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="id000323">
                <pb pagenum="15" xlink:href="015/01/034.jpg"/>
              tertiam, ad productum ex aggregato tertiæ & omiotatæ ad ſecun­
                <lb/>
              dam in ipſam quartam.</s>
            </p>
            <p type="main">
              <s id="id000324">Hæc magis reducit confuſam proportionem ad notitiam, quàm,
                <lb/>
              præcedens, quia reducit ad proportionem
                <expan abbr="productã">productam</expan>
              , quę operatio
                <lb/>
              eſt ſimpliciſsima, ſiue per multiplicationem quantitatum fiat, duæ
                <lb/>
              ſunt tantum multiplicationes, ſiue per eundem terminum ſufficit
                <lb/>
              alium addere. </s>
              <s id="id000325">Summatur ergo a b, c, d & e, & non ſit maior propor­
                <lb/>
              tio d ad e, quàm a b ad c, & ſtatuatur tunc prima a b, ſecunda c, ter­
                <lb/>
              tia d, quarta e, & poſtquam non eſt minor ratio a b ad c, quàm d ad
                <lb/>
              e, ſumatur a f ad c, ut d ad e. </s>
              <s id="id000326">licet enim hoc facere. </s>
              <s id="id000327">Dico quod pro­
                <lb/>
              portio confuſa a b & d ad c & e eſt uelut producti ex aggregato a b
                <lb/>
              & d in d ad productum ex aggregato a f & d in e. </s>
              <s id="id000328">Statuatur aggre­
                <lb/>
                <arrow.to.target n="marg49"/>
                <lb/>
              gatum a b & d linea a d prima quantitas, & aggregatum a f & d,
                <lb/>
                <figure id="id.015.01.034.1.jpg" xlink:href="015/01/034/1.jpg" number="27"/>
                <lb/>
              a d ſecunda quantitas, & d tertia,
                <lb/>
              & c quarta, & ex a b in d fiat g, ex
                <lb/>
              a d in e fiat h, erit ergo per pri­
                <lb/>
              mam propoſitionem g ad h pro­
                <lb/>
                <arrow.to.target n="marg50"/>
                <lb/>
              ducta ex proportionibus a b d ad
                <lb/>
              a f d, & d ad c. </s>
              <s id="id000329">Sed proportio a f d
                <lb/>
              ad aggregatum c e, eſt uelut d ad
                <lb/>
              e. </s>
              <s id="id000330">Proportio uerò a b d ad a f d, &
                <lb/>
              a f d ad e c producunt proportio­
                <lb/>
              nem a b d ad c & e per ſecundam propoſitionem, harum igitur con­
                <lb/>
              fuſa a b ad c, & d ad e, & eſt proportio a b d ad c & e, producuntur
                <lb/>
              ex proportionibus a b d ad a f d, & d ad e. </s>
              <s id="id000331">Ergo proportio g ad h
                <lb/>
              eſt confuſa ex a b ad e, & d ad e, quod erat demonſtrandum.</s>
            </p>
            <p type="margin">
              <s id="id000332">
                <margin.target id="marg49"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              10. P
                <emph type="italics"/>
              et.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000333">
                <margin.target id="marg50"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              13. P
                <emph type="italics"/>
              et.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id000334">Propoſitio decima ſeptima.</s>
            </p>
            <p type="main">
              <s id="id000335">Omnes duę proportiones conuerſæ producunt æqualem pro­
                <lb/>
              portionem.
                <lb/>
                <arrow.to.target n="table12"/>
              </s>
            </p>
            <table>
              <table.target id="table12"/>
              <row>
                <cell>a</cell>
              </row>
              <row>
                <cell>-----</cell>
              </row>
              <row>
                <cell>b</cell>
              </row>
              <row>
                <cell>---</cell>
              </row>
              <row>
                <cell>c</cell>
              </row>
              <row>
                <cell>----</cell>
              </row>
            </table>
            <p type="main">
              <s id="id000336">Sint duæ proportiones a ad b & b ad a conuerſa,
                <lb/>
                <figure id="id.015.01.034.2.jpg" xlink:href="015/01/034/2.jpg" number="28"/>
                <arrow.to.target n="marg51"/>
                <lb/>
              dico, quòd producunt proportionem æqualem. </s>
              <s id="id000337">fiat
                <lb/>
              enim b ad c, ut b ad a, erit igitur a æqualis c & b c con
                <lb/>
                <arrow.to.target n="marg52"/>
                <lb/>
              uerſa eius quæ eſt a ad b, ſed per ſecundam harum
                <lb/>
              proportiones a ad b, & b ad c producunt propor­
                <lb/>
              tionem a ad c, igitur proportiones etiam a ad b & b ad a produ­
                <lb/>
              cunt eandem.</s>
            </p>
            <p type="margin">
              <s id="id000338">
                <margin.target id="marg51"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="margin">
              <s id="id000339">
                <margin.target id="marg52"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              6. A
                <emph type="italics"/>
              ni­
                <lb/>
              mi
                <expan abbr="communẽ">communem</expan>
                <lb/>
              ſententiam.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id000340">Propoſitio decima octaua.</s>
            </p>
            <p type="main">
              <s id="id000341">Si fuerint quotlibet quantitates in continua proportione multi­
                <lb/>
              plici præter ultimam: proportio uerò penultimæ ad ultimam qua­
                <lb/>
              lis reſidui primæ ad ſecundam, erit primæ ad aggregatum reliqua­
                <lb/>
              rum uelut penultimæ ad </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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