Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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        <body>
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              <subchap2>
                <p type="main">
                  <s>
                    <pb xlink:href="039/01/099.jpg" pagenum="71"/>
                    <arrow.to.target n="note47"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note47"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA XX.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si Parallelogrammum quodvis
                    <emph.end type="italics"/>
                  ASPQ
                    <emph type="italics"/>
                  angulis duobus oppoſitis
                    <emph.end type="italics"/>
                  A
                    <emph type="italics"/>
                  &
                    <emph.end type="italics"/>
                    <lb/>
                  P
                    <emph type="italics"/>
                  tangit ſectionem quamvis Conicam in punctis
                    <emph.end type="italics"/>
                  A
                    <emph type="italics"/>
                  &
                    <emph.end type="italics"/>
                  P;
                    <emph type="italics"/>
                  &, lateri­
                    <lb/>
                  bus unius angulorum illorum infinite productis
                    <emph.end type="italics"/>
                  AQ, AS,
                    <emph type="italics"/>
                  occurrit
                    <lb/>
                  eidem ſectioni Conicæ in
                    <emph.end type="italics"/>
                  B
                    <emph type="italics"/>
                  &
                    <emph.end type="italics"/>
                  C;
                    <emph type="italics"/>
                  a punctis autem occurſuum
                    <emph.end type="italics"/>
                  B
                    <emph type="italics"/>
                  &
                    <emph.end type="italics"/>
                    <lb/>
                  C
                    <emph type="italics"/>
                  ad quintum quodvis ſectionis Conicæ punctum
                    <emph.end type="italics"/>
                  D
                    <emph type="italics"/>
                  agantur rec­
                    <lb/>
                  tæ duæ
                    <emph.end type="italics"/>
                  BD, CD
                    <emph type="italics"/>
                  occurrentes alteris duobus infinite productis pa­
                    <lb/>
                  rallelogrammi lateribus
                    <emph.end type="italics"/>
                  PS, PQ
                    <emph type="italics"/>
                  in
                    <emph.end type="italics"/>
                  T
                    <emph type="italics"/>
                  &
                    <emph.end type="italics"/>
                  R:
                    <emph type="italics"/>
                  erunt ſemper abſciſſæ
                    <lb/>
                  laterum partes
                    <emph.end type="italics"/>
                  PR
                    <emph type="italics"/>
                  &
                    <emph.end type="italics"/>
                  PT
                    <emph type="italics"/>
                  adinvicem in data ratione. </s>
                  <s>Et contra, ſi
                    <lb/>
                  partes illæ abſciſſæ ſunt ad invicem in data ratione, punctum
                    <emph.end type="italics"/>
                  D
                    <emph type="italics"/>
                  tan­
                    <lb/>
                  get Sectionem Conicam per puncta quatuor
                    <emph.end type="italics"/>
                  A, B, C, P
                    <emph type="italics"/>
                  tranſeuntem.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  1. Jungantur
                    <emph type="italics"/>
                  BP, CP
                    <emph.end type="italics"/>
                  & a puncto
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  agantur rectæ duæ
                    <lb/>
                    <emph type="italics"/>
                  DG, DE,
                    <emph.end type="italics"/>
                  quarum prior
                    <lb/>
                    <figure id="id.039.01.099.1.jpg" xlink:href="039/01/099/1.jpg" number="45"/>
                    <lb/>
                    <emph type="italics"/>
                  DG
                    <emph.end type="italics"/>
                  ipſi
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  parallela ſit &
                    <lb/>
                  occurrat
                    <emph type="italics"/>
                  PB, PQ, CA
                    <emph.end type="italics"/>
                  in
                    <lb/>
                    <emph type="italics"/>
                  H, I, G
                    <emph.end type="italics"/>
                  ; altera
                    <emph type="italics"/>
                  DE
                    <emph.end type="italics"/>
                  paral­
                    <lb/>
                  lela ſit ipfi
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  & occurrat
                    <lb/>
                    <emph type="italics"/>
                  PC, PS, AB
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  F, K, E:
                    <emph.end type="italics"/>
                    <lb/>
                  & erit (per Lemma XVII.) re­
                    <lb/>
                  ctangulum
                    <emph type="italics"/>
                  DEXDF
                    <emph.end type="italics"/>
                  ad re­
                    <lb/>
                  ctangulum
                    <emph type="italics"/>
                  DGXDH
                    <emph.end type="italics"/>
                  in ra­
                    <lb/>
                  tione data. </s>
                  <s>Sed eſt
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                    <emph type="italics"/>
                  DE
                    <emph.end type="italics"/>
                  (ſeu
                    <emph type="italics"/>
                  IQ
                    <emph.end type="italics"/>
                  ) ut
                    <emph type="italics"/>
                  PB
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  HB,
                    <emph.end type="italics"/>
                    <lb/>
                  adeoque ut
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  DH
                    <emph.end type="italics"/>
                  ; &
                    <lb/>
                  viciſſim
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  DE
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  DH.
                    <emph.end type="italics"/>
                  Eſt &
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  DF
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  RC
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  DC,
                    <emph.end type="italics"/>
                  adeoque ut (
                    <emph type="italics"/>
                  IG
                    <emph.end type="italics"/>
                  vel)
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  DG,
                    <emph.end type="italics"/>
                  & viciſſim
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                    <lb/>
                  ut
                    <emph type="italics"/>
                  DF
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  DG
                    <emph.end type="italics"/>
                  ; & conjunctis rationibus fit rectangulum
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                    <lb/>
                  ad rectangulum
                    <emph type="italics"/>
                  PSXPT
                    <emph.end type="italics"/>
                  ut rectangulum
                    <emph type="italics"/>
                  DEXDF
                    <emph.end type="italics"/>
                  ad rectan­
                    <lb/>
                  gulum
                    <emph type="italics"/>
                  DGXDH,
                    <emph.end type="italics"/>
                  atque adeo in data ratione. </s>
                  <s>Sed dantur
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                    <lb/>
                  &
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  & propterea ratio
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  datur.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  2. Quod ſi
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  ponantur in data ratione ad invi­
                    <lb/>
                  cem, tum ſimili ratiocinio regrediendo, ſequetur eſſe rectangulum
                    <lb/>
                    <emph type="italics"/>
                  DEXDF
                    <emph.end type="italics"/>
                  ad rectangulum
                    <emph type="italics"/>
                  DGXDH
                    <emph.end type="italics"/>
                  in ratione data, adeoque
                    <lb/>
                  punctum
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  (per Lemma XVIII.) contingere Conicam ſectionem
                    <lb/>
                  tranſeuntem per puncta
                    <emph type="italics"/>
                  A, B, C, P. Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
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