Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/111.jpg" pagenum="83"/>
                  ta, & eadem pro radio ordinato primo adhibita, tranſmutetur fi­
                    <lb/>
                    <arrow.to.target n="note59"/>
                  gura (per Lem. </s>
                  <s>XXII) in figuram novam, & tangentes binæ, quæ ad
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                  radium ordinatum primum concurrebant, jam evadent parallelæ. </s>
                  <s>Sun­
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                  to illæ
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                  hi
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                  &
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                  kl, ik
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                  &
                    <emph type="italics"/>
                  hl
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                  continentes parallelogrammum
                    <emph type="italics"/>
                  hikl.
                    <emph.end type="italics"/>
                  Sit­
                    <lb/>
                  que
                    <emph type="italics"/>
                  p
                    <emph.end type="italics"/>
                  punctum in hac nova figura, puncto in figura prima dato
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                  reſpondens. </s>
                  <s>Per figuræ centrum
                    <emph type="italics"/>
                  O
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                  agatur
                    <emph type="italics"/>
                  pq,
                    <emph.end type="italics"/>
                  & exiſtente
                    <emph type="italics"/>
                  Oq
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                  æ­
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                  quali
                    <emph type="italics"/>
                  Op,
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                  erit
                    <emph type="italics"/>
                  q
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                  punctum alterum per quod ſectio Conica in hac
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                  figura nova tranſire debet. </s>
                  <s>Per Lemmatis XXII operationem in­
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                  verſam transferatur hoc punctum in figuram primam, & ibi habe­
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                  buntur puncta duo per quæ Trajectoria deſcribenda eſt. </s>
                  <s>Per ea­
                    <lb/>
                  dem vero deſcribi poteſt Trajectoria illa per Prob. </s>
                  <s>XVII.
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                  q.E.F.
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                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note59"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA XXIII.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si rectæ duæ poſitione datæ
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                  AC, BD
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                  ad data puncta
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                  A, B,
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                  ter­
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                  minentur, datamque habeant rationem ad invicem, & recta
                    <emph.end type="italics"/>
                    <lb/>
                  CD,
                    <emph type="italics"/>
                  qua puncta indeterminata
                    <emph.end type="italics"/>
                  C, D
                    <emph type="italics"/>
                  junguntur, ſecetur in ra­
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                  tione data in
                    <emph.end type="italics"/>
                  K:
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                  dico quod punctum
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                  K
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                  locabitur in recta poſi­
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                  tione data.
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                  </s>
                </p>
                <p type="main">
                  <s>Concurrant enim rectæ
                    <emph type="italics"/>
                  AC,
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                    <lb/>
                    <figure id="id.039.01.111.1.jpg" xlink:href="039/01/111/1.jpg" number="57"/>
                    <lb/>
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  E,
                    <emph.end type="italics"/>
                  & in
                    <emph type="italics"/>
                  BE
                    <emph.end type="italics"/>
                  capiatur
                    <emph type="italics"/>
                  BG
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  AE
                    <emph.end type="italics"/>
                  ut eſt
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  AC,
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                  ſit­
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                  que
                    <emph type="italics"/>
                  FD
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                  ſemper æqualis datæ
                    <lb/>
                    <emph type="italics"/>
                  EG
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                  ; & erit ex conſtructione
                    <lb/>
                    <emph type="italics"/>
                  EC
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  GD,
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                  hoc eſt, ad
                    <emph type="italics"/>
                  EF
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  BD,
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                  adeoQ.E.I. ratione
                    <lb/>
                  data, & propterea dabitur ſpecie
                    <lb/>
                  triangulum
                    <emph type="italics"/>
                  EFC.
                    <emph.end type="italics"/>
                  Secetur
                    <emph type="italics"/>
                  CF
                    <emph.end type="italics"/>
                    <lb/>
                  in
                    <emph type="italics"/>
                  L
                    <emph.end type="italics"/>
                  ut ſit
                    <emph type="italics"/>
                  CL
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CF
                    <emph.end type="italics"/>
                  in ratio­
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                  ne
                    <emph type="italics"/>
                  CK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CD
                    <emph.end type="italics"/>
                  ; &, ob datam il­
                    <lb/>
                  lam rationem, dabitur etiam ſpecie triangulum
                    <emph type="italics"/>
                  EFL
                    <emph.end type="italics"/>
                  ; proindeque
                    <lb/>
                  punctum
                    <emph type="italics"/>
                  L
                    <emph.end type="italics"/>
                  locabitur in recta
                    <emph type="italics"/>
                  EL
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                  poſitione data. </s>
                  <s>Junge
                    <emph type="italics"/>
                  LK,
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                  &
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                  ſimilia erunt triangula
                    <emph type="italics"/>
                  CLK, CFD
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                  ; &, ob datam
                    <emph type="italics"/>
                  FD
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                  & datam
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                  rationem
                    <emph type="italics"/>
                  LK
                    <emph.end type="italics"/>
                  ad
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                  FD,
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                  dabitur
                    <emph type="italics"/>
                  LK.
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                  Huic æqualis capiatur
                    <emph type="italics"/>
                  EH,
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                    <lb/>
                  & erit ſemper
                    <emph type="italics"/>
                  ELKH
                    <emph.end type="italics"/>
                  parallelogrammum. </s>
                  <s>Locatur igitur punc­
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                  tum
                    <emph type="italics"/>
                  K
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                  in parallelogrammi illius latere poſitione dato
                    <emph type="italics"/>
                  HK. Q.E.D.
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                  </s>
                </p>
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