Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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          <chap>
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              <subchap2>
                <pb xlink:href="039/01/057.jpg" pagenum="29"/>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA IX.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si recta
                    <emph.end type="italics"/>
                  AE
                    <emph type="italics"/>
                  & curva
                    <emph.end type="italics"/>
                  ABC
                    <emph type="italics"/>
                  poſitione datæ ſe mutuo ſecent in
                    <lb/>
                  angulo dato
                    <emph.end type="italics"/>
                  A,
                    <emph type="italics"/>
                  & ad rectam illam in alio dato angulo ordina­
                    <lb/>
                  tim applicentur
                    <emph.end type="italics"/>
                  BD, CE,
                    <emph type="italics"/>
                  curvæ occurrentes in
                    <emph.end type="italics"/>
                  B, C;
                    <emph type="italics"/>
                  dein
                    <lb/>
                  puncta
                    <emph.end type="italics"/>
                  B, C
                    <emph type="italics"/>
                  ſimul accedant ad punctum
                    <emph.end type="italics"/>
                  A:
                    <emph type="italics"/>
                  dico quod areæ tri­
                    <lb/>
                  angulorum
                    <emph.end type="italics"/>
                  ABD, ACE
                    <emph type="italics"/>
                  erunt ultimo ad invicem in duplicata
                    <lb/>
                  ratione laterum.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Etenim dum puncta
                    <emph type="italics"/>
                  B, C
                    <emph.end type="italics"/>
                  acce­
                    <lb/>
                    <figure id="id.039.01.057.1.jpg" xlink:href="039/01/057/1.jpg" number="11"/>
                    <lb/>
                  dunt ad punctum
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  intelligatur
                    <lb/>
                  ſemper
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  produci ad puncta lon­
                    <lb/>
                  ginqua
                    <emph type="italics"/>
                  d
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  e,
                    <emph.end type="italics"/>
                  ut ſint
                    <emph type="italics"/>
                  Ad, Ae
                    <emph.end type="italics"/>
                  ip­
                    <lb/>
                  ſis
                    <emph type="italics"/>
                  AD, AE
                    <emph.end type="italics"/>
                  proportionales, & e­
                    <lb/>
                  rigantur ordinatæ
                    <emph type="italics"/>
                  db, ec
                    <emph.end type="italics"/>
                  ordina­
                    <lb/>
                  tis
                    <emph type="italics"/>
                  DB, EC
                    <emph.end type="italics"/>
                  parallelæ quæ occur­
                    <lb/>
                  rant ipſis
                    <emph type="italics"/>
                  AB, AC
                    <emph.end type="italics"/>
                  productis in
                    <lb/>
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  c.
                    <emph.end type="italics"/>
                  Duci intelligatur, tum curva
                    <lb/>
                    <emph type="italics"/>
                  Abc
                    <emph.end type="italics"/>
                  ipſi
                    <emph type="italics"/>
                  ABC
                    <emph.end type="italics"/>
                  ſimilis, tum recta
                    <lb/>
                    <emph type="italics"/>
                  Ag,
                    <emph.end type="italics"/>
                  quæ tangat curvam utramque
                    <lb/>
                  in
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  & ſecet ordinatim applica­
                    <lb/>
                  tas
                    <emph type="italics"/>
                  DB, EC, db, ec
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  F, G, f, g.
                    <emph.end type="italics"/>
                    <lb/>
                  Tum manente longitudine
                    <emph type="italics"/>
                  Ae
                    <emph.end type="italics"/>
                  coeant puncta
                    <emph type="italics"/>
                  B, C
                    <emph.end type="italics"/>
                  cum puncto
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  ;
                    <lb/>
                  & angulo
                    <emph type="italics"/>
                  cAg
                    <emph.end type="italics"/>
                  evaneſcente, coincident areæ curvilineæ
                    <emph type="italics"/>
                  Abd, Ace
                    <emph.end type="italics"/>
                    <lb/>
                  cum rectilineis
                    <emph type="italics"/>
                  Afd, Age:
                    <emph.end type="italics"/>
                  adeoque (per Lemma v) erunt in dupli­
                    <lb/>
                  cata ratione laterum
                    <emph type="italics"/>
                  Ad, Ae:
                    <emph.end type="italics"/>
                  Sed his areis proportionales ſemper
                    <lb/>
                  ſunt areæ
                    <emph type="italics"/>
                  ABD, ACE,
                    <emph.end type="italics"/>
                  & his lateribus latera
                    <emph type="italics"/>
                  AD, AE.
                    <emph.end type="italics"/>
                  Ergo &
                    <lb/>
                  areæ
                    <emph type="italics"/>
                  ABD, ACE
                    <emph.end type="italics"/>
                  ſunt ultimo in duplicata ratione laterum
                    <emph type="italics"/>
                  AD,
                    <lb/>
                  AE.
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA X.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Spatia quæ corpus urgente quacunque Vi finita deſcribit, five Vis
                    <lb/>
                  illa determinata & immutabilis ſit, five eadem continuo auge­
                    <lb/>
                  atur vel continuo diminuatur, ſunt ipſo motus initio in duplica­
                    <lb/>
                  ta ratione Temporum.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Exponantur tempora per lineas
                    <emph type="italics"/>
                  AD, AE,
                    <emph.end type="italics"/>
                  & velocitates genitæ
                    <lb/>
                  per ordinatas
                    <emph type="italics"/>
                  DB, EC
                    <emph.end type="italics"/>
                  ; & ſpatia his velocitatibus deſcripta, erunt
                    <lb/>
                  ut areæ
                    <emph type="italics"/>
                  ABD, ACE
                    <emph.end type="italics"/>
                  his ordinatis deſcriptæ, hoc eſt, ipſo motus
                    <lb/>
                  initio (per Lemma IX) in duplicata ratione temporum
                    <emph type="italics"/>
                  AD, AE.
                    <lb/>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
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