Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/094.jpg" pagenum="66"/>
                    <arrow.to.target n="note42"/>
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                <p type="margin">
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                  DE MOTU
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                  CORPORUM</s>
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                <p type="main">
                  <s>
                    <emph type="center"/>
                  SECTIO V.
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                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Inventio Orbium ubi umbilicus neuter datur.
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                    <emph.end type="center"/>
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                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA XVII.
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                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si a datæ Conicæ Sectionis puncto quovis
                    <emph.end type="italics"/>
                  P,
                    <emph type="italics"/>
                  ad Trapezii alicujus
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                    <lb/>
                  ABDC,
                    <emph type="italics"/>
                  in Conica illa ſectione inſcripti, latera quatuor infinite
                    <lb/>
                  producta
                    <emph.end type="italics"/>
                  AB, CD, AC, DB,
                    <emph type="italics"/>
                  totidem rectæ
                    <emph.end type="italics"/>
                  PQ, PR, PS, PT
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                    <emph type="italics"/>
                  in datis angulis ducantur, ſingulæ ad ſingula: rectangulum duc­
                    <lb/>
                  tarum ad oppoſita duo latera
                    <emph.end type="italics"/>
                  PQXPR,
                    <emph type="italics"/>
                  erit ad rectangulum duc­
                    <lb/>
                  tarum ad alia duo latera oppoſita
                    <emph.end type="italics"/>
                  PSXPT
                    <emph type="italics"/>
                  in data ratione.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  1. Ponamus primo lineas ad
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                    <figure id="id.039.01.094.1.jpg" xlink:href="039/01/094/1.jpg" number="38"/>
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                  oppoſita latera ductas parallelas eſ­
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                  ſe alterutri reliquorum laterum,
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                  puta
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  lateri
                    <emph type="italics"/>
                  AC,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                    <lb/>
                  ac
                    <emph type="italics"/>
                  PT
                    <emph.end type="italics"/>
                  lateri
                    <emph type="italics"/>
                  AB.
                    <emph.end type="italics"/>
                  SintQ.E.I.ſuper
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                  latera duo ex oppoſitis, puta
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                    <lb/>
                  &
                    <emph type="italics"/>
                  BD,
                    <emph.end type="italics"/>
                  ſibi invicem paralle­
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                  la. </s>
                  <s>Et recta quæ biſecat paralle­
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                  la illa latera erit una ex diametris
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                  Conicæ ſectionis, & biſecabit eti­
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                  am
                    <emph type="italics"/>
                    <expan abbr="Rq.">Rque</expan>
                    <emph.end type="italics"/>
                  Sit
                    <emph type="italics"/>
                  O
                    <emph.end type="italics"/>
                  punctum in quo
                    <lb/>
                    <emph type="italics"/>
                  RQ
                    <emph.end type="italics"/>
                  biſecatur, & erit
                    <emph type="italics"/>
                  PO
                    <emph.end type="italics"/>
                  ordinatim applicata ad diametrum illam. </s>
                  <s>
                    <lb/>
                  Produc
                    <emph type="italics"/>
                  PO
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  ut ſit
                    <emph type="italics"/>
                  OK
                    <emph.end type="italics"/>
                  æqualis
                    <emph type="italics"/>
                  PO,
                    <emph.end type="italics"/>
                  & erit
                    <emph type="italics"/>
                  OK
                    <emph.end type="italics"/>
                  ordinatim
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                  applicata ad contrarias partes diametri. </s>
                  <s>Cum igitur puncta
                    <emph type="italics"/>
                  A, B,
                    <lb/>
                  P
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  ſint ad Conicam ſectionem, &
                    <emph type="italics"/>
                  PK
                    <emph.end type="italics"/>
                  ſecet
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  in dato an­
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                  gulo, erit (per Prop.17 & 18 Lib. </s>
                  <s>III Conieorum
                    <emph type="italics"/>
                  Apollonii
                    <emph.end type="italics"/>
                  ) rectangu­
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                  lum
                    <emph type="italics"/>
                  PQK
                    <emph.end type="italics"/>
                  ad rectangulum
                    <emph type="italics"/>
                  AQB
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                  in data ratione. </s>
                  <s>Sed
                    <emph type="italics"/>
                  QK
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                  &
                    <emph type="italics"/>
                  PR
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                    <lb/>
                  æquales ſunt, utpote æqualium
                    <emph type="italics"/>
                  OK, OP,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  OQ, OR
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                  differentiæ,
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                  & inde etiam rectangula
                    <emph type="italics"/>
                  PQK
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                  æqualia ſunt; at­
                    <lb/>
                  que adeo rectangulum
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                  eſt ad rectangulum
                    <emph type="italics"/>
                  AQB,
                    <emph.end type="italics"/>
                  hoc
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                  eſt ad rectangulum
                    <emph type="italics"/>
                  PSXPT
                    <emph.end type="italics"/>
                  in data ratione.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
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