Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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corpus cadendo deſcribit lineolam
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DE,
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ut lineola illa directe &
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velocitas V inverſe, eſtque vis ut velocitatis incrementum I directe
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& tempus inverſe, adeoque ſi primæ naſcentium rationes ſuman
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tur, ut (IXV/
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DE
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), hoc eſt, ut longitudo
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DF.
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Ergo Vis ipſi
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DF
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vel
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EG
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proportionalis facit ut corpus ea cum Velocitate deſcendat quæ ſit
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ut areæ
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ABGE
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latus quadratum.
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E. D.
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LIBER
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PRIMUS.</
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>Porro cum tempus, quo quælibet longitudinis datæ lineola
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DE
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deſcribatur, ſit ut velocitas inverſe adeoque ut areæ
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ABFD
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latus
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quadratum inverſe; ſitque
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DL,
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atque adeo area naſcens
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DLME,
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ut idem latus quadratum inverſe: erit tempus ut area
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DLME,
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&
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ſumma omnium temporum ut ſumma omnium arearum, hoc eſt
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(per Corol. </
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<
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>Lem. </
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<
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>IV) Tempus totum quo linea
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AE
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deſcribitur ut
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area tota
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AME.
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E. D.
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Corol.
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1. Si
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P
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ſit locus de quo corpus cadere debet, ut, urgen
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te aliqua uniformi vi centripeta nota (qualis vulgo ſupponitur
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Gravitas) velocitatem acquirat in loco
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D
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æqualem velocitati
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quam corpus aliud vi quacunque cadens acquiſivit eodem loco
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D,
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& in perpendiculari
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DF
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capiatur
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DR,
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quæ ſit ad
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DF
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ut vis illa
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uniformis ad vim alteram in loco
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D,
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& compleatur rectangulum
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PDRQ,
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eique æqualis abſcindatur area
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ABFD;
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erit
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A
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locus
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de quo corpus alterum cecidit. </
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>Namque completo rectangulo
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DRSE,
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cum ſit area
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ABFD
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ad aream
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DFGE
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ut VV ad
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2VI, adeoque ut 1/2 V ad I, id eſt, ut ſemiſſis velocitatis totius
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ad incrementum velocitatis corporis vi inæquabili cadentis; & ſi
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militer area
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PQRD
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ad aream
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DRSE
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ut ſemiſſis velocitatis to
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tius ad incrementum velocitatis corporis uniformi vi cadentis;
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ſintQ.E.I.crementa illa (ob æqualitatem temporum naſcentium)
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ut vires generatrices, id eſt, ut ordinatim applicatæ
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DF, DR,
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adeoque ut areæ naſcentes
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DFGE, DRSE
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; erunt (ex æquo)
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areæ totæ
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ABFD, PQRD
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ad invicem ut ſemiſſes totarum ve
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locitatum, & propterea (ob æqualitatem velocitatum) æquantur. </
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Corol.
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2. Unde ſi corpus quodlibet de loco quocunque
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D
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data
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cum velocitate vel ſurſum vel deorſum projiciatur, & detur lex vis
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centripetæ, invenietur velocitas ejus in alio quovis loco
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e,
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erigen
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do ordinatam
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eg,
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& capiendo velocitatem illam ad velocitatem in
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loco
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D
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ut eſt latus quadratum rectanguli
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PQRD
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area curvili
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nea
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DFge
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vel aucti, ſi locus
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e
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eſt loco
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D
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inferior, vel diminuti,
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ſi is ſuperior eſt, ad latus quadratum rectanguli ſolius
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PQRD,
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id
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eſt, ut √
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PQRD
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+vel-
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DFge
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ad √
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PQRD.
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