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="id001178">
                <pb pagenum="62" xlink:href="015/01/081.jpg"/>
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                <lb/>
              miles, ſed unam perſæpe in utriſ que eſſe uult. </s>
              <s id="id001179">Sed & hoc Archime­
                <lb/>
              des dicere uidetur: lineæ ductæ à uertice coniſcaleni ad perpendi­
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              culum ſuper baſes ſingulas omnium triangulorum per axem coni
                <lb/>
              tranſeuntium in peripheriam unius circuli cadunt.</s>
            </p>
            <p type="margin">
              <s id="id001180">
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              8</s>
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            <p type="margin">
              <s id="id001181">
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              9</s>
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              10</s>
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              11</s>
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              <s id="id001184">
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              12</s>
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              13</s>
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              <s id="id001186">
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              14</s>
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              <s id="id001187">
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              15</s>
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              <s id="id001188">
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              16</s>
            </p>
            <p type="margin">
              <s id="id001189">
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              17</s>
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            <p type="margin">
              <s id="id001190">
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              18</s>
            </p>
            <p type="main">
              <s id="id001191">Propoſitio ſeptuageſima.</s>
            </p>
            <p type="main">
              <s id="id001192">Si fuerint tres quantitates in continua proportione, aliæque toti­
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              dem in continua proportione, poterunt conſtituere tres quantita­
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              tes in æquali differentia peruerſim copulatæ.
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              </s>
            </p>
            <p type="margin">
              <s id="id001193">
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              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              m.</s>
            </p>
            <p type="main">
              <s id="id001194">Velut ſint a b c primi ordi­
                <lb/>
                <figure id="id.015.01.081.1.jpg" xlink:href="015/01/081/1.jpg" number="77"/>
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              nis, & d ef ſecundi, & ſit 28, </s>
            </p>
            <p type="main">
              <s id="id001195">
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                <lb/>
              b 4, c 2, & d 2 1/4, e 1 1/2, f 1, tunc
                <lb/>
              iunctis a & e fit 9 1/2, & b & d b
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              1/4, & e cum f 3, at 3 & 6 1/4 & 9 1/2
                <lb/>
              æqualiter diſtant, nam diffe­
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              rentia eſt 3 1/4. At ſi iungatur
                <lb/>
              cum e, & b cum f, & c cum d
                <lb/>
              idem poterit contingere: ut in
                <lb/>
              figura uides, nam a e eſt 8 1/2,
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              p: <02> 1 1/4, & b f 7, & c d 5 1/2, m: <02> 1 1/4, & differentia b f ab utro que com­
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              poſito, eſt 1 1/2 p: <02> 1 1/4, qua excedit & exceditur. </s>
              <s id="id001196">Dico modo, quaſi
                <lb/>
              ex ordine coniungantur qualeſcunque proportiones fuerint, modo
                <lb/>
              non ſint ambæ æqualitatis 1, ut b iungatur cum c, & reliquæ ut li­
                <lb/>
              bet, uelut a cum d, & c cum f, uel a cum f, & e cum d, nunquam fient
                <lb/>
                <arrow.to.target n="marg251"/>
                <lb/>
              æquales exceſſus, nam de primo eſt clarum: nam ſi a cum d iun­
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              gatur, & ambæ fuerint maximæ, maior eſt differentia a ad b, quàm
                <lb/>
              b ad c, & maior etiam d ad e quàm e ad f, ideo maior erit differentia
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              a & d ad b e quàm b e ad c f, quod erat probandum. </s>
              <s id="id001197">Eodem modo
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              ſed laborioſius demonſtratur reliquus modus ſcilicet, quod con­
                <lb/>
              iunctio a f ad b e eſt maior aut minor quàm b e ad c d, ex hoc ſe­
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              quuntur corrolaria.</s>
            </p>
            <p type="margin">
              <s id="id001198">
                <margin.target id="marg250"/>
              16</s>
            </p>
            <p type="margin">
              <s id="id001199">
                <margin.target id="marg251"/>
              17</s>
            </p>
            <p type="main">
              <s id="id001200">Primum, tres æquales quantitates non poſſunt diuidi in tres, &
                <lb/>
              tres quantitates in continua proportione ordinatè, ut dixi, niſi u­
                <lb/>
              triuſque ordinis tres, ac tres inuicem ſint æquales.</s>
            </p>
            <p type="main">
              <s id="id001201">Secundum, tres quantitates in æquali exceſſu ordinate, ut dixi,
                <lb/>
              non poſſunt diuidi in tres, & tres quantitates, quæ ſint in eadem
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              proportione quantumcunque proportiones illæ duorum ordinum
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              fint diuerſæ.</s>
            </p>
            <p type="main">
              <s id="id001202">Tertium, tres quantitates, quæ ſint in eadem proportione non
                <lb/>
              poſſunt diuidi ordinate in tres ac tres, quæ ſint in continua propor
                <lb/>
              tione niſi ſint ambæ proportiones eædem cum proportione ipſa­
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              rum quantitatum.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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