo">numerando</expan>
              uerſus a.
                <lb/>
              </s>
              <s id="id000405">Rurſus dico, quod productum multiplicis cuiuslibet
                <expan abbr="quãtitatis">quantitatis</expan>
              in
                <lb/>
              minimam, ſeu quadratum eiuſdem quantitatis ęquale eſt producto
                <lb/>
              eiuſdem quantitatis, & dupli omnium ſequentium primi ordinis in
                <lb/>
              ipſam minimam quantitatem, uelut quadratum a eſt æquale produ
                <lb/>
              cto ex h in a, & in duplum b c d e f g h, hoc
                <expan abbr="autẽ">autem</expan>
              facile eſt probare in
                <lb/>
              his quantitatibus, quia ſi quadratum a eſt æquale producto h in o­
                <lb/>
              mnes quantitates ſecundi ordinis, & omnes quantitates ſecundi or
                <lb/>
              dinis ſimul ſumptæ ſunt ęquales ipſi a, & duplo
                <expan abbr="reliquarũ">reliquarum</expan>
              primi or
                <lb/>
              dinis, quia tales quantitates ſunt æquales ſuis ſupplementis uiciſ­
                <lb/>
              ſim, ut h cum i, k cum g, f cum l, e
                <expan abbr="cũ">cum</expan>
              m, ergo tam ſupplementa, quàm
                <lb/>
              quantitates primi ordinis ſunt dimidium quantitatum ſecundi or­
                <lb/>
              dinis, ergo duplum quantitatum primi ordinis eſt dimidium quan
                <lb/>
              titatum ſecundi ordinis, uerùm de b dico idem accidere, quia qua­
                <lb/>
              dratum b eſt ęquale producto ex h in b, & in duplum reliquarum à
                <lb/>
              b, ſcilicet duplum c d e f g h, & hoc eſt oſtendere, quod iſtę quantita
                <lb/>
              tes ſunt dimidium totidem quantitatum æqualium b, nam c eſt mi­
                <lb/>
              nor b in h, & ſupplementum p quod eſt æquale ipſi b, ſi tota h p fiat
                <lb/>
              æqualis ipſi b, ut pote h q erit ipſa q dempta h æqualis ipſi c, ergo
                <lb/>
              quantitates primi ordinis ſemper ſunt æquales ſupplementis non
                <lb/>
              ueris, ſed prioris quantitatis aſſumptæ, ſeu in comparatione ad il­
                <lb/>
              lam, quadratum igitur b eſt æquale producto ex h in b, & in duplum
                <lb/>
              c d e f g h, & ſimiliter per eadem, quadratum c eſt æquale producto
                <lb/>
              ex h in c, & in duplum d e f g h, & ſic de alijs. </s>
              <s id="id000406">Habemus ergo, quod
                <lb/>
              quadrata a b c d e f g h ſimul iuncta ſunt æqualia producto ex h in
                <lb/>
              a, & in duplum reliquarum, & ex h in b, & in duplum reliquarum
                <lb/>
              ſequentium, & producto ex h in c ſemel, & in duplum ſequentium
                <lb/>
              uſque ad h, & ita de reliquis. </s>
              <s id="id000407">hoc enim eſt, quod nuper demonſtraui­
                <lb/>
              mus. </s>
              <s id="id000408">Antea quo que
                <expan abbr="demõſtratum">demonſtratum</expan>
              eſt, quod duplum b in i, c in k, d in
                <lb/>
              l, e in m, f in n, g in o, h in p,
                <expan abbr="cũ">cum</expan>
              producto h in
                <expan abbr="aggregatũ">aggregatum</expan>
              a b c d e f g h
                <lb/>
              erat ęquale productis ex h in a ſemel, & in b ter, & in c quinquies, in
                <lb/>
              d ſepties, in e nouies, in fundecies, in g tredecies, in ſe ipſam h quin­
                <lb/>
              decies, detractis ergo p
                <expan abbr="ordinẽ">ordinem</expan>
              , q̊d fit ex h in a ab utro que aggregato,
                <lb/>
              & ex h in b c d e f g h bis
                <expan abbr="relinquet̃">relinquetur</expan>
              ex una parte, quae fit ex h in b ſemel </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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