Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/463.jpg" pagenum="435"/>
                  -
                    <emph type="italics"/>
                  NMXmC
                    <emph.end type="italics"/>
                  : & harum differentia
                    <emph type="italics"/>
                  LNXMm-NMX—MC+mC,
                    <emph.end type="italics"/>
                    <lb/>
                    <arrow.to.target n="note464"/>
                  eſt vis particularum ambarum ſimul ſumptarum ad Terram
                    <lb/>
                  rotandam. </s>
                  <s>Hujus differentiæ pars affirmativa
                    <emph type="italics"/>
                  LNXMm
                    <emph.end type="italics"/>
                  ſeu
                    <lb/>
                  2
                    <emph type="italics"/>
                  LNXNX,
                    <emph.end type="italics"/>
                  eſt ad particularum duarum ejuſdem magnitudi­
                    <lb/>
                  nis in
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  conſiſtentium vim 2
                    <emph type="italics"/>
                  AHXHC,
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  LXq
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                    <expan abbr="ACq.">ACque</expan>
                    <emph.end type="italics"/>
                    <lb/>
                  Et pars negativa
                    <emph type="italics"/>
                  NMX—MC+mC
                    <emph.end type="italics"/>
                  ſeu 2
                    <emph type="italics"/>
                  XYXCY,
                    <emph.end type="italics"/>
                  ad parti­
                    <lb/>
                  cularum earundem in
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  conſiſtentium vim 2
                    <emph type="italics"/>
                  AHXHC,
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                    <expan abbr="ACq.">ACque</expan>
                    <emph.end type="italics"/>
                  Ac proinde partium differentia, id eſt, par­
                    <lb/>
                  ticularum duarum
                    <emph type="italics"/>
                  L
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  l
                    <emph.end type="italics"/>
                  ſimul ſumptarum vis ad Terram rotan­
                    <lb/>
                  dam, eſt ad vim particularum duarum iiſdem æqualium & in loco
                    <lb/>
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  conſiſtentium, ad Terram itidem rotandam, ut
                    <emph type="italics"/>
                  LXq-CXq
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                    <expan abbr="ACq.">ACque</expan>
                    <emph.end type="italics"/>
                  Sed ſi circuli
                    <emph type="italics"/>
                  IK
                    <emph.end type="italics"/>
                  circumferentia
                    <emph type="italics"/>
                  IK
                    <emph.end type="italics"/>
                  dividatur in par­
                    <lb/>
                  ticulas innumeras æquales
                    <emph type="italics"/>
                  L,
                    <emph.end type="italics"/>
                  erunt omnes
                    <emph type="italics"/>
                  LXq
                    <emph.end type="italics"/>
                  ad totidem
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                    <lb/>
                  ut 1 ad 2, (per Lem. </s>
                  <s>I.) atque ad totidem
                    <emph type="italics"/>
                  ACq,
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                  2
                    <emph type="italics"/>
                  ACq
                    <emph.end type="italics"/>
                  ; & totidem
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad totidem
                    <emph type="italics"/>
                  ACq
                    <emph.end type="italics"/>
                  ut 2
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad 2
                    <emph type="italics"/>
                    <expan abbr="ACq.">ACque</expan>
                    <emph.end type="italics"/>
                    <lb/>
                  Quare vires conjunctæ particularum omnium in circuitu circuli
                    <lb/>
                    <emph type="italics"/>
                  IK,
                    <emph.end type="italics"/>
                  ſunt ad vires conjunctas particularum totidem in loco
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  -2
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad 2
                    <emph type="italics"/>
                  ACq
                    <emph.end type="italics"/>
                  : & propterea (per Lem. </s>
                  <s>I.) ad vires
                    <lb/>
                  conjunctas particularum totidem in circuitu circuli
                    <emph type="italics"/>
                  AE,
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  -2
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                    <expan abbr="ACq.">ACque</expan>
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note464"/>
                  LIBER
                    <lb/>
                  TERTIUS.</s>
                </p>
                <p type="main">
                  <s>Jam vero ſi Sphæræ diameter
                    <emph type="italics"/>
                  Pp
                    <emph.end type="italics"/>
                  dividatur in partes innume­
                    <lb/>
                  ras æquales, quibus inſiſtant circuli totidem
                    <emph type="italics"/>
                  IK
                    <emph.end type="italics"/>
                  ; materia in peri­
                    <lb/>
                  metro circuli cujuſque
                    <emph type="italics"/>
                  IK
                    <emph.end type="italics"/>
                  erit ut
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  : ideoque vis materiæ
                    <lb/>
                  illius ad Terram rotandam, erit ut
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  -2
                    <emph type="italics"/>
                    <expan abbr="CXq.">CXque</expan>
                    <emph.end type="italics"/>
                  Et
                    <lb/>
                  vis materiæ ejuſdem, ſi in circuli
                    <emph type="italics"/>
                  AE
                    <emph.end type="italics"/>
                  perimetro conſiſteret, eſſet
                    <lb/>
                  ut
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                    <expan abbr="ACq.">ACque</expan>
                    <emph.end type="italics"/>
                  Et propterea vis particularum omnium ma­
                    <lb/>
                  teriæ totius, extra globum in perimetris circulorum omnium con­
                    <lb/>
                  ſiſtentis, eſt ad vim particularum totidem in perimetro circuli
                    <lb/>
                  maximi
                    <emph type="italics"/>
                  AE
                    <emph.end type="italics"/>
                  conſiſtentis, ut omnia
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  -2
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                  totidem
                    <emph type="italics"/>
                  IXq
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  ACq,
                    <emph.end type="italics"/>
                  hoc eſt, ut omnia
                    <emph type="italics"/>
                  ACq-CXq
                    <emph.end type="italics"/>
                  in
                    <lb/>
                    <emph type="italics"/>
                  ACq
                    <emph.end type="italics"/>
                  -3
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad totidem
                    <emph type="italics"/>
                  ACq-CXq
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  ACq,
                    <emph.end type="italics"/>
                  id eſt, ut
                    <lb/>
                  omnia
                    <emph type="italics"/>
                  ACqq
                    <emph.end type="italics"/>
                  -4
                    <emph type="italics"/>
                  ACqXCXq
                    <emph.end type="italics"/>
                  +3
                    <emph type="italics"/>
                  CXqq
                    <emph.end type="italics"/>
                  ad totidem
                    <emph type="italics"/>
                  ACqq
                    <lb/>
                  -ACqXCXq,
                    <emph.end type="italics"/>
                  hoc eſt, ut tota quantitas fluens cujus fluxio
                    <lb/>
                  eſt
                    <emph type="italics"/>
                  ACqq
                    <emph.end type="italics"/>
                  -4
                    <emph type="italics"/>
                  ACqXCXq
                    <emph.end type="italics"/>
                  +3
                    <emph type="italics"/>
                  CXqq,
                    <emph.end type="italics"/>
                  ad totam quantitatem flu­
                    <lb/>
                  entem cujus fluxio eſt
                    <emph type="italics"/>
                  ACqq-ACqXCXq
                    <emph.end type="italics"/>
                  ; ac proinde per Me­
                    <lb/>
                  thodum Fluxionum, ut
                    <emph type="italics"/>
                  ACqqXCX
                    <emph.end type="italics"/>
                  -4/3
                    <emph type="italics"/>
                  ACqxCXcub
                    <emph.end type="italics"/>
                  +3/5
                    <emph type="italics"/>
                  CXqc
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  ACqqXCX
                    <emph.end type="italics"/>
                  -1/3
                    <emph type="italics"/>
                  ACqXCXcub,
                    <emph.end type="italics"/>
                  id eſt, ſi pro
                    <emph type="italics"/>
                  CX
                    <emph.end type="italics"/>
                  ſcribatur
                    <lb/>
                  tota
                    <emph type="italics"/>
                  Cp
                    <emph.end type="italics"/>
                  vel
                    <emph type="italics"/>
                  AC,
                    <emph.end type="italics"/>
                  ut (4/15)
                    <emph type="italics"/>
                  ACqc
                    <emph.end type="italics"/>
                  ad 2/3
                    <emph type="italics"/>
                  ACqc,
                    <emph.end type="italics"/>
                  hoc eſt, ut duo ad
                    <lb/>
                  quinque.
                    <emph type="italics"/>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
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