Schott, Gaspar, Mechanica hydraulico-pneumatica. Pars I. Mechanicae Hydraulico-pnevmaticae Theoriam continet. , 1657

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            <p type="main">
              <s>
                <pb xlink:href="051/01/149.jpg" pagenum="118"/>
              quemadmodum ratio quæcunque triplicata, quadruplicata, &c.
                <lb/>
              eſt ratio quæcunque ſimplex bis, ter &c. repetita, ſeu ter,
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              quater continuè ſumpta. </s>
              <s>Exemplum. </s>
              <s>Inter 2 & 1 reperitur
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              ratio dupla; hæc ratio ſi ſemel repetatur, ſeu adhuc ſemel
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              accipiatur, hoc eſt, ſi bis continuè ſumatur hoc modo, 4,
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              2, 1; erit inter 4 & 1 ratio ſeu proportio duplicata illius pro­
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              portionis, quæ eſt inter 2 & 1, quandoquidem inter 4 & 1
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              reperitur ratio dupla ſemel repetita, ſeu bis continuè
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              ſumpta, hoc eſt, duplicata, ſcilicet ſemel inter 4 & 2, & ite­
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              rum inter 2 & 1. </s>
              <s>Similiter inter 8 & 1 eſt ratio triplicata il­
                <lb/>
              lius, quæ eſt inter 2 & 1, quia inter 8 & 1, intercedit ter
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              ratio dupla, nempe 8 ad 4, 4 ad 2, 2 ad 1. </s>
              <s>Sic 16 ad 1 ha­
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              bet rationem quadruplicatam, & 32 ad 1 rationem quintu­
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              plicatam rationis illius, quam habet 2 ad 1. </s>
              <s>Aliud exemplum. </s>
              <s>
                <lb/>
              Inter 6 ad 4 reperitur ratio ſeſquialtera ſimplex; hæc ratio
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              duplicatur, ſi adhuc ſemel repetatur, ſeu ſi bis continuè ſuma­
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              tur, ut apparet in his numeris 9, 6, 4: nam quia ut 6 ad 4,
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              ita 9 ad 6; ideo inter 9 & 4 bis reperitur ratio ſeſquialtera. </s>
              <s>
                <lb/>
              Si verò eadem ratio ſeſquialtera bis repetatur, ſeu ter conti­
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              nuè ponatur; erit inter extremos terminos ratio ſeſqui altera
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              triplicata, ut apparet in his numeris, 13 1/4, 9, 6, 4; quam pro­
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              portionem abſque fractione habebis, ſi duplicaveris hoſce
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              numeros ſic, 27, 18, 12, 8: nam ut 12 continet 8 ſemel cum
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              dimidio, ita 18 continet 12 ſemel cum dimidio, & 27 etiam
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              continet 18 ſemel cum dimidio.
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                <arrow.to.target n="marg215"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg214"/>
                <emph type="italics"/>
              Duplicata
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              proportio
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              quæ.
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              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg215"/>
                <emph type="italics"/>
              Subduplica­
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              ta proportio
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              quæ.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Ex ratione duplicata, triplicata, quadruplicata, &c. facilè
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              intelligitur ratio ſubduplicata, ſubtriplicata, ſubquadruplica­
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              ta, &c. </s>
              <s>Nam per rationem ſubduplicatam intelligimus
                <expan abbr="dimidiũ">dimidium</expan>
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              rationis duplicatæ. </s>
              <s>Verbi gratia, 4 ad 1 habet rationem dupli­
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              catam rationis duplæ; 2 ad 1, aut 4 ad 2, conſtituunt dimidium
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              rationis 4 ad 1; ideo 2 ad 1, & 4 ad 2, habent rationem ſub­
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              duplicatam. </s>
              <s>Similiter 9 ad 4 habet rationem duplicatam
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              rationis ſeſquialteræ; dimidium talis rationis eſt 9 ad 6, vel
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              6 ad 4; ideo 9 ad 6, & 6 ad 4 habent rationem ſubduplica­
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              tam prædictæ rationis ſeſquialteræ. </s>
            </p>
          </chap>
        </body>
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