Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/145.jpg" pagenum="117"/>
                    <arrow.to.target n="note93"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note93"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XLI. PROBLEMA XXVIII.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Poſita cujuſcunque generis Vi centripeta & conceſſis Figurarum
                    <lb/>
                  curvilinearum quadraturis, requiruntur tum Trajectoriæ in qui­
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                  bus corpora movebuntur, tum Tempora motuum in Trajectoriis
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                  inventis.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Tendat vis quælibet ad centrum
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  & invenienda ſit Trajectoria
                    <lb/>
                    <emph type="italics"/>
                  VITKk.
                    <emph.end type="italics"/>
                  Detur Circulus
                    <emph type="italics"/>
                  VXY
                    <emph.end type="italics"/>
                  centro
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  intervallo quovis
                    <emph type="italics"/>
                  CV
                    <emph.end type="italics"/>
                    <lb/>
                  deſcriptus, centroque eodem deſcribantur alii quivis circuli
                    <emph type="italics"/>
                  ID,
                    <lb/>
                  KE
                    <emph.end type="italics"/>
                  Trajectoriam ſecantes in
                    <emph type="italics"/>
                  I
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  rectamque
                    <emph type="italics"/>
                  CV
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  E.
                    <emph.end type="italics"/>
                    <lb/>
                  Age tum rectam
                    <emph type="italics"/>
                  CNIX
                    <emph.end type="italics"/>
                  ſecantem circulos
                    <emph type="italics"/>
                  KE, VY
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  N
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  X,
                    <emph.end type="italics"/>
                    <lb/>
                  tum rectam
                    <emph type="italics"/>
                  CKY
                    <emph.end type="italics"/>
                  occurrentem circulo
                    <emph type="italics"/>
                  VXY
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  Y.
                    <emph.end type="italics"/>
                  Sint autem
                    <lb/>
                  puncta
                    <emph type="italics"/>
                  I
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  ſibi invicem viciniſſima, & pergat corpus ab
                    <emph type="italics"/>
                  V
                    <emph.end type="italics"/>
                  per
                    <lb/>
                    <emph type="italics"/>
                  I, T
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  k;
                    <emph.end type="italics"/>
                  ſitque punctum
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  locus ille de quo corpus aliud
                    <lb/>
                  cadere debet ut in loco
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  velocitatem acquirat æqualem veloci­
                    <lb/>
                  tati corporis prioris in
                    <emph type="italics"/>
                  I
                    <emph.end type="italics"/>
                  ; & ſtantibus quæ in Propoſitione XXXIX,
                    <lb/>
                  lineola
                    <emph type="italics"/>
                  IK,
                    <emph.end type="italics"/>
                  dato tempore quam minimo deſcripta, erit ut ve­
                    <lb/>
                  locitas atque adeo ut latus quadratum areæ
                    <emph type="italics"/>
                  ABFD,
                    <emph.end type="italics"/>
                  & triangu­
                    <lb/>
                  lum
                    <emph type="italics"/>
                  ICK
                    <emph.end type="italics"/>
                  tempori proportionale dabitur, adeoque
                    <emph type="italics"/>
                  KN
                    <emph.end type="italics"/>
                  erit reci­
                    <lb/>
                  proce ut altitudo
                    <emph type="italics"/>
                  IC,
                    <emph.end type="italics"/>
                  id eſt, ſi detur quantitas aliqua Q, & alti­
                    <lb/>
                  tudo
                    <emph type="italics"/>
                  IC
                    <emph.end type="italics"/>
                  nominetur A, ut Q/A. </s>
                  <s>Hanc quantitatem Q/A nominemus Z,
                    <lb/>
                  & ponamus eam eſſe magnitudinem ipſius Q ut ſit in aliquo
                    <lb/>
                  caſu √
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  ad Z ut eſt
                    <emph type="italics"/>
                  IK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KN,
                    <emph.end type="italics"/>
                  & erit in omni caſu
                    <lb/>
                  √
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  ad Z ut
                    <emph type="italics"/>
                  IK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KN,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  ad ZZ ut
                    <emph type="italics"/>
                    <expan abbr="IKq.">IKque</expan>
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                    <expan abbr="KNq.">KNque</expan>
                    <emph.end type="italics"/>
                    <lb/>
                  & diviſim
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  -ZZ ad ZZ ut
                    <emph type="italics"/>
                  IN quad
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KN quad,
                    <emph.end type="italics"/>
                  ad­
                    <lb/>
                  eoque √
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  -ZZ ad (Z ſeu)Q/A ut
                    <emph type="italics"/>
                  IN
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KN,
                    <emph.end type="italics"/>
                  & propterea
                    <lb/>
                  AX
                    <emph type="italics"/>
                  KN
                    <emph.end type="italics"/>
                  æquale (QX
                    <emph type="italics"/>
                  IN/√ABFD
                    <emph.end type="italics"/>
                  -ZZ). Unde cum
                    <emph type="italics"/>
                  YXXXC
                    <emph.end type="italics"/>
                  ſit ad
                    <lb/>
                  AX
                    <emph type="italics"/>
                  KN
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  CXq
                    <emph.end type="italics"/>
                  ad AA, erit rectangulum
                    <emph type="italics"/>
                  YXXXC
                    <emph.end type="italics"/>
                  æquale
                    <lb/>
                  (QX
                    <emph type="italics"/>
                  INXCX quad.
                    <emph.end type="italics"/>
                  /AA√
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  -ZZ). Igitur ſi in perpendiculo
                    <emph type="italics"/>
                  DF
                    <emph.end type="italics"/>
                  capiantur
                    <lb/>
                  ſemper
                    <emph type="italics"/>
                  Db, Dc
                    <emph.end type="italics"/>
                  ipſis (Q/2√
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  -ZZ) & (QX
                    <emph type="italics"/>
                  CX quad.
                    <emph.end type="italics"/>
                  /2AA√
                    <emph type="italics"/>
                  ABFD
                    <emph.end type="italics"/>
                  -ZZ)
                    <lb/>
                  æquales reſpective, & deſcribantur curvæ lineæ
                    <emph type="italics"/>
                  ab, cd
                    <emph.end type="italics"/>
                  quas </s>
                </p>
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        </body>
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