Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/165.jpg" pagenum="137"/>
                  id eſt, ratio mutationum momentanearum curvæ
                    <emph type="italics"/>
                  AP,
                    <emph.end type="italics"/>
                  rectæ
                    <lb/>
                    <arrow.to.target n="note113"/>
                    <emph type="italics"/>
                  CP,
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                  arcus circularis
                    <emph type="italics"/>
                  BP,
                    <emph.end type="italics"/>
                  ac rectæ
                    <emph type="italics"/>
                  VP,
                    <emph.end type="italics"/>
                  eadem erit quæ linea­
                    <lb/>
                  rum
                    <emph type="italics"/>
                  PV, PF, PG, PI
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                  reſpective. </s>
                  <s>Cum autem
                    <emph type="italics"/>
                  VF
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CF
                    <emph.end type="italics"/>
                  &
                    <lb/>
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CV
                    <emph.end type="italics"/>
                  perpendiculares ſunt, angulique
                    <emph type="italics"/>
                  HVG, VCF
                    <emph.end type="italics"/>
                  prop­
                    <lb/>
                  terea æquales; & angulus
                    <emph type="italics"/>
                  VHG
                    <emph.end type="italics"/>
                  (ob angulos quadrilateri
                    <emph type="italics"/>
                  HVEP
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  V
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  rectos) angulo
                    <emph type="italics"/>
                  CEP
                    <emph.end type="italics"/>
                  æqualis eſt, ſimilia erunt tri­
                    <lb/>
                  angula
                    <emph type="italics"/>
                  VHG, CEP
                    <emph.end type="italics"/>
                  ; & inde fiet ut
                    <emph type="italics"/>
                  EP
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  ita
                    <emph type="italics"/>
                  HG
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  HV
                    <emph.end type="italics"/>
                    <lb/>
                  ſeu
                    <emph type="italics"/>
                  HP
                    <emph.end type="italics"/>
                  & ita
                    <emph type="italics"/>
                  KI
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KP,
                    <emph.end type="italics"/>
                  & compoſite vel diviſim ut
                    <emph type="italics"/>
                  CB
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  ita
                    <emph type="italics"/>
                  PI
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PK,
                    <emph.end type="italics"/>
                  & duplicatis conſequentibus ut
                    <emph type="italics"/>
                  CB
                    <emph.end type="italics"/>
                  ad 2
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                    <lb/>
                  ita
                    <emph type="italics"/>
                  PI
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PV,
                    <emph.end type="italics"/>
                  atQ.E.I.a adeo
                    <emph type="italics"/>
                  Pq
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Pm.
                    <emph.end type="italics"/>
                  Eſt igitur decremen­
                    <lb/>
                  tum lineæ
                    <emph type="italics"/>
                  VP,
                    <emph.end type="italics"/>
                  id eſt, incrementum lineæ
                    <emph type="italics"/>
                  BV-VP
                    <emph.end type="italics"/>
                  ad incremen­
                    <lb/>
                  tum lineæ curvæ
                    <emph type="italics"/>
                  AP
                    <emph.end type="italics"/>
                  in data ratione
                    <emph type="italics"/>
                  CB
                    <emph.end type="italics"/>
                  ad 2
                    <emph type="italics"/>
                  CE,
                    <emph.end type="italics"/>
                  & prop­
                    <lb/>
                  terea (per Corol. </s>
                  <s>Lem. </s>
                  <s>IV.) longitudines
                    <emph type="italics"/>
                  BV-VP
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  AP,
                    <emph.end type="italics"/>
                  in­
                    <lb/>
                  crementis illis genitæ, ſunt in eadem ratione. </s>
                  <s>Sed, exiſtente
                    <emph type="italics"/>
                  BV
                    <emph.end type="italics"/>
                  ra­
                    <lb/>
                  dio, eſt
                    <emph type="italics"/>
                  VP
                    <emph.end type="italics"/>
                  co-ſinus anguli
                    <emph type="italics"/>
                  BVP
                    <emph.end type="italics"/>
                  ſeu 1/2
                    <emph type="italics"/>
                  BEP,
                    <emph.end type="italics"/>
                  adeoque
                    <emph type="italics"/>
                  BV-VP
                    <emph.end type="italics"/>
                    <lb/>
                  ſinus verſus ejuſdem anguli; & propterea in hac Rota, cujus radius
                    <lb/>
                  eſt 1/2
                    <emph type="italics"/>
                  BV,
                    <emph.end type="italics"/>
                  erit
                    <emph type="italics"/>
                  BV-VP
                    <emph.end type="italics"/>
                  duplus ſinus verſus arcus 1/2
                    <emph type="italics"/>
                  BP.
                    <emph.end type="italics"/>
                  Ergo
                    <lb/>
                    <emph type="italics"/>
                  AP
                    <emph.end type="italics"/>
                  eſt ad duplum ſinum verſum arcus 1/2
                    <emph type="italics"/>
                  BP
                    <emph.end type="italics"/>
                  ut 2
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CB.
                    <lb/>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note113"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>Lineam autem
                    <emph type="italics"/>
                  AP
                    <emph.end type="italics"/>
                  in Propoſitione priore Cycloidem extra
                    <lb/>
                  Globum, alteram in poſteriore Cycloidem intra Globum diſtincti­
                    <lb/>
                  onis gratia nominabimus. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  1. Hinc ſi deſcribatur Cyclois integra
                    <emph type="italics"/>
                  ASL
                    <emph.end type="italics"/>
                  & biſecetur
                    <lb/>
                  ea in
                    <emph type="italics"/>
                  S,
                    <emph.end type="italics"/>
                  erit longitudo partis
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  ad longitudinem
                    <emph type="italics"/>
                  VP
                    <emph.end type="italics"/>
                  (quæ du­
                    <lb/>
                  plus eſt ſinus anguli
                    <emph type="italics"/>
                  VBP,
                    <emph.end type="italics"/>
                  exiſtente
                    <emph type="italics"/>
                  EB
                    <emph.end type="italics"/>
                  radio) ut 2
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CB,
                    <emph.end type="italics"/>
                    <lb/>
                  atque adeo in ratione data. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  2. Et longitudo ſemiperimetri Cycloidis
                    <emph type="italics"/>
                  AS
                    <emph.end type="italics"/>
                  æquabitur
                    <lb/>
                  lineæ rectæ quæ eſt ad Rotæ diametrum
                    <emph type="italics"/>
                  BV,
                    <emph.end type="italics"/>
                  ut 2
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CB.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO L. PROBLEMA XXXIII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Facere ut Corpus pendulum oſcilletur in Cycloide data.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Intra Globum
                    <emph type="italics"/>
                  QVS,
                    <emph.end type="italics"/>
                  centro
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  deſcriptum, detur Cyclois
                    <emph type="italics"/>
                  QRS
                    <emph.end type="italics"/>
                    <lb/>
                  biſecta in
                    <emph type="italics"/>
                  R
                    <emph.end type="italics"/>
                  & punctis ſuis extremis
                    <emph type="italics"/>
                  Q
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  ſuperficiei Globi hinc
                    <lb/>
                  inde occurrens. </s>
                  <s>Agatur
                    <emph type="italics"/>
                  CR
                    <emph.end type="italics"/>
                  biſecans arcum
                    <emph type="italics"/>
                  QS
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  O,
                    <emph.end type="italics"/>
                  & produca­
                    <lb/>
                  tur ea ad
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  ut ſit
                    <emph type="italics"/>
                  CA
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CO
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  CO
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CR.
                    <emph.end type="italics"/>
                  Centro
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  in-</s>
                </p>
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