Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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evaneſcentium ſummas & rationes, primaſque naſcentium, id eſt,
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ad limites ſummarum & rationum deducere; & propterea limitum
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illorum demonſtrationes qua potui brevitate præmittere. </
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>His enim
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idem præſtatur quod per methodum Indiviſibilium; & principiis de
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monſtratis jam tutius utemur. </
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<
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>Proinde in ſequentibus, ſiquando
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quantitates tanquam ex particulis conſtantes conſideravero, vel ſi
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pro rectis uſurpavero lineolas curvas; nolim indiviſibilia, ſed eva
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neſcentia diviſibilia, non ſummas & rationes partium determinata
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rum, ſed ſummarum & rationum limites ſemper intelligi; vimque
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talium demonſtrationum ad methodum præcedentium Lemmatum
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ſemper revocari. </
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>Objectio eſt, quod quantitatum evaneſcentium nulla ſit ultima
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proportio; quippe quæ, antequam evanuerunt, non eſt ultima, ubi
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evanuerunt, nulla eſt. </
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>Sed & eodem argumento æque contendi poſſet
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nullam eſſe corporis ad certum locum pervenientis velocitatem ul
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timam: hanc enim, antequam corpus attingit locum, non eſſe ulti
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mam, ubi attingit, nullam eſſe. </
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>Et reſponſio facilis eſt: Per velocita
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tem ultimam intelligi eam, qua corpus movetur neque antequam
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attingit locum ultimum & motus ceſſat, neque poſtea, ſed tunc
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cum attingit; id eſt, illam ipſam velocitatem quacum corpus attin
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git locum ultimum & quacum motus ceſſat. </
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>Et ſimiliter per ulti
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mam rationem quantitatum evaneſcentium, intelligendam eſſe ratio
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nem quantitatum non antequam evaneſcunt, non poſtea, ſed qua
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cum evaneſcunt. </
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>Pariter & ratio prima naſcentium eſt ratio qua
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cum naſcuntur. </
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>Et ſumma prima & ultima eſt quacum eſſe (vel
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augeri & minui) incipiunt & ceſſant. </
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>Extat limes quem velocitas
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in fine motus attingere poteſt, non autem tranſgredi. </
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>Hæc eſt
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velocitas ultima. </
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>Et par eſt ratio limitis quantitatum & propor
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tionum omnium incipientium & ceſſantium. </
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>Cumque hic limes
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ſit certus & definitus, Problema eſt vere Geometricum eundem de
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terminare. </
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>Geometrica vero omnia in aliis Geometricis determi
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nandis ac demonſtrandis legitime uſurpantur. </
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<
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>Contendi etiam poteſt, quod ſi dentur ultimæ quantitatum eva
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neſcentium rationes, dabuntur & ultimæ magnitudines: & ſic quan
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titas omnis conſtabit ex Indiviſibilibus, contra quam
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Euclides
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de
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Incommenſurabilibus, in libro decimo Elementorum, demonſtravit. </
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Verum hæc Objectio falſæ innititur hypotheſi. </
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>Ultimæ rationes
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illæ quibuſcum quantitates evaneſcunt, revera non ſunt rationes
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quantitatum ultimarum, ſed limites ad quos quantitatum ſine limi
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te decreſcentium rationes ſemper appropinquant; & quas propius
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aſſequi poſſunt quam pro data quavis differentia, nunquam vero </
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