Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  DE MOTU
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                  CORPORUM</s>
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                  PROPOSITIO II. THEOREMA II.
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                  Si Corpori reſiſtitur in ratione velocitatis, & idem ſola vi inſita
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                  per Medium ſimilare moveatur, ſumantur autem tempora æqua­
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                  lia: velocitates in principiis ſingulorum temporum ſunt in pro­
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                  greſſione Geometrica, & ſpatia ſingulis temporibus deſcripta
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                  ſunt ut velocitates.
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                  Cas.
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                  1. Dividatur tempus in particulas æquales; & ſi ipſis parti­
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                  cularum initiis agat vis reſiſtentiæ impulſo unico, quæ ſit ut velo­
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                  citas: erit decrementum velocitatis ſingulis temporis particulis ut
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                  eadem velocitas. </s>
                  <s>Sunt ergo velocitates differentiis ſuis proportio­
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                  nales, & propterea (per Lem. </s>
                  <s>I. Lib. </s>
                  <s>II.) continue proportionales. </s>
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                  Proinde ſi ex æquali particularum numero componantur tempora
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                  quælibet æqualia, erunt velocitates ipſis temporum initiis, ut ter­
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                  mini in progreſſione continua, qui per ſaltum capiuntur, omiſſo
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                  paſſim æquali terminorum intermediorum numero. </s>
                  <s>Componuntur
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                  autem horum terminorum rationes ex æqualibus rationibus termi­
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                  norum intermediorum æqualiter repetitis, & propterea ſunt æqua­
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                  les. </s>
                  <s>Igitur velocitates, his terminis proportionales, ſunt in pro­
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                  greſſione Geometrica. </s>
                  <s>Minuantur jam æquales illæ temporum par­
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                  ticulæ, & augeatur earum numerus in infinitum, eo ut reſiſtentiæ
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                  impulſus reddatur continuus; & velocitates in principiis æqualium
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                  temporum, ſemper continue proportionales, erunt in hoc etiam
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                  caſu continue proportionales.
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                  Q.E.D.
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                  Cas.
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                  2. Et diviſim velocitatum differentiæ, hoc eſt, earum partes
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                  ſingulis temporibus amiſſæ, ſunt ut totæ: Spatia autem ſingulis
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                  temporibus deſcripta ſunt ut velocitatum partes amiſſæ, (per Prop. </s>
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                  I. </s>
                  <s>Lib II.) & propterea etiam ut totæ.
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                    <expan abbr="q.">que</expan>
                  E. D.
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                  Corol.
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                  Hinc ſi Aſymptotis rectangulis
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                  ADC, CH
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                  deſcribatur
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                  Hyperbola
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                  BG,
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                  ſintque
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                  AB, DG
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                  ad Aſymptoton
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                  AC
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                  perpen­
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                  diculares, & exponatur tum corporis velocitas tum reſiſtentia Me­
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                  dii, ipſo motus initio, per lineam quam­
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                  vis datam
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                  AC,
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                  elapſo autem tempore ali­
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                  quo per lineam indefinitam
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                  DC:
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                  exponi
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                  poteſt tempus per aream
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                  ABGD,
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                  & ſpa­
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                  tium eo tempore deſcriptum per lineam
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                    <emph type="italics"/>
                  AD.
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                  Nam ſi area illa per motum puncti
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                    <emph type="italics"/>
                  D
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                  augeatur uniformiter ad modum tempo-</s>
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