Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <pb xlink:href="039/01/095.jpg" pagenum="67"/>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  2. Ponamus jam Trapezii latera oppoſita
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  non
                    <lb/>
                    <arrow.to.target n="note43"/>
                  eſſe parallela. </s>
                  <s>Age
                    <emph type="italics"/>
                  Bd
                    <emph.end type="italics"/>
                  parallelam
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  & occurrentem tum rectæ
                    <lb/>
                    <emph type="italics"/>
                  ST
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  t,
                    <emph.end type="italics"/>
                  tum Conicæ ſectioni in
                    <emph type="italics"/>
                  d.
                    <emph.end type="italics"/>
                  Junge
                    <emph type="italics"/>
                  Cd
                    <emph.end type="italics"/>
                  ſecantem
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  r,
                    <emph.end type="italics"/>
                    <lb/>
                  & ipſi
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  parallelam age
                    <emph type="italics"/>
                  DM
                    <emph.end type="italics"/>
                    <lb/>
                    <figure id="id.039.01.095.1.jpg" xlink:href="039/01/095/1.jpg" number="39"/>
                    <lb/>
                  ſecantem
                    <emph type="italics"/>
                  Cd
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  M
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  N.
                    <emph.end type="italics"/>
                    <lb/>
                  Jam ob ſimilia triangula
                    <emph type="italics"/>
                  BTt,
                    <lb/>
                  DBN
                    <emph.end type="italics"/>
                  ; eſt
                    <emph type="italics"/>
                  Bt
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Tt
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                    <emph type="italics"/>
                  DN
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  NB.
                    <emph.end type="italics"/>
                  Sic &
                    <emph type="italics"/>
                  Rr
                    <emph.end type="italics"/>
                  eſt ad
                    <lb/>
                    <emph type="italics"/>
                  AQ
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  DM
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  AN.
                    <emph.end type="italics"/>
                    <lb/>
                  Ergo, ducendo antecedentes in
                    <lb/>
                  antecedentes & conſequentes in
                    <lb/>
                  conſequentes, ut rectangulum
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                    <lb/>
                  in
                    <emph type="italics"/>
                  Rr
                    <emph.end type="italics"/>
                  eſt ad rectangulum
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  in
                    <lb/>
                    <emph type="italics"/>
                  Tt,
                    <emph.end type="italics"/>
                  ita rectangulum
                    <emph type="italics"/>
                  NDM
                    <emph.end type="italics"/>
                  eſt
                    <lb/>
                  ad rectangulum
                    <emph type="italics"/>
                  ANB,
                    <emph.end type="italics"/>
                  & (per Caſ.1) ita rectangulum
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  Pr
                    <emph.end type="italics"/>
                  eſt
                    <lb/>
                  ad rectangulum
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  Pt,
                    <emph.end type="italics"/>
                  ac diviſim ita rectangulum
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                    <lb/>
                  eſt ad rectangulum
                    <emph type="italics"/>
                  PSXPT. Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note43"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  3. Ponamus denique lineas
                    <lb/>
                    <figure id="id.039.01.095.2.jpg" xlink:href="039/01/095/2.jpg" number="40"/>
                    <lb/>
                  quatuor
                    <emph type="italics"/>
                  PQ, PR, PS, PT
                    <emph.end type="italics"/>
                  non
                    <lb/>
                  eſſe parallelas lateribus
                    <emph type="italics"/>
                  AC, AB,
                    <emph.end type="italics"/>
                    <lb/>
                  ſed ad ea utcunQ.E.I.clinatas. </s>
                  <s>Ea­
                    <lb/>
                  rum vice age
                    <emph type="italics"/>
                  Pq, Pr
                    <emph.end type="italics"/>
                  parallelas
                    <lb/>
                  ipſi
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ; &
                    <emph type="italics"/>
                  Ps, Pt
                    <emph.end type="italics"/>
                  parallelas
                    <lb/>
                  ipſi
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  ; & propter datos angu­
                    <lb/>
                  los triangulorum
                    <emph type="italics"/>
                  PQq, PRr,
                    <lb/>
                  PSs, PTt,
                    <emph.end type="italics"/>
                  dabuntur rationes
                    <lb/>
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Pq, PR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Pr, PS
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  Ps,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Pt
                    <emph.end type="italics"/>
                  ; atque adeo rationes compoſitæ
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  PqXPr,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PSXPT
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PsXPt.
                    <emph.end type="italics"/>
                  Sed, per ſuperius de­
                    <lb/>
                  monſtrata, ratio
                    <emph type="italics"/>
                  PqXPr
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PsXPt
                    <emph.end type="italics"/>
                  data eſt: Ergo & ratio
                    <lb/>
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PSXPT. Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA XVIII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Iiſdem poſitis, ſi rectangulum ductarum ad oppoſita duo latera Tra­
                    <lb/>
                  pezii
                    <emph.end type="italics"/>
                  PQXPR
                    <emph type="italics"/>
                  ſit ad rectangulum ductarum ad reliqua duo late­
                    <lb/>
                  ra
                    <emph.end type="italics"/>
                  PSXPT
                    <emph type="italics"/>
                  in data ratione; punctum
                    <emph.end type="italics"/>
                  P,
                    <emph type="italics"/>
                  a quo lineæ ducuntur,
                    <lb/>
                  tanget Conicam ſectionem circa Trapezium deſcriptam.
                    <emph.end type="italics"/>
                  </s>
                </p>
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