Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
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                <p type="margin">
                  <s>
                    <margin.target id="note475"/>
                  DE MUNDI
                    <lb/>
                  SYSTEMATE</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA V.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Invenire lineam curvam generis Parabolici, quæ per data
                    <lb/>
                  quotcunque puncta tranſibit.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Sunto puncta illa
                    <emph type="italics"/>
                  A, B, C, D, E, F,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>& ab iiſdem ad rectam
                    <lb/>
                  quamvis poſitione datam
                    <emph type="italics"/>
                  HN
                    <emph.end type="italics"/>
                  demitte perpendicula quotcunque
                    <lb/>
                    <emph type="italics"/>
                  AH, BI, CK, DL, EM, FN.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Caſ.
                    <emph.end type="italics"/>
                  1. Si punctorum
                    <emph type="italics"/>
                  H, I, K, L, M, N
                    <emph.end type="italics"/>
                  æqualia ſunt inter­
                    <lb/>
                  valla
                    <emph type="italics"/>
                  HI, IK, KL,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>collige perpendiculorum
                    <emph type="italics"/>
                  AH, BI,
                    <lb/>
                  CK,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>differentias primas
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  2
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  3
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  4
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  5
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>ſecundas
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  2
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                    <lb/>
                  3
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  4
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>tertias
                    <emph type="italics"/>
                  d,
                    <emph.end type="italics"/>
                  2
                    <emph type="italics"/>
                  d,
                    <emph.end type="italics"/>
                  3
                    <emph type="italics"/>
                  d,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>id eſt, ita ut ſit
                    <emph type="italics"/>
                  AH-BI=b,
                    <lb/>
                  BI-CK=2b, CK-DL=3b, DL+EM=4b,-EM+FN=5b,
                    <emph.end type="italics"/>
                    <lb/>
                    <figure id="id.039.01.474.1.jpg" xlink:href="039/01/474/1.jpg" number="227"/>
                    <lb/>
                  &c. </s>
                  <s>dein
                    <emph type="italics"/>
                  b-2b=c,
                    <emph.end type="italics"/>
                  &c.
                    <lb/>
                  </s>
                  <s>& ſic pergatur ad diffe­
                    <lb/>
                  rentiam ultimam quæ hic
                    <lb/>
                  eſt
                    <emph type="italics"/>
                  f.
                    <emph.end type="italics"/>
                  Deinde erecta qua­
                    <lb/>
                  cunque perpendiculari
                    <lb/>
                    <emph type="italics"/>
                  RS,
                    <emph.end type="italics"/>
                  quæ fuerit ordina­
                    <lb/>
                  tim applicata ad curvam
                    <lb/>
                  quæſitam: ut inveniatur
                    <lb/>
                  hujus longitudo, pone
                    <lb/>
                  intervalla
                    <emph type="italics"/>
                  HI, IK, KL,
                    <lb/>
                  LM,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>unitates eſſe,
                    <lb/>
                  & dic
                    <emph type="italics"/>
                  AH=a,-HS=p,
                    <lb/>
                  1/2p
                    <emph.end type="italics"/>
                  in -
                    <emph type="italics"/>
                  IS=q, 1/3q
                    <emph.end type="italics"/>
                  in
                    <lb/>
                  +
                    <emph type="italics"/>
                  SK=r, 1/4r
                    <emph.end type="italics"/>
                  in +
                    <emph type="italics"/>
                  SL=s, 1/5s
                    <emph.end type="italics"/>
                  in +
                    <emph type="italics"/>
                  SM=t
                    <emph.end type="italics"/>
                  ; pergendo videlicet
                    <lb/>
                  ad uſque penultimum perpendiculum
                    <emph type="italics"/>
                  ME,
                    <emph.end type="italics"/>
                  & præponendo ſigna
                    <lb/>
                  negativa terminis
                    <emph type="italics"/>
                  HS, IS,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>qui jacent ad partes puncti
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  ver­
                    <lb/>
                  ſus
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  & ſigna affirmativa terminis
                    <emph type="italics"/>
                  SK, SL,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>qui jacent
                    <lb/>
                  ad alteras partes puncti
                    <emph type="italics"/>
                  S.
                    <emph.end type="italics"/>
                  Et ſignis probe obſervatis, erit
                    <lb/>
                    <emph type="italics"/>
                  RS=a+bp+cq+dr+es+ft,
                    <emph.end type="italics"/>
                  &c. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Caſ.
                    <emph.end type="italics"/>
                  2. Quod ſi punctorum
                    <emph type="italics"/>
                  H, I, K, L,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>inæqualia ſint inter­
                    <lb/>
                  valla
                    <emph type="italics"/>
                  HI, IK,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>collige perpendiculorum
                    <emph type="italics"/>
                  AH, BI, CK,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>
                    <lb/>
                  differentias primas per intervalla perpendiculorum diviſas
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  2
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                    <lb/>
                  3
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  4
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                  5
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  ; ſecundas per intervalla bina diviſas
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  2
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  3
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  4
                    <emph type="italics"/>
                  c,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>
                    <lb/>
                  tertias per intervalla terna diviſas
                    <emph type="italics"/>
                  d,
                    <emph.end type="italics"/>
                  2
                    <emph type="italics"/>
                  d,
                    <emph.end type="italics"/>
                  3
                    <emph type="italics"/>
                  d,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>quartas per </s>
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