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="id003243">
                <pb pagenum="187" xlink:href="015/01/206.jpg"/>
              autem aliquid ſuperſit, aberraſti. </s>
              <s id="id003244">Vt au­
                <lb/>
                <figure id="id.015.01.206.1.jpg" xlink:href="015/01/206/1.jpg" number="202"/>
                <arrow.to.target n="table26"/>
                <lb/>
              tem habeas numeros ſingulorum or­
                <lb/>
              dinum, in quauis multitudine, deduci­
                <lb/>
              to numerum ordinis à primo, & diui­
                <lb/>
              de per numerum ordinis ipſius reli­
                <lb/>
              quum, & illud quod prouenit, duci­
                <lb/>
              to in numerum maximum præceden­
                <lb/>
              tis ordinis, & habebis numerum quæ­
                <lb/>
              ſitum. </s>
              <s id="id003245">Velut ſi ſint undecim, uolo ſci­
                <lb/>
              re breuiter numeros, qui fiunt ex ua­
                <lb/>
              riatione trium. </s>
              <s id="id003246">Primum deduco pro
                <lb/>
              ſecundo ordine 1 ex 11 fit 10, diuido per
                <lb/>
              2 numerum ordinis, exit 5, duco in 11 fit
                <lb/>
              55 numerus ſecundi ordinis. </s>
              <s id="id003247">Inde detra
                <lb/>
              ho 2, qui eſt numerus differentiæ ordi­
                <lb/>
              nis tertij à primo ex 11, relinquitur 9, di­
                <lb/>
              uido 9 per 3
                <expan abbr="numerũ">numerum</expan>
              ordinis exit 3, du­
                <lb/>
              co 3 in 55 numerum ſecundi fit 165, nu­
                <lb/>
              merus tertij ordinis. </s>
              <s id="id003248">Similiter uolo nu
                <lb/>
              merum uariationum quatuor, deduco
                <lb/>
              3 differentiam 4 à primo ordine ab 11,
                <lb/>
              relinquitur 8. diuido 8 per 4 numerum ordinis, exit 2, duc 2 in 195
                <lb/>
              fit 330. numerus quarti ordinis. </s>
              <s id="id003249">Similiter pro quinto detraho 4 dif­
                <lb/>
              ferentiam à primo ordine, relinquitur 7, diuido per 5 numerum or­
                <lb/>
              dinis exit 1 2/5, duco in 330 numerum præcedentis ordinis, fit 462
                <lb/>
              numerus quinti ordinis.</s>
            </p>
            <table>
              <table.target id="table26"/>
              <row>
                <cell>1</cell>
                <cell>1</cell>
              </row>
              <row>
                <cell>2</cell>
                <cell>2</cell>
              </row>
              <row>
                <cell>3</cell>
                <cell>4</cell>
              </row>
              <row>
                <cell>4</cell>
                <cell>8</cell>
              </row>
              <row>
                <cell>5</cell>
                <cell>16</cell>
              </row>
              <row>
                <cell>6</cell>
                <cell>32</cell>
              </row>
              <row>
                <cell>7</cell>
                <cell>64</cell>
              </row>
              <row>
                <cell>8</cell>
                <cell>128</cell>
              </row>
              <row>
                <cell>9</cell>
                <cell>256</cell>
              </row>
              <row>
                <cell>10</cell>
                <cell>512</cell>
              </row>
              <row>
                <cell>11</cell>
                <cell>1024</cell>
              </row>
              <row>
                <cell>12</cell>
                <cell>2048</cell>
              </row>
              <row>
                <cell>13</cell>
                <cell>4096</cell>
              </row>
              <row>
                <cell>14</cell>
                <cell>8192</cell>
              </row>
              <row>
                <cell>15</cell>
                <cell>16384</cell>
              </row>
              <row>
                <cell>16</cell>
                <cell>32768</cell>
              </row>
              <row>
                <cell>17</cell>
                <cell>65536</cell>
              </row>
              <row>
                <cell>18</cell>
                <cell>131072</cell>
              </row>
              <row>
                <cell>19</cell>
                <cell>262144</cell>
              </row>
              <row>
                <cell>20</cell>
                <cell>524288</cell>
              </row>
              <row>
                <cell>21</cell>
                <cell>1048576</cell>
              </row>
              <row>
                <cell>22</cell>
                <cell>2097152</cell>
              </row>
              <row>
                <cell>23</cell>
                <cell>4194304</cell>
              </row>
              <row>
                <cell>24</cell>
                <cell>8388608</cell>
              </row>
              <row>
                <cell>25</cell>
                <cell>16777216</cell>
              </row>
            </table>
            <p type="main">
              <s id="id003250">Ex hoc colligitur manifeſtè modus conuertendi proportionem </s>
            </p>
            <p type="main">
              <s id="id003251">
                <arrow.to.target n="marg594"/>
                <lb/>
              arithmeticam in proportionem miſtam: dico miſtam, quia opor­
                <lb/>
              tet addere monadem in priore numero: dein de quia numerum
                <lb/>
              terminorum oportet ſumere iuxta numerum aſsignatum, ſcilicet
                <lb/>
              addita monade: demum, quia oportet detrahere monadem ipſam.
                <lb/>
              </s>
              <s id="id003252">Eſt tamen ſumpta à proportione Geometrica ut liquet, ſcilicet con­
                <lb/>
              tinua dupla.</s>
            </p>
            <p type="margin">
              <s id="id003253">
                <margin.target id="marg594"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}. 2.</s>
            </p>
            <p type="main">
              <s id="id003254">Propoſitio centeſima ſeptuageſima prima.</s>
            </p>
            <p type="main">
              <s id="id003255">Propoſitis duobus quibuslibet numeris, quotuis alios, ſeu in
                <lb/>
              continuum, ſeu medios in continua proportione arithmetica, geo­
                <lb/>
              metrica & muſica inuenire.</s>
            </p>
            <p type="main">
              <s id="id003256">Hæc tota propoſitio pendet ex intellectu diffinitionis earum.
                <lb/>
                <arrow.to.target n="marg595"/>
                <lb/>
              Sint ergo propoſiti duo numeri 2 & 3, & uelim tertium in conti­
                <lb/>
                <arrow.to.target n="marg596"/>
                <lb/>
              nua proportione arithmetica, duplico quemuis, ut pote 3 fit 6, </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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