Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                    <pb xlink:href="039/01/249.jpg" pagenum="221"/>
                  quadrata ſunt ut earundem differentiæ; & idcirco cum quadrata ve­
                    <lb/>
                    <arrow.to.target n="note197"/>
                  locitatum fuerint etiam ut ipſarum differentiæ, ſimilis erit amba­
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                  rum progreſſio. </s>
                  <s>Quo demonſtrato, conſequens eſt etiam ut areæ
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                  his lineis deſcriptæ ſint in progreſſione conſimili cum ſpatiis quæ
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                  velocitatibus deſcribuntur. </s>
                  <s>Ergo ſi velocitas initio primi tempo­
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                  ris
                    <emph type="italics"/>
                  AK
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                  exponatur per lineam
                    <emph type="italics"/>
                  AB,
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                  & velocitas initio ſecundi
                    <emph type="italics"/>
                  KL
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                    <lb/>
                  per lineam
                    <emph type="italics"/>
                  Kk,
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                  & longitudo primo tempore deſcripta per aream
                    <lb/>
                    <emph type="italics"/>
                  AKkB
                    <emph.end type="italics"/>
                  ; velocitates omnes ſubſequentes exponentur per lineas
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                  ſubſequentes
                    <emph type="italics"/>
                  Ll, Mm,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>& longitudines deſcriptæ per areas
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                    <emph type="italics"/>
                  Kl, Lm,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>Et compoſite, ſi tempus totum exponatur per ſum­
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                  mam partium ſuarum
                    <emph type="italics"/>
                  AM,
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                  longitudo tota deſcripta exponetur per
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                  ſummam partium ſuarum
                    <emph type="italics"/>
                  AMmB.
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                  Concipe jam tempus
                    <emph type="italics"/>
                  AM
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                  ita
                    <lb/>
                  dividi in partes
                    <emph type="italics"/>
                  AK, KL, LM,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>ut ſint
                    <emph type="italics"/>
                  CA, CK, CL, CM,
                    <emph.end type="italics"/>
                    <lb/>
                  &c. </s>
                  <s>in progreſſione Geometrica; & erunt partes illæ in eadem pro­
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                  greſſione, & velocitates
                    <emph type="italics"/>
                  AB, Kk, Ll, Mm,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>in progreſſione ea­
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                  dem inverſa, atque ſpatia deſcripta
                    <emph type="italics"/>
                  Ak, Kl, Lm,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>æqualia.
                    <lb/>
                    <emph type="italics"/>
                  Q.E.D.
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                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note197"/>
                  LIBER
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                  SECUNDUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  1. Pater ergo quod, ſi tempus exponatur per Aſymptoti
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                  partem quamvis
                    <emph type="italics"/>
                  AD,
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                  & velocitas in principio temporis per ordi­
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                  natim applicatam
                    <emph type="italics"/>
                  AB
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                  ; velocitas in fine temporis exponetur per
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                  ordinatam
                    <emph type="italics"/>
                  DG,
                    <emph.end type="italics"/>
                  & ſpatium totum deſcriptum per aream Hyper­
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                  bolicam adjacentem
                    <emph type="italics"/>
                  ABGD
                    <emph.end type="italics"/>
                  ; necnon ſpatium quod corpus ali­
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                  quod eodem tempore
                    <emph type="italics"/>
                  AD,
                    <emph.end type="italics"/>
                  velocitate prima
                    <emph type="italics"/>
                  AB,
                    <emph.end type="italics"/>
                  in Medio non
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                  reſiſtente deſcribere poſſet, per rectangulum
                    <emph type="italics"/>
                  ABXAD.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  2. Unde datur ſpatium in Medio reſiſtente deſcriptum, ca­
                    <lb/>
                  piendo illud ad ſpatium quod velocitate uniformi
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  in medio non
                    <lb/>
                  reſiſtente ſimul deſcribi poſſet, ut eſt area Hyperbolica
                    <emph type="italics"/>
                  ABGD
                    <emph.end type="italics"/>
                    <lb/>
                  ad rectangulum
                    <emph type="italics"/>
                  ABXAD.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  3. Datur etiam reſiſtentia Medii, ſtatuendo eam ipſo mo­
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                  tus initio æqualem eſſe vi uniformi centripetæ, quæ in cadente cor­
                    <lb/>
                  pore, tempore
                    <emph type="italics"/>
                  AC,
                    <emph.end type="italics"/>
                  in Medio non reſiſtente, generare poſſet velo­
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                  citatem
                    <emph type="italics"/>
                  AB.
                    <emph.end type="italics"/>
                  Nam ſi ducatur
                    <emph type="italics"/>
                  BT
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                  quæ tangat Hyperbolam in
                    <emph type="italics"/>
                  B,
                    <emph.end type="italics"/>
                    <lb/>
                  & occurrat Aſymptoto in
                    <emph type="italics"/>
                  T
                    <emph.end type="italics"/>
                  ; recta
                    <emph type="italics"/>
                  AT
                    <emph.end type="italics"/>
                  æqualis erit ipſi
                    <emph type="italics"/>
                  AC,
                    <emph.end type="italics"/>
                  &
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                  tempus exponet quo reſiſtentia prima uniformiter continuata tolle­
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                  re poſſet velocitatem totam
                    <emph type="italics"/>
                  AB.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol
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                  4. Et inde datur etiam proportio hujus reſiſtentiæ ad vim
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                  gravitatis, aliamve quamvis datam vim centripetam. </s>
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                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  5. Et viceverſa, ſi datur proportio reſiſtentiæ ad datam
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                  quamvis vim centripetam; datur tempus
                    <emph type="italics"/>
                  AC,
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                  quo vis centripeta
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                  reſiſtentiæ æqualis generare poſſit velocitatem quamvis
                    <emph type="italics"/>
                  AB
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                  ; & in-</s>
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