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="242" xlink:href="015/01/261.jpg"/>
            <p type="margin">
              <s id="id004129">
                <margin.target id="marg814"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <table>
              <table.target id="table30"/>
              <row>
                <cell>1 a f in a h</cell>
                <cell>f c in a h bis</cell>
              </row>
              <row>
                <cell>2 a f in h d</cell>
                <cell>f e in d k</cell>
              </row>
              <row>
                <cell>3 a f in d k</cell>
                <cell/>
              </row>
              <row>
                <cell>4 f c in d k</cell>
                <cell/>
              </row>
              <row>
                <cell>5 c e in d k</cell>
                <cell/>
              </row>
              <row>
                <cell>1 a f in a h</cell>
                <cell>4 f c in d k</cell>
              </row>
              <row>
                <cell>2 a f in d h</cell>
                <cell>5 c e in d k</cell>
              </row>
              <row>
                <cell>3 a f in d k</cell>
                <cell/>
              </row>
            </table>
            <p type="head">
              <s id="id004130">SCHOLIVM.</s>
            </p>
            <p type="main">
              <s id="id004131">Dico etiam, quòd duæ lineæ b e & af ſunt minores duabus a c,
                <lb/>
              c b ſimul iunctis, nam quia d b, e b, c b, ſunt in eadem proportione,
                <lb/>
              & d b eſt maior e b, erit maior differentia d b ad e b, quam e b ad
                <lb/>
                <arrow.to.target n="marg815"/>
                <lb/>
              c b, igitur maior d e quam e c, quare e c eſt minor medietate d c, &
                <lb/>
              ideo multo minor medietate a c. </s>
              <s id="id004132">Et ſimiliter, quia a c eſt maior af, &
                <lb/>
              a c, a f, a d ſunt in continua proportione, maior erit c f quam
                <lb/>
              fd, & ideò conſtat quamuis longum eſſet, ſi quis uellet demon­
                <lb/>
              ſtrare perfectè, quod b e & a f iunctæ ſunt minores tota a b ſeu du­
                <lb/>
              plo a c.</s>
            </p>
            <p type="margin">
              <s id="id004133">
                <margin.target id="marg815"/>
              P
                <emph type="italics"/>
              er conuer­
                <lb/>
              ſam quaſi
                <emph.end type="italics"/>
              8.
                <lb/>
                <emph type="italics"/>
              quinti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id004134">Exemplum, ſint h b & h a 25, & a e, c b 5, producta mutua 250,
                <lb/>
              ſitqúe g d 49, & erit b e 7, ſit autem d k 1, & erit a f 1, quia ergo a f
                <lb/>
              eſt 1, a e 5, erit f c 4, & quia e b eſt 7, & b c 5, erit e c 2, quare etiam ef2,
                <lb/>
              productum ergo ex e b in d k eſt 7, & ex a f in d g 49, totum ag­
                <lb/>
              gregatum 56, differentia a 250, eſt 194, qui ſit ex duplo fc, quod
                <lb/>
              eſt 8 in d h, quæ eſt 24, & fit 192, & ex fe, quæ eſt 2, in d k, quæ eſt 1,
                <lb/>
              & fit: quod additum ad 192 facit 194. Similiter capio 450, cuius di­
                <lb/>
              midium eſt 225, c g & c k 225, & c a & c b 15 ſingulæ. </s>
              <s id="id004135">Et ponatur
                <lb/>
              d g 441, eritqúe e b 21, & d k 9, & erit a f 3, igitur cum b e ſit 21,
                <lb/>
              & b c 15, erit c e 6, a f uerò eſt 3, igitur f e eſt 6. Producta mu­
                <lb/>
              tua æqualia 6750, inæqualia 1521, differentia 5238, quia er­
                <lb/>
              go f c eſt 12, duplum eius eſt 24, ductum in d h, quæ eſt
                <lb/>
              216, nam d k ex ſuppoſito eſt 9, fiet ergo 5184, cui ſi addam, quod
                <lb/>
              fit ex f e, quæ eſt 6, in d k, quæ eſt 9, fitqúe 54, erit totum 5238, quod
                <lb/>
              erat propoſitum.
                <lb/>
                <arrow.to.target n="marg816"/>
              </s>
            </p>
            <p type="margin">
              <s id="id004136">
                <margin.target id="marg816"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id004137">Ex hac demonſtratione liquet, quod ſi linea in duas partes æ­
                <lb/>
              quales diuidatur, & duas inæquales, quòd parallelipeda æqua­
                <lb/>
              lium ſectionum pariter accepta excedent parallelipeda inæqua­
                <lb/>
              lium ſectionum, ſimul iuncta in eo quod fit ex tota linea in quadra­
                <lb/>
              tum differentiæ partium æqualium ab inæ qualibus.</s>
            </p>
            <p type="main">
              <s id="id004138">Propoſitio ducenteſima decima.</s>
            </p>
            <p type="main">
              <s id="id004139">Si duæ lineæ ad æquales angulos ab eodem puncto peripheriæ
                <lb/>
              circuli reflectantur, neceſſe eſt angulos cum dimetiente factos æ­
                <lb/>
              quales eſſe. </s>
              <s id="id004140">Vnde manifeſtum eſt protractam diametrum angu­
                <lb/>
              lum ſuppoſitum per æqualia diuidere.</s>
            </p>
            <p type="main">
              <s id="id004141">
                <arrow.to.target n="marg817"/>
              </s>
            </p>
            <p type="margin">
              <s id="id004142">
                <margin.target id="marg817"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id004143">Reſiliat radius d b c ad æquales angulos, ut fert natura rerum </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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