Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/218.jpg" pagenum="190"/>
                    <arrow.to.target n="note166"/>
                  cem ut
                    <emph type="italics"/>
                  SP quad
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SA quad:
                    <emph.end type="italics"/>
                  Si in quadruplicata, ut
                    <emph type="italics"/>
                  SP cub
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                    <emph type="italics"/>
                  SA cub.
                    <emph.end type="italics"/>
                  Unde cum attractio in
                    <emph type="italics"/>
                  P,
                    <emph.end type="italics"/>
                  in hoc ultimo caſu, inventa
                    <lb/>
                  fuit reciproce ut
                    <emph type="italics"/>
                  PS cubXPI,
                    <emph.end type="italics"/>
                  attractio in
                    <emph type="italics"/>
                  I
                    <emph.end type="italics"/>
                  erit reciproce ut
                    <lb/>
                    <emph type="italics"/>
                  SA cubXPI,
                    <emph.end type="italics"/>
                  id eſt (ob datum
                    <emph type="italics"/>
                  SA cub
                    <emph.end type="italics"/>
                  ) reciproce ut
                    <emph type="italics"/>
                  PI.
                    <emph.end type="italics"/>
                  Et
                    <lb/>
                  ſimilis eſt progreſſus in infinitum. </s>
                  <s>Theorema vero ſic demon­
                    <lb/>
                  ſtratur. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note166"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>Stantibus jam ante conſtructis, & exiſtente corpore in loco
                    <lb/>
                  quovis
                    <emph type="italics"/>
                  P,
                    <emph.end type="italics"/>
                  ordinatim applicata
                    <emph type="italics"/>
                  DN
                    <emph.end type="italics"/>
                  inventa fuit ut (
                    <emph type="italics"/>
                  DEqXPS/PEXV
                    <emph.end type="italics"/>
                  ).
                    <lb/>
                  Ergo ſi agatur
                    <emph type="italics"/>
                  IE,
                    <emph.end type="italics"/>
                  ordinata illa ad alium quemvis locum
                    <emph type="italics"/>
                  I,
                    <emph.end type="italics"/>
                  mu­
                    <lb/>
                  tatis mutandis, evadet ut (
                    <emph type="italics"/>
                  DEqXIS/IEXV
                    <emph.end type="italics"/>
                  ). Pone vires centripetas, e
                    <lb/>
                  Sphæræ puncto quovis
                    <emph type="italics"/>
                  E
                    <emph.end type="italics"/>
                  manantes, eſſe ad invicem in diſtantiis
                    <lb/>
                    <emph type="italics"/>
                  IE, PE,
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  PE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  IE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                  ,
                    <emph.end type="italics"/>
                  (ubi numerus
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  deſignet indicem
                    <lb/>
                  poteſtatum
                    <emph type="italics"/>
                  PE
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  IE
                    <emph.end type="italics"/>
                  ) & ordinatæ illæ fient ut (
                    <emph type="italics"/>
                  DEqXPS/PEXPE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ) &
                    <lb/>
                  (
                    <emph type="italics"/>
                  DEqXIS/IEXIE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ), quarum ratio ad invicem eſt ut
                    <emph type="italics"/>
                  PSXIEXIE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                    <emph type="italics"/>
                  ISXPEXPE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                  .
                    <emph.end type="italics"/>
                  Quoniam ob ſimilia triangula
                    <emph type="italics"/>
                  SPE, SEI,
                    <emph.end type="italics"/>
                  fit
                    <lb/>
                    <emph type="italics"/>
                  IE
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PE
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  IS
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SE
                    <emph.end type="italics"/>
                  vel
                    <emph type="italics"/>
                  SA
                    <emph.end type="italics"/>
                  ; pro ratione
                    <emph type="italics"/>
                  IE
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PE
                    <emph.end type="italics"/>
                  ſcribe
                    <lb/>
                  rationem
                    <emph type="italics"/>
                  IS
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SA
                    <emph.end type="italics"/>
                  ; & ordinatarum ratio evadet
                    <emph type="italics"/>
                  PSXIE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                    <emph type="italics"/>
                  SAXPE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                  .
                    <emph.end type="italics"/>
                  Sed
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SA
                    <emph.end type="italics"/>
                  ſubduplicata eſt ratio diſtantiarum
                    <lb/>
                    <emph type="italics"/>
                  PS, SI
                    <emph.end type="italics"/>
                  ; &
                    <emph type="italics"/>
                  IE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PE
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ſubduplicata eſt ratio virium in diſtan­
                    <lb/>
                  tiis
                    <emph type="italics"/>
                  PS, IS.
                    <emph.end type="italics"/>
                  Ergo ordinatæ, & propterea areæ quas ordinatæ
                    <lb/>
                  deſcribunt, hiſque proportionales attractiones, ſunt in ratione com­
                    <lb/>
                  poſita ex ſubduplicatis illis rationibus.
                    <emph type="italics"/>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO LXXXIII. PROBLEMA XLII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Invenire vim qua corpuſculum in centro Sphæræ locatum ad ejus
                    <lb/>
                  Segmentum quodcunque attrahitur.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Sit
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  corpus in centro Sphæræ, &
                    <emph type="italics"/>
                  RBSD
                    <emph.end type="italics"/>
                  Segmentum ejus
                    <lb/>
                  plano
                    <emph type="italics"/>
                  RDS
                    <emph.end type="italics"/>
                  & ſuperficie Sphærica
                    <emph type="italics"/>
                  RBS
                    <emph.end type="italics"/>
                  contentum. </s>
                  <s>Superfi­
                    <lb/>
                  cie Sphærica
                    <emph type="italics"/>
                  EFG
                    <emph.end type="italics"/>
                  centro
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  deſcripta ſecetur
                    <emph type="italics"/>
                  DB
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  F,
                    <emph.end type="italics"/>
                  ac di­
                    <lb/>
                  ſtinguatur Segmentum in partes
                    <emph type="italics"/>
                  BREFGS, FEDG.
                    <emph.end type="italics"/>
                  Sit
                    <lb/>
                  autem ſuperficies illa non pure Mathematica, ſed Phyſica, pro­
                    <lb/>
                  funditatem habens quam minimam. </s>
                  <s>Nominetur iſta profundi-</s>
                </p>
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