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="id000242">
                <pb pagenum="11" xlink:href="015/01/030.jpg"/>
              portionem productam inter ſe fuerint æquales reducetur hæc pro­
                <lb/>
              portio ad quatuor quantitates omologas, ſciliter abiectis amba­
                <lb/>
              bus æqualibus. </s>
              <s id="id000243">Sit gratia exempli prima æqualis quintæ: & quia
                <lb/>
              in octauo modo proportio
                <expan abbr="ſecũdi">ſecundi</expan>
              ad quartum producitur ex pro­
                <lb/>
              portione primi ad quintum, & ſexti ad tertium, ergo per expoſita
                <lb/>
              proportio ſecundi ad quartum, ut ſexti ad tertium, & ita permutan­
                <lb/>
              do, & conuertendo ſecundi ad ſextum, ut quarti ad tertium, & tertij </s>
            </p>
            <p type="main">
              <s id="id000244">
                <arrow.to.target n="marg26"/>
                <lb/>
              ad quartum, ut ſexti ad ſecundum.</s>
            </p>
            <p type="margin">
              <s id="id000245">
                <margin.target id="marg26"/>
              V
                <emph type="italics"/>
              ndecima
                <lb/>
              petitione.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id000246">Propoſitio octaua.</s>
            </p>
            <p type="main">
              <s id="id000247">Si duarum
                <expan abbr="proportionũ">proportionum</expan>
              ſuperiores numeri alternatim cum infe
                <lb/>
              rioribus multiplicentur, atque coniungantur: erit proportio aggre­
                <lb/>
              gati ad productum ex inferioribus inuicem proportio ex primis
                <lb/>
              proportionibus compoſita.</s>
            </p>
            <figure id="id.015.01.030.1.jpg" xlink:href="015/01/030/1.jpg" number="17"/>
            <p type="main">
              <s id="id000248">Sit proportio una a ad b, alia c ad d, ducatur b in
                <lb/>
                <arrow.to.target n="marg27"/>
                <lb/>
              c, fiatque e & a in d, & fiat f, iunganturque e & f & fiat h,
                <lb/>
              & ducatur b in d et fiat g: dico
                <expan abbr="proportionẽ">proportionem</expan>
              h g com­
                <lb/>
              poſitam eſſe ex proportione a ad b, & c ad d. </s>
              <s id="id000249">Quia
                <lb/>
                <arrow.to.target n="marg28"/>
                <lb/>
              enim ex b in c fit e, & ex b in d fit g, erit proportio e
                <lb/>
              ad g, ut c ad d, & ſimiliter, quia ex d in a fit f, & ex d in b fit g, erit f ad
                <lb/>
              g ut a ad b. </s>
              <s id="id000250">Sed e & f componunt h, igitur proportio h ad g eſt com
                <lb/>
              poſita ex proportionibus e & f ad g, igitur per communem animi
                <lb/>
              ſententiam, & diffinitionem compoſitæ proportionis, proportio h
                <lb/>
                <arrow.to.target n="marg29"/>
                <lb/>
              ad g compoſita eſt ex proportionibus a ad b, & c ad d, quod eſt
                <lb/>
              propoſitum.</s>
            </p>
            <p type="margin">
              <s id="id000251">
                <margin.target id="marg27"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="margin">
              <s id="id000252">
                <margin.target id="marg28"/>
              E
                <emph type="italics"/>
              x
                <emph.end type="italics"/>
              13
                <emph type="italics"/>
              peti­
                <lb/>
              tione.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000253">
                <margin.target id="marg29"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              14
                <emph type="italics"/>
              diffi
                <lb/>
              nitionem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id000254">Propoſitio nona.</s>
            </p>
            <p type="main">
              <s id="id000255">Si duarum proportionum ſuperiores numeri alternatim cum
                <lb/>
              inferioribus multiplicentur, minusque productum ex maiore detra­
                <lb/>
              hatur, erit reſidui ad productum ex inferioribus proportio uelut
                <lb/>
              illa, quæ relinquitur detracta minore proportione ex maiore.</s>
            </p>
            <p type="main">
              <s id="id000256">Hæc eodem modo probatur, ut præcedens, niſi quod h fit de­
                <lb/>
                <arrow.to.target n="marg30"/>
                <lb/>
              tracto è minore: gratia exempli ex f, & ita ex diffinitione patet pro­
                <lb/>
              poſitum.</s>
            </p>
            <p type="margin">
              <s id="id000257">
                <margin.target id="marg30"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              _{m}.
                <lb/>
              152.</s>
            </p>
            <p type="main">
              <s id="id000258">Propoſitio decima.</s>
            </p>
            <p type="main">
              <s id="id000259">Si fuerit alicuius quantitatis ad unam partem proportio uelut al
                <lb/>
              terius partis ad
                <expan abbr="ſecũdam">ſecundam</expan>
              quantitatem erit proportio cuiuſuis quan
                <lb/>
              titatis eiuſdem generis ad ſecundam compoſita proportio ex pro­
                <lb/>
              portionibus eiuſdem quantitatis aſſumptæ ad utran que partem pri­
                <lb/>
              mæ quantitatis ſeorſum.
                <lb/>
                <arrow.to.target n="marg31"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000260">
                <margin.target id="marg31"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <figure id="id.015.01.030.2.jpg" xlink:href="015/01/030/2.jpg" number="18"/>
            <p type="main">
              <s id="id000261">Sit a b quantitas diuiſa in c, & ſi c ut a b ad a c,
                <lb/>
              ita b c ad d: eritque iterum permutando a b ad b c,
                <lb/>
              ut a c ad d, & ſumatur quædam quantitas e </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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