Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <pb xlink:href="039/01/101.jpg" pagenum="73"/>
                <p type="main">
                  <s>
                    <arrow.to.target n="note49"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note49"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>Nam in recta
                    <emph type="italics"/>
                  MN
                    <emph.end type="italics"/>
                  detur punctum
                    <emph type="italics"/>
                  N,
                    <emph.end type="italics"/>
                  & ubi punctum mobile
                    <lb/>
                    <emph type="italics"/>
                  M
                    <emph.end type="italics"/>
                  incidit in immotum
                    <emph type="italics"/>
                  N,
                    <emph.end type="italics"/>
                  incidat punctum mobile
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  in immo­
                    <lb/>
                  tum
                    <emph type="italics"/>
                  P,
                    <emph.end type="italics"/>
                  Junge
                    <emph type="italics"/>
                  CN, BN,
                    <emph.end type="italics"/>
                    <lb/>
                    <figure id="id.039.01.101.1.jpg" xlink:href="039/01/101/1.jpg" number="47"/>
                    <lb/>
                    <emph type="italics"/>
                  CP, BP,
                    <emph.end type="italics"/>
                  & a puncto
                    <lb/>
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  age rectas
                    <emph type="italics"/>
                  PT, PR
                    <emph.end type="italics"/>
                    <lb/>
                  occurrentes ipſis
                    <emph type="italics"/>
                  BD,
                    <lb/>
                  CD
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  T
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  R,
                    <emph.end type="italics"/>
                  & fa­
                    <lb/>
                  cientes angulum
                    <emph type="italics"/>
                  BPT
                    <emph.end type="italics"/>
                    <lb/>
                  æqualem angulo dato
                    <lb/>
                    <emph type="italics"/>
                  BNM,
                    <emph.end type="italics"/>
                  & angulum
                    <lb/>
                    <emph type="italics"/>
                  CPR
                    <emph.end type="italics"/>
                  æqualem angu­
                    <lb/>
                  gulo dato
                    <emph type="italics"/>
                  CNM.
                    <emph.end type="italics"/>
                  Cum
                    <lb/>
                  ergo (ex Hypotheſi)
                    <lb/>
                  æquales ſint anguli
                    <lb/>
                    <emph type="italics"/>
                  MBD, NBP,
                    <emph.end type="italics"/>
                  ut &
                    <lb/>
                  anguli
                    <emph type="italics"/>
                  MCD, NCP
                    <emph.end type="italics"/>
                  ;
                    <lb/>
                  aufer communes
                    <emph type="italics"/>
                  NBD
                    <emph.end type="italics"/>
                    <lb/>
                  &
                    <emph type="italics"/>
                  NCD,
                    <emph.end type="italics"/>
                  & reſtabunt
                    <lb/>
                  æquales
                    <emph type="italics"/>
                  NBM
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PBT,
                    <lb/>
                  NCM
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PCR:
                    <emph.end type="italics"/>
                  adeoque triangula
                    <emph type="italics"/>
                  NBM, PBT
                    <emph.end type="italics"/>
                  ſimilia ſunt, ut
                    <lb/>
                  & triangula
                    <emph type="italics"/>
                  NCM, PCR.
                    <emph.end type="italics"/>
                  Quare
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  eſt ad
                    <emph type="italics"/>
                  NM
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  PB
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                    <emph type="italics"/>
                  NB,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  NM
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  PC
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  NC.
                    <emph.end type="italics"/>
                  Sunt autem puncta
                    <emph type="italics"/>
                  B, C, N, P
                    <emph.end type="italics"/>
                    <lb/>
                  immobilia. </s>
                  <s>Ergo
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  datam habent rationem ad
                    <emph type="italics"/>
                  NM,
                    <emph.end type="italics"/>
                  pro­
                    <lb/>
                  indeQ.E.D.tam rationem inter ſe; atque adeo, per Lemma xx,
                    <lb/>
                  punctum
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  (perpetuus rectarum mobilium
                    <emph type="italics"/>
                  BT
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  CR
                    <emph.end type="italics"/>
                  concurſus)
                    <lb/>
                  contingit ſectionem Conicam, per puncta
                    <emph type="italics"/>
                  B, C, P
                    <emph.end type="italics"/>
                  tranſeuntem.
                    <lb/>
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Et contra, ſi punctum mobile
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  contingat ſectionem Conicam
                    <lb/>
                  tranſeuntem per data puncta
                    <emph type="italics"/>
                  B, C, A,
                    <emph.end type="italics"/>
                  & ſit angulus
                    <emph type="italics"/>
                  DBM
                    <emph.end type="italics"/>
                  ſemper
                    <lb/>
                  æqualis angulo dato
                    <emph type="italics"/>
                  ABC,
                    <emph.end type="italics"/>
                  & angulus
                    <emph type="italics"/>
                  DCM
                    <emph.end type="italics"/>
                  ſemper æqualis angu­
                    <lb/>
                  lo dato
                    <emph type="italics"/>
                  ACB,
                    <emph.end type="italics"/>
                  & ubi punctum
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  incidit ſucceſſive in duo quævis ſe­
                    <lb/>
                  ctionis puncta immobilia
                    <emph type="italics"/>
                  p, P,
                    <emph.end type="italics"/>
                  punctum mobile
                    <emph type="italics"/>
                  M
                    <emph.end type="italics"/>
                  incidat ſucceſſive
                    <lb/>
                  in puncta duo immobilia
                    <emph type="italics"/>
                  n, N:
                    <emph.end type="italics"/>
                  per eadem
                    <emph type="italics"/>
                  n, N
                    <emph.end type="italics"/>
                  agatur Recta
                    <emph type="italics"/>
                  n N,
                    <emph.end type="italics"/>
                    <lb/>
                  & hæc erit Locus perpetuus puncti illius mobilis
                    <emph type="italics"/>
                  M.
                    <emph.end type="italics"/>
                  Nam, ſi fieri
                    <lb/>
                  poteſt, verſetur punctum
                    <emph type="italics"/>
                  M
                    <emph.end type="italics"/>
                  in linea aliqua Curva. </s>
                  <s>Tanget ergo
                    <lb/>
                  punctum
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  ſectionem Conicam per puncta quinque
                    <emph type="italics"/>
                  B, CA, p, P,
                    <emph.end type="italics"/>
                    <lb/>
                  tranſeuntem, ubi punctum
                    <emph type="italics"/>
                  M
                    <emph.end type="italics"/>
                  perpetuo tangit lineam Curvam. </s>
                  <s>Sed
                    <lb/>
                  & ex jam demonſtratis tanget etiam punctum
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  ſectionem CoNI­
                    <lb/>
                  cam per eadem quinque puncta
                    <emph type="italics"/>
                  B, C, A, p, P
                    <emph.end type="italics"/>
                  tranſeuntem, ubi pun-</s>
                </p>
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