Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 1: Opera mechanica

Table of Notes

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            <s xml:id="echoid-s5091" xml:space="preserve">Rem ita ſe habere oſtendam, & </s>
            <s xml:id="echoid-s5092" xml:space="preserve">ut mutatam ejus objectio-
              <lb/>
            nem ſolvam, demonſtrabo principium, quod ponit, ve-
              <lb/>
            rum eſſe non poſſe. </s>
            <s xml:id="echoid-s5093" xml:space="preserve">Etiam falſum eſſe oſtendam alterum ejus
              <lb/>
            principium generale, quo utitur in ſuâ verâ ſolutione Mathe-
              <lb/>
            matica Problematis de centris Oſcillationis, & </s>
            <s xml:id="echoid-s5094" xml:space="preserve">tandem ambo
              <lb/>
            hæc principia ſibi mutuò contrariare: </s>
            <s xml:id="echoid-s5095" xml:space="preserve">non deſpero fore ut
              <lb/>
            ipſe Abbas Catelanus mecum conveniat, ſi ad ſequentia at-
              <lb/>
            tendat.</s>
            <s xml:id="echoid-s5096" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5097" xml:space="preserve">Quæſtio ſecundum illum ad hanc propoſitionem redit. </s>
            <s xml:id="echoid-s5098" xml:space="preserve">Si
              <lb/>
            habeamus duas magnitudines inæquales a a & </s>
            <s xml:id="echoid-s5099" xml:space="preserve">b b, & </s>
            <s xml:id="echoid-s5100" xml:space="preserve">ſummam
              <lb/>
            harum radicum dividamus in duas partes, quæ ſint inter ſe
              <lb/>
            ut a a ad b b, quæ partes ideo ſunt {a3+a a b/a a+b b} & </s>
            <s xml:id="echoid-s5101" xml:space="preserve">{b3+a b b/a a+b b},
              <lb/>
            ut facile per Algebram invenitur, demonſtrare, ſummam
              <lb/>
            magnitudinum a a & </s>
            <s xml:id="echoid-s5102" xml:space="preserve">b b, quæ repræſentant altitudines, un-
              <lb/>
            de deſcendunt pondera duo æqualia eidem Pendulo alligata,
              <lb/>
            non poſſe æquari ſummæ quadratorum partium {a3+a a b/a a + b b} & </s>
            <s xml:id="echoid-s5103" xml:space="preserve">
              <lb/>
            {b3+a b b/a a+b b}, quarum quadrata repræſentant altitudines, ad quas
              <lb/>
            pondera, percuſſione ſeparata, redeunt, niſi pars a a æque-
              <lb/>
            tur b b; </s>
            <s xml:id="echoid-s5104" xml:space="preserve">id eſt (quoniam quantitates in quæſtione propoſita
              <lb/>
            ſunt inæquales) niſi pars æqualis ſit toti.</s>
            <s xml:id="echoid-s5105" xml:space="preserve"/>
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            <s xml:id="echoid-s5106" xml:space="preserve">Hæc eſt propoſitio Abbatis Catelani, quam tantum cla-
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            rius exprimere conatus ſum, quâ demonſtratâ, ut facile fit
              <lb/>
            comparando duas illas ſummas per calculum Algebraicum,
              <lb/>
            contendit, fundamentalem meam de centris Oſcillationis
              <lb/>
            propoſitionem ruere.</s>
            <s xml:id="echoid-s5107" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5108" xml:space="preserve">Sed etiam relictâ Algebrâ demonſtrari poteſt illius propo-
              <lb/>
            ſitio; </s>
            <s xml:id="echoid-s5109" xml:space="preserve">nam ſi ponatur a a æquale eſſe 1; </s>
            <s xml:id="echoid-s5110" xml:space="preserve">& </s>
            <s xml:id="echoid-s5111" xml:space="preserve">b b æquale 4;
              <lb/>
            </s>
            <s xml:id="echoid-s5112" xml:space="preserve">ſumma radicum a + b eſt 3, & </s>
            <s xml:id="echoid-s5113" xml:space="preserve">partes proportionales hujus
              <lb/>
            ſummæ ſunt {3/5} & </s>
            <s xml:id="echoid-s5114" xml:space="preserve">{12/5}, faciunt enim junctim {15/5} hoc eſt 3. </s>
            <s xml:id="echoid-s5115" xml:space="preserve">Qua-
              <lb/>
            drata earundem partium ſunt {9/52} & </s>
            <s xml:id="echoid-s5116" xml:space="preserve">{144/25}. </s>
            <s xml:id="echoid-s5117" xml:space="preserve">Hoc igitur ſolum re-
              <lb/>
            ſtaret demonſtrandum, ſummam 1. </s>
            <s xml:id="echoid-s5118" xml:space="preserve">& </s>
            <s xml:id="echoid-s5119" xml:space="preserve">4. </s>
            <s xml:id="echoid-s5120" xml:space="preserve">non eſſe æqualem
              <lb/>
            ſummæ, quæ prodit ex additione {9/25} ad {144/25}, ſive 5 & </s>
            <s xml:id="echoid-s5121" xml:space="preserve">6{3/25} non
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            eſſe æqualia inter ſe; </s>
            <s xml:id="echoid-s5122" xml:space="preserve">quod ſane per ſe clarum eſt.</s>
            <s xml:id="echoid-s5123" xml:space="preserve"/>
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            <s xml:id="echoid-s5124" xml:space="preserve">Vera ergo eſſet Abbatis Propoſitio niſi affirmaret quadra-
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            ta partium {a3+a a b/a a + b b} & </s>
            <s xml:id="echoid-s5125" xml:space="preserve">{b3 + a b b/a a + b b}, quæ hic ſunt {9/25} & </s>
            <s xml:id="echoid-s5126" xml:space="preserve">{144/25}, </s>
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