Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <pb xlink:href="039/01/313.jpg" pagenum="285"/>
                <p type="main">
                  <s>Deſignet jam V velocitatem maximam in oſcillatione quavis,
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                    <arrow.to.target n="note261"/>
                  ſintque A, B, C quantitates datæ, & fingamus quod differentia
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                  arcuum ſit AV+BV 1/2+CV
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                  2
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                  . </s>
                  <s>Cum velocitates maximæ ſint in
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                  Cycloide ut ſemiſſes arcuum oſcillando deſcriptorum, in Circu­
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                  lo vero ut ſemiſſium arcuum illorum chordæ; adeoque paribus
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                  arcubus majores ſint in Cycloide quam in Circulo, in ratione
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                  ſemiſſium arcuum ad eorundem chordas; tempora autem in Cir­
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                  culo ſint majora quam in Cycloide in velocitatis ratione reci­
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                  proca; patet arcuum differentias (quæ ſunt ut reſiſtentia & qua­
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                  dratum temporis conjunctim) eaſdem fore, quamproxime, in utra­
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                  que Curva: deberent enim differentiæ illæ in Cycloide augeri, una
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                  cum reſiſtentia, in duplicata circiter ratione arcus ad chordam, ob
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                  velocitatem in ratione illa ſimplici auctam; & diminui, una cum
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                  quadrato temporis, in eadem duplicata ratione. </s>
                  <s>Itaque ut reductio
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                  fiat ad Cycloidem, eædem ſumendæ ſunt arcuum differentiæ quæ
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                  fuerunt in Circulo obſervatæ, velocitates vero maximæ ponen­
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                  dæ ſunt arcubus dimidiatis vel integris, hoc eſt, numeris 1/2, 1, 2,
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                  4, 8, 16 analogæ. </s>
                  <s>Scribamus ergo in caſu ſecundo, quarto & ſex­
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                  to numeros 1, 4 & 16 pro V; & prodibit arcuum differentia
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                  (1/2/121)=A+B+C in caſu ſecundo; (2/35 1/2)=4A+8B+16C in caſu
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                  quarto; & (8/9 2/3)=16A+64B+256C in caſu ſexto. </s>
                  <s>Et ex his æ­
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                  quationibus, per debitam collationem & reductionem Analyticam,
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                  fit A=0,0000916, B=0,0010847, & C=0,0029558. Eſt igitur
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                  differentia arcuum ut 0,0000916V+0,0010847V1/2+0,0029558V
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                  2
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                  :
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                  & propterea cum (per Corollarium Propoſitionis XXX) reſiſtentia
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                  Globi in medio arcus oſcillando deſcripti, ubi velocitas eſt V,
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                  ſit ad ipſius pondus ut (7/11)AV+(16/23)BV1/2+1/4CV
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                  2
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                  ad longitudinem
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                  Penduli; ſi pro A, B & C ſcribantur numeri inventi, fiet reſiſtentia
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                  Globi ad ejus pondus, ut 0,0000583V+0,0007546V1/2+0,0022169V
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                  2
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                  ad longitudinem Penduli inter centrum ſuſpenſionis & Regulam,
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                  id eſt, ad 121 digitos. </s>
                  <s>Unde cum V in caſu ſecundo deſignet 1,
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                  in quarto 4, in ſexto 16: erit reſiſtentia ad pondus Globi in caſu
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                  ſecundo ut 0,0030298 ad 121, in quarto ut 0,0417402 ad 121, in
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                  ſexto ut 0,61675 ad 121. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note261"/>
                  LIBER
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                  SECUNDUS.</s>
                </p>
                <p type="main">
                  <s>Arcus quem punctum in filo notatum in caſu ſexto deſcripſit,
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                  erat 120-(8/9 2/3) ſeu (119 5/29) digitorum. </s>
                  <s>Et propterea cum radius eſſet
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                  121 digitorum, & longitudo Penduli inter punctum ſuſpenſionis </s>
                </p>
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          </chap>
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