Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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latera recta principalia ſunt æqualia, velocitas corporis eſt reciproce
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ut perpendiculum demiſſum ab umbilico ad tangentem. </
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DE MOTU
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CORPORUM</
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Corol.
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6. In Parabola, velocitas eſt reciproce in ſubduplicata ra
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tione diſtantiæ corporis ab umbilico figuræ; in Ellipſi magis varia
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tur, in Hyperbola minus, quam in hac ratione. </
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>Nam (per Corol. </
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2. Lem. </
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>XIV.) perpendiculum demiſſum ab umbilico ad tangentem
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Parabolæ eſt in ſubduplicata ratione diſtantiæ. </
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<
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>In Hyperbola per
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pendiculum minus variatur, in Ellipſi magis. </
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Corol.
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7. In Parabola, velocitas corporis ad quamvis ab umbili
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co diſtantiam, eſt ad velocitatem corporis revolventis in Circulo
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ad eandem a centro diſtantiam, in ſubduplicata ratione numeri bi
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narii ad unitatem; in Ellipſi minor eſt, in Hyperbola major quam
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in hac ratione. </
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>Nam per hujus Corollarium ſecundum, velocitas
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in vertice Parabolæ eſt in hac ratione, & per Corollaria ſexta hu
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jus & Propoſitionis quartæ, ſervatur eadem proportio in omnibus
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diſtantiis. </
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>Hinc etiam in Parabola velocitas ubique æqualis eſt ve
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locitati corporis revolventis in Circulo ad dimidiam diſtantiam, in
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Ellipſi minor eſt, in Hyperbola major. </
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Corol.
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8. Velocitas gyrantis in Sectione quavis Conica eſt ad ve
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locitatem gyrantis in Circulo in diſtantia dimidii lateris recti princi
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palis Sectionis, ut diſtantia illa ad perpendiculum ab umbilico in
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tangentem Sectionis demiſſum. </
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<
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Corol.
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9. Unde cum (per Corol. </
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>6. Prop. </
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>IV.) velocitas gyrantis
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in hoc Circulo ſit ad velocitatem gyrantis in Circulo quovis alio,
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reciproce in ſubduplicata ratione diſtantiarum; fiet ex æquo velo
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citas gyrantis in Conica ſectione ad velocitatem gyrantis in Circulo
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in eadem diſtantia, ut media proportionalis inter diſtantiam illam
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communem & ſemiſſem principalis lateris recti ſectionis, ad per
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pendiculum ab umbilico communi in tangentem ſectionis de
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miſſum. </
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PROPOSITIO XVII. PROBLEMA. IX.
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Poſito quod vis centripeta ſit reciproce proportionalis quadrato diſtan
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ſtantiæ loeorum a centro, & quod vis illius quantitas abſoluta ſit
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cognita; requiritur Linea quam corpus deſcribit, de loco dato, cum
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data velocitate, ſecundum datam rectam egrediens.
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S
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ea ſit qua corpus
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in or
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bita quavis data
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pq
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gyretur, & cognoſcatur hujus velocitas in loco
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p.
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