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="id000341">
                <pb pagenum="16" xlink:href="015/01/035.jpg"/>
                <arrow.to.target n="marg53"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000342">
                <margin.target id="marg53"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id000343">Sint quantitates a b c d in continua proportione multiplici, ſed
                <lb/>
              d ad e ſit uelut reſidui a & b ad b, dico proportionem a ad b c d e
                <lb/>
              eſſe ut d ad e. </s>
              <s id="id000344">Quia enim eſt gnomonis e ad quadratum d, ut d ad e
                <lb/>
              ex ſuppoſito erit per coniunctam proportionem c & d ad d & e, ut</s>
            </p>
            <p type="main">
              <s id="id000345">
                <arrow.to.target n="marg54"/>
                <lb/>
              d ad e, ſed e gnomo cum quadrato d efficit qua­
                <lb/>
                <figure id="id.015.01.035.1.jpg" xlink:href="015/01/035/1.jpg" number="29"/>
                <lb/>
              dratum e, igitur ut c quadrati ad d & eiuncta, ita
                <lb/>
              d ad e. </s>
              <s id="id000346">Rurſus, quia b quadrati ad c quadratum,
                <lb/>
                <arrow.to.target n="marg55"/>
                <lb/>
              ut c ad d erit gnomonis b ad quadratum c, ut
                <lb/>
              gnomonis c ad quadratum d, & ita d ad e, igitur
                <lb/>
                <arrow.to.target n="marg56"/>
                <lb/>
              gnomonum b c cum quadrato d ad aggrega­
                <lb/>
              tum c d e quadratorum, ut d ad e, ſed c gno­
                <lb/>
              mo cum d quadrato perficit c quadratum,
                <lb/>
              & c quadratum cum gnomone b perficit
                <lb/>
              quadratum b, igitur proportio quadrati b
                <lb/>
              ad quadrata c d e, ut d quadrati a d e. </s>
              <s id="id000347">Et ita
                <lb/>
              repetendo de quotuis quantitatibus in infi
                <lb/>
              nitum uſque. </s>
              <s id="id000348">Hæc proponitur ab Archimede in libro de quadrato
                <lb/>
              æquali parabolæ, & minus generaliter & pluribus demonſtratur.
                <lb/>
              </s>
              <s id="id000349">Ego tamen quia eſt generalis, deſcribam illam per corrolarium: ad­
                <lb/>
              damque aliud quod ex hoc ſequitur.
                <lb/>
                <arrow.to.target n="marg57"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000350">
                <margin.target id="marg54"/>
              13. P
                <emph type="italics"/>
              ropoſ.
                <lb/>
              quinti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000351">
                <margin.target id="marg55"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              19.
                <emph type="italics"/>
              quin
                <lb/>
              ti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000352">
                <margin.target id="marg56"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              12.
                <emph type="italics"/>
              quin
                <lb/>
              ti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lement.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000353">
                <margin.target id="marg57"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}. 1.</s>
            </p>
            <p type="main">
              <s id="id000354">Si fuerint quotlibet
                <expan abbr="quãtitates">quantitates</expan>
              omnes analogæ præter ultimam,
                <lb/>
              ſit autem penultima ad ultimam qualis reſidui primæ & ſecundæ
                <lb/>
              ad ſecundam, erit proportio primæ ad aggregatum omnium alia­
                <lb/>
              rum ueluti penultimæ ad ultimam.</s>
            </p>
            <p type="main">
              <s id="id000355">
                <arrow.to.target n="marg58"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000356">
                <margin.target id="marg58"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id000357">Hæc enim eſt euidens, quia conuenit ei demonſtratio propoſita.
                <lb/>
                <figure id="id.015.01.035.2.jpg" xlink:href="015/01/035/2.jpg" number="30"/>
                <lb/>
              exemplo autem in numeris à latere
                <lb/>
              poſito uides declarationem. </s>
              <s id="id000358">nam
                <lb/>
              proportio 16 ad 32 eſt uelut 27 reſi
                <lb/>
              dui primæ & ſecundæ ad ipſam ſe­
                <lb/>
              cundam ſcilicet ad 54.</s>
            </p>
            <p type="main">
              <s id="id000359">
                <arrow.to.target n="marg59"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000360">
                <margin.target id="marg59"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}. 2.</s>
            </p>
            <p type="main">
              <s id="id000361">Ex hoc patet etiam quòd aſſumptis omnibus, ſub multiplicibus
                <lb/>
              analogiæ uſque in infinitum prima quantitas eſt multiplex aggre­
                <lb/>
              gati omnium reliquarum numero 1 m: quo prima eſt multiplex
                <lb/>
              ſecundæ.</s>
            </p>
            <p type="main">
              <s id="id000362">
                <arrow.to.target n="marg60"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000363">
                <margin.target id="marg60"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}. 3.</s>
            </p>
            <p type="main">
              <s id="id000364">Si fuerint quotlibet quantitates in ſuper particulari proportio­
                <lb/>
              ne analogæ, erit proportio primæ ad aggregatum omnium in infi­
                <lb/>
              nitum iuxta proportionem multiplicem conuerſam illius partis.</s>
            </p>
            <p type="main">
              <s id="id000365">
                <arrow.to.target n="marg61"/>
              </s>
            </p>
            <p type="margin">
              <s id="id000366">
                <margin.target id="marg61"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id000367">Velut collectæ in ſeſquialtera duplæ in ſexquitertia triplæ in
                <lb/>
              ſexquiſeptima ſeptuplæ. </s>
              <s id="id000368">Vt capio 512 448 392 343, & ita deinceps
                <lb/>
              uſque in infinitum aggregatum omnium earum erit 3584. </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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