Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/255.jpg" pagenum="227"/>
                  tibus [
                    <emph type="italics"/>
                  Data Æquatione quotcunque Fluentes quantitates invelven-
                    <emph.end type="italics"/>
                    <lb/>
                    <arrow.to.target n="note203"/>
                    <emph type="italics"/>
                  te, Fluxiones invenire, & vice verſa
                    <emph.end type="italics"/>
                  ] eandem celarem: reſcripſit
                    <lb/>
                  Vir Clariſſimus ſe quoQ.E.I. ejuſmodi methodum incidiſſe, & me­
                    <lb/>
                  thodum ſuam communicavit a mea vix abludentem præterquam in
                    <lb/>
                  verborum & notarum formulis, & Idea generationis quantitatum. </s>
                  <s>
                    <lb/>
                  Utriuſque fundamentum continetur in hoc Lemmate. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note203"/>
                  LIBER
                    <lb/>
                  SECUNDUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO VIII. THEOREMA VI.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si corpus in Medio uniformi, Gravitate uniformiter agente, recta
                    <lb/>
                  aſcendat vel deſcendat, & ſpatium totum deſcriptum diſtingua­
                    <lb/>
                  tur in partes æquales, inque principiis ſingularum partium
                    <lb/>
                  (addendo reſiſtentiam Medii ad vim gravitatis, quando cor­
                    <lb/>
                  pus aſcendit, vel ſubducendo ipſam quando corpus deſcendit)
                    <lb/>
                  colligantur vires abſolutæ; dico quod vires illæ abſolutæ ſunt
                    <lb/>
                  in progreſſione Geometrica.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Exponatur enim vis gravitatis per datam lineam
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ; reſiſten­
                    <lb/>
                  tia per lineam indefinitam
                    <emph type="italics"/>
                  AK
                    <emph.end type="italics"/>
                  ; vis abſoluta in deſcenſu corporis
                    <lb/>
                  per differentiam
                    <emph type="italics"/>
                  KC
                    <emph.end type="italics"/>
                  ; velocitas corporis per lineam
                    <emph type="italics"/>
                  AP
                    <emph.end type="italics"/>
                  (quæ ſit
                    <lb/>
                  media proportionalis inter
                    <emph type="italics"/>
                  AK
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  AC,
                    <emph.end type="italics"/>
                  ideoQ.E.I. ſubduplicata
                    <lb/>
                  ratione reſiſtentiæ;) incrementum reſiſtentiæ data temporis particu­
                    <lb/>
                  la factum per lineolam
                    <emph type="italics"/>
                  KL,
                    <emph.end type="italics"/>
                  & contemporaneum velocitatis incre­
                    <lb/>
                  mentum per lineolam
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  ; & centro
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  Aſymptotis rectangulis
                    <lb/>
                    <emph type="italics"/>
                  CA, CH
                    <emph.end type="italics"/>
                  deſcribatur Hyperbola quævis
                    <emph type="italics"/>
                  BNS,
                    <emph.end type="italics"/>
                  erectis perpendi­
                    <lb/>
                  culis
                    <emph type="italics"/>
                  AB, KN, LO, PR, QS
                    <emph.end type="italics"/>
                  occurrens in
                    <emph type="italics"/>
                  B, N, O, R, S.
                    <emph.end type="italics"/>
                  Quo­
                    <lb/>
                  niam
                    <emph type="italics"/>
                  AK
                    <emph.end type="italics"/>
                  eſt ut
                    <emph type="italics"/>
                  APq,
                    <emph.end type="italics"/>
                  erit hujus momentum
                    <emph type="italics"/>
                  KL
                    <emph.end type="italics"/>
                  ut illius mo­
                    <lb/>
                  mentum 2
                    <emph type="italics"/>
                  APQ,
                    <emph.end type="italics"/>
                  id eſt, ut
                    <emph type="italics"/>
                  AP
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  KC.
                    <emph.end type="italics"/>
                  Nam velocitatis incre­
                    <lb/>
                  mentum
                    <emph type="italics"/>
                  PQ,
                    <emph.end type="italics"/>
                  (per motus Leg.11.) proportionale eſt vi generanti
                    <emph type="italics"/>
                  KC.
                    <emph.end type="italics"/>
                    <lb/>
                  Componatur ratio ipſius
                    <emph type="italics"/>
                  KL
                    <emph.end type="italics"/>
                  cum ratione ipſius
                    <emph type="italics"/>
                  KN,
                    <emph.end type="italics"/>
                  & fiet rect­
                    <lb/>
                  angulum
                    <emph type="italics"/>
                  KLXKN
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  APXKCXKN
                    <emph.end type="italics"/>
                  ; hoc eſt, ob datum rect­
                    <lb/>
                  angulum
                    <emph type="italics"/>
                  KCXKN,
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  AP.
                    <emph.end type="italics"/>
                  Atqui areæ Hyperbolicæ
                    <emph type="italics"/>
                  KNOL
                    <emph.end type="italics"/>
                    <lb/>
                  ad rectangulum
                    <emph type="italics"/>
                  KLXKN
                    <emph.end type="italics"/>
                  ratio ultima, ubi coeunt puncta
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  L,
                    <emph.end type="italics"/>
                    <lb/>
                  eſt æqualitatis. </s>
                  <s>Ergo area illa Hyperbolica evaneſcens eſt ut
                    <emph type="italics"/>
                  AP.
                    <emph.end type="italics"/>
                    <lb/>
                  Componitur igitur area tota Hyperbolica
                    <emph type="italics"/>
                  ABOL
                    <emph.end type="italics"/>
                  ex particulis
                    <lb/>
                    <emph type="italics"/>
                  KNOL
                    <emph.end type="italics"/>
                  velocitati
                    <emph type="italics"/>
                  AP
                    <emph.end type="italics"/>
                  ſemper proportionalibus, & propterea
                    <lb/>
                  ſpatio velocitate iſta deſcripto proportionalis eſt. </s>
                  <s>Dividatur jam
                    <lb/>
                  area illa in partes æquales
                    <emph type="italics"/>
                  ABMI, IMNK, KNOL,
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>& vi-</s>
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