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="id003305">
                <pb pagenum="186 [=192]" xlink:href="015/01/211.jpg"/>
              rentiæ ſecundæ à tertia ad 1 differentiam ſecundæ à prima. </s>
              <s id="id003306">Manife­
                <lb/>
              ſtum eſt autem quod in uera harmonica proportio differentiarum
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              eſt primæ & ſecundæ ad illam quæ ſecundæ & tertiæ.</s>
            </p>
            <p type="margin">
              <s id="id003307">
                <margin.target id="marg609"/>
              1 2</s>
            </p>
            <p type="margin">
              <s id="id003308">
                <margin.target id="marg610"/>
              6 5 3</s>
            </p>
            <p type="main">
              <s id="id003309">Secunda notha harmonica eſt, ut ſit propor­
                <lb/>
                <figure id="id.015.01.211.1.jpg" xlink:href="015/01/211/1.jpg" number="203"/>
                <lb/>
              tio primæ ad tertiam, uelut differentiæ primæ à
                <lb/>
              tertia ad differentiam ſecundæ à tertia, ponatur
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              25, prima 21, ſecunda 15, tertia proportio 25 ad 15
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              eſt uelut 10 differentiæ primę à tertia ad b differen
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              tiam ſecundæ à tertia.</s>
            </p>
            <p type="main">
              <s id="id003310">Tertia eſt ſimilis priori, niſi quod ſumitur dif­
                <lb/>
                <figure id="id.015.01.211.2.jpg" xlink:href="015/01/211/2.jpg" number="204"/>
                <lb/>
              ferentia primæ à ſecunda pro ultimo termino. </s>
              <s id="id003311">Ex­
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              emplum, 25 primus terminus, 19 ſecundus, 15 ter­
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              tius, proportio 25 ad 15 eſt uelut 10 differentiæ pri­
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              mæ a tertia ad b, differentiam primæ à ſecunda.
                <lb/>
              </s>
              <s id="id003312">Has proportiones quanquàm exiguæ utilitatis, proponere uo­
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              lui, ut excogitatis aliquibus demonſtrationibus, uelut ſuperius
                <lb/>
              diximus, pulchra theoremata & problemata tradi poſſent.</s>
            </p>
            <p type="main">
              <s id="id003313">Propoſitio centeſima ſeptuageſima tertia.</s>
            </p>
            <p type="main">
              <s id="id003314">Circulum ſuper centro ſuo mouere æqualiter, ita quòd omnia
                <lb/>
              illius puncta per rectam lineam moueantur ultro citro que.
                <lb/>
                <arrow.to.target n="marg611"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003315">
                <margin.target id="marg611"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003316">Sit a centrum circuli b c, & æqualis ei
                <lb/>
                <figure id="id.015.01.211.3.jpg" xlink:href="015/01/211/3.jpg" number="205"/>
                <lb/>
              circulus d e, centrum eius b in circumfe­
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              rentia circuli b c, fixum ita ut ibi mouea­
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              tur ad motum circuli b c: & moueatur b
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              uerſus c æqualiter, & e contrario motu
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              etiam regulariter, & duplo uelocius ex e
                <lb/>
              uerſus d, dico omnia puncta d e moue­
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              ri in linea recta, & primum capio pun­
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              ctum d, quod ſit in linea recta centro­
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              rum: & moueatur b ad c, & ſi circulus d e
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              eſſet immobilis, palam eſt quòd pun­
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              ctum d cum ſit in una linea a b, cum b
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              perueniret in c, d eſſet in linea a c, putà in
                <lb/>
              h ſecundum quantitatem, ergo b d ex </s>
            </p>
            <p type="main">
              <s id="id003317">
                <arrow.to.target n="marg612"/>
                <lb/>
              centro c, deſcribo circuli portionem h k,
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              duco etiam c k, erit ergo angulus h c k
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              duplus a, quare arcus h k duplus b c,
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              nam conſiſtunt in centris circulorum æ­
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              qualium: igitur cum ex h motu conuerſo, & duplo ueloci in codem
                <lb/>
              tempore feratur d perueniet in k, & ita ſecundum rectam lineam
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              erit motum eadem ratione ex d in k, quod erat demonſtrandum.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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