Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/264.jpg" pagenum="236"/>
                    <arrow.to.target n="note212"/>
                  termini ſubſequentes evadent infinite minores tertio, ideoque neg­
                    <lb/>
                  ligi poſſunt. </s>
                  <s>Terminus quartus determinat variationem curva­
                    <lb/>
                  turæ, quintus variationem variationis, & ſic deinceps. </s>
                  <s>Unde obi­
                    <lb/>
                  ter patet uſus non contemnendus harum Serierum in ſolutione
                    <lb/>
                  Problematum quæ pendent a tangentibus & curvatura curvarum. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note212"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>Conferatur jam ſeries
                    <emph type="italics"/>
                  e-(ao/e)-(nnoo/2e
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  )-(anno
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  /2e
                    <emph type="sup"/>
                  5
                    <emph.end type="sup"/>
                  )
                    <emph.end type="italics"/>
                  -&c, cum ſerie
                    <lb/>
                  P-Q
                    <emph type="italics"/>
                  o
                    <emph.end type="italics"/>
                  -R
                    <emph type="italics"/>
                  oo
                    <emph.end type="italics"/>
                  -S
                    <emph type="italics"/>
                  o
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  -&c. </s>
                  <s>& perinde pro P, Q, R & S ſcribatur
                    <lb/>
                    <emph type="italics"/>
                  e, (a/e), (nn/2e
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  )
                    <emph.end type="italics"/>
                  & (
                    <emph type="italics"/>
                  ann/2e
                    <emph type="sup"/>
                  5
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ), & pro √1+QQ ſcribatur √1+(
                    <emph type="italics"/>
                  aa/ee
                    <emph.end type="italics"/>
                  ) ſeu
                    <emph type="italics"/>
                  n/e,
                    <emph.end type="italics"/>
                  &
                    <lb/>
                  prodibit Medii denſitas ut (
                    <emph type="italics"/>
                  a/ne
                    <emph.end type="italics"/>
                  ), hoc eſt, (ob datam
                    <emph type="italics"/>
                  n,
                    <emph.end type="italics"/>
                  ) ut
                    <emph type="italics"/>
                  a/e,
                    <emph.end type="italics"/>
                  ſeu
                    <lb/>
                  (
                    <emph type="italics"/>
                  AC/CH
                    <emph.end type="italics"/>
                  ), id eſt, ut tangentis longitudo illa
                    <emph type="italics"/>
                  HT
                    <emph.end type="italics"/>
                  quæ ad ſemidiame­
                    <lb/>
                  trum
                    <emph type="italics"/>
                  AF
                    <emph.end type="italics"/>
                  ipſi
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  normaliter inſiſtentem terminatur: & reſiſten­
                    <lb/>
                  tia erit ad gravitatem ut 3
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  ad 2
                    <emph type="italics"/>
                  n,
                    <emph.end type="italics"/>
                  id eſt, ut 3
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ad Circuli
                    <lb/>
                  diametrum
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  : velocitas autem erit ut √
                    <emph type="italics"/>
                  CH.
                    <emph.end type="italics"/>
                  Quare ſi corpus
                    <lb/>
                  juſta cum velocitate ſecundum lineam ipſi
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  parallelam exeat
                    <lb/>
                  de loco
                    <emph type="italics"/>
                  F,
                    <emph.end type="italics"/>
                  & Medii denſitas in ſingulis locis
                    <emph type="italics"/>
                  H
                    <emph.end type="italics"/>
                  ſit ut longi­
                    <lb/>
                  tudo tangentis
                    <emph type="italics"/>
                  HT,
                    <emph.end type="italics"/>
                  & reſiſtentia etiam in loco aliquo
                    <emph type="italics"/>
                  H
                    <emph.end type="italics"/>
                  ſit ad
                    <lb/>
                  vim gravitatis ut 3
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PQ,
                    <emph.end type="italics"/>
                  corpus illud deſcribet Circuli
                    <lb/>
                  quadrantem
                    <emph type="italics"/>
                  FHQ. Q.E.I.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>At ſi corpus idem de loco
                    <emph type="italics"/>
                  P,
                    <emph.end type="italics"/>
                  ſecundum lineam ipſi
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  per­
                    <lb/>
                  pendicularem egrederetur, & in arcu ſemicirculi
                    <emph type="italics"/>
                  PFQ
                    <emph.end type="italics"/>
                  moveri
                    <lb/>
                  inciperet, ſumenda eſſet
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  ad contrarias partes centri
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                    <lb/>
                  & propterea ſignum ejus mutandum eſſet & ſcribendum -
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  pro
                    <lb/>
                  +
                    <emph type="italics"/>
                  a.
                    <emph.end type="italics"/>
                  Quo pacto prodiret Medii denſitas ut -
                    <emph type="italics"/>
                  a/e
                    <emph.end type="italics"/>
                  . </s>
                  <s>Negativam
                    <lb/>
                  autem denſitatem, hoc eſt, quæ motus corporum accelerat, Na­
                    <lb/>
                  tura non admittit: & propterea naturaliter fieri non poteſt, ut
                    <lb/>
                  corpus aſcendendo a
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  deſcribat Circuli quadrantem
                    <emph type="italics"/>
                  PF.
                    <emph.end type="italics"/>
                  Ad
                    <lb/>
                  hunc effectum deberet corpus a Medio impellente accelerari, non
                    <lb/>
                  a reſiſtente impediri. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Exempl.
                    <emph.end type="italics"/>
                  2. Sit linea
                    <emph type="italics"/>
                  PFHQ
                    <emph.end type="italics"/>
                  Parabola, axem habens
                    <emph type="italics"/>
                  AF
                    <emph.end type="italics"/>
                  ho­
                    <lb/>
                  rizonti
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  perpendicularem, & requiratur Medii denſitas quæ
                    <lb/>
                  faciat ut Projectile in ipſa moveatur. </s>
                </p>
                <p type="main">
                  <s>Ex natura Parabolæ, rectangulum
                    <emph type="italics"/>
                  PDQ
                    <emph.end type="italics"/>
                  æquale eſt rectan­
                    <lb/>
                  gulo ſub ordinata
                    <emph type="italics"/>
                  DI
                    <emph.end type="italics"/>
                  & recta aliqua data: hoc eſt, ſi dicantur </s>
                </p>
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