Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/369.jpg" pagenum="341"/>
                  tus iidem peragantur ac prius, augebitur in ſubduplicata ratione
                    <lb/>
                    <arrow.to.target n="note349"/>
                  denſitatis, ac diminuetur in ſubduplicata ratione vis Elaſticæ. </s>
                  <s>Et
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                  propterea velocitas pulſuum erit in ratione compoſita ex ratione
                    <lb/>
                  ſubduplicata denſitatis Medii inverſe & ratione ſubduplicata vis
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                  Elaſticæ directe.
                    <emph type="italics"/>
                    <expan abbr="q.">que</expan>
                  E. D.
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                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note349"/>
                  LIBER
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                  SECUNDUS.</s>
                </p>
                <p type="main">
                  <s>Hæc Propoſitio ulterius patebit ex conſtructione ſequentis. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XLIX. PROBLEMA XI.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Datis Medii denſitate & vi Elaſtica, invenire velocitatem pul­
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                  ſuum.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Fingamus Medium ab incumbente pondere, pro more Aeris
                    <lb/>
                  noſtri comprimi; ſitque A altitudo Medii homogenei, cujus pon­
                    <lb/>
                  dus adæquet pondus incumbens, & cujus denſitas eadem ſit cum
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                  denſitate Medii compreſſi, in quo pulſus propagantur. </s>
                  <s>Conſti­
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                  tui autem intelligatur Pendulum, cujus longitudo inter punctum
                    <lb/>
                  ſuſpenſionis & centrum oſcillationis ſit A: & quo tempore Pen­
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                  dulum illud oſcillationem integram ex itu & reditu compoſitam
                    <lb/>
                  peragit, eodem pulſus eundo conficiet ſpatium circumferentiæ
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                  circuli radio A deſcripti æquale. </s>
                </p>
                <p type="main">
                  <s>Nam ſtantibus quæ in Propoſitione XLVII conſtructa ſunt,
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                  ſi linea quævis Phyſica
                    <emph type="italics"/>
                  EF,
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                  ſingulis vibrationibus deſcribendo
                    <lb/>
                  ſpatium
                    <emph type="italics"/>
                  PS,
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                  urgeatur in extremis itus & reditus cujuſque locis
                    <lb/>
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  S,
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                  a vi Elaſtica quæ ipſius ponderi æquetur; peraget hæc
                    <lb/>
                  vibrationes ſingulas quo tempore eadem in Cycloide, cujus peri­
                    <lb/>
                  meter tota longitudini
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  æqualis eſt, oſcillari poſſet: id adeo
                    <lb/>
                  quia vires æquales æqualia corpuſcula per æqualia ſpatia ſimul im­
                    <lb/>
                  pellent. </s>
                  <s>Quare cum oſcillationum tempora ſint in ſubduplicata
                    <lb/>
                  ratione longitudinis Pendulorum, & longitudo Penduli æquetur
                    <lb/>
                  dimidio arcui Cycloidis totius; foret tempus vibrationis unius ad
                    <lb/>
                  tempus oſcillationis Penduli cujus longitudo eſt A, in ſubdupli­
                    <lb/>
                  cata ratione longitudinis 1/2
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  PO
                    <emph.end type="italics"/>
                  ad longitudinem A. </s>
                  <s>Sed
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                  vis Elaſtica qua lineola Phyſica
                    <emph type="italics"/>
                  EG,
                    <emph.end type="italics"/>
                  in locis ſuis extremis
                    <emph type="italics"/>
                  P, S
                    <emph.end type="italics"/>
                    <lb/>
                  exiſtens, urgetur, erat (in demonſtratione Propoſitionis XLVII)
                    <lb/>
                  ad ejus vim totam Elaſticam ut
                    <emph type="italics"/>
                  HL-KN
                    <emph.end type="italics"/>
                  ad V, hoc eſt
                    <lb/>
                  (cum punctum
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  jam incidat in
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  ) ut
                    <emph type="italics"/>
                  HK
                    <emph.end type="italics"/>
                  ad V: & vis illa
                    <lb/>
                  tota, hoc eſt pondus incumbens, quo lineola
                    <emph type="italics"/>
                  EG
                    <emph.end type="italics"/>
                  comprimitur,
                    <lb/>
                  eſt ad pondus lineolæ ut ponderis incumbentis altitudo A ad line­
                    <lb/>
                  olæ longitudinem
                    <emph type="italics"/>
                  EG
                    <emph.end type="italics"/>
                  ; adeoque ex æquo, vis qua lineola
                    <emph type="italics"/>
                  EG
                    <emph.end type="italics"/>
                  in
                    <lb/>
                  locis ſuis
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  urgetur, eſt ad lineolæ illius pondus ut
                    <emph type="italics"/>
                  HK
                    <emph.end type="italics"/>
                  XA
                    <lb/>
                  ad VX
                    <emph type="italics"/>
                  EG,
                    <emph.end type="italics"/>
                  ſive ut
                    <emph type="italics"/>
                  PO
                    <emph.end type="italics"/>
                  XA ad VV, nam
                    <emph type="italics"/>
                  HK
                    <emph.end type="italics"/>
                  erat ad
                    <emph type="italics"/>
                  EG
                    <emph.end type="italics"/>
                  ut </s>
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