Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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DE MOTU
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CORPORUM</
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PROPOSITIO XXXIX. PROBLEMA XXVII.
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Poſita cujuſcunque generis Vi centripeta, & conceſſis figurarum
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curvilinearum quadraturis, requiritu, corporis recta aſcenden
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tis vel deſcendentis tum Velocitas in locis ſingulis, tum Tempus
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quo corpus ad locum quemvis perveniet: Et contra.
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A
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in recta
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ADEC
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cadat corpus
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E,
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deque loco
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ejus
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E
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erigatur ſemper perpendicularis
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EG,
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vi centripetæ in loco
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illo ad centrum
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C
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tendenti proportio
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nalis: Sitque
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BFG
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linea curva quam
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punctum
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G
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perpetuo tangit. </
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dat autem
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EG
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ipſo motus initio cum
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perpendiculari
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AB,
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& erit corporis Ve
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locitas in loco quovis
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E
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ut areæ cur
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vilineæ
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ABGE
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latus quadratum.
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E. I.
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<
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EG
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capiatur
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EM
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lateri quadra
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to areæ
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ABGE
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reciproce proportio
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nalis, & ſit
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ALM
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linea curva quam
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punctum
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M
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perpetuotangit, & erit Tem
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pus quo corpus cadendo deſcribit li
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neam
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AE
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ut area curvilinea
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ALME.
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E. I.
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<
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AE
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capiatur linea
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quam minima
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DE
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datæ longitudinis,
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ſitque
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DLF
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locus lineæ
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EMG
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ubi
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corpus verſabatur in
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D
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; & ſi ea ſit vis centripeta, ut areæ
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ABGE
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latus quadratum ſit ut deſcendentis velocitas, erit area ipſa in du
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plicata ratione velocitatis, id eſt, ſi pro velocitatibus in
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D
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&
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E
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ſcribantur V & V+I, erit area
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ABFD
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ut VV, & area
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ABGE
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ut
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VV+2 VI+II, & diviſim area
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DFGE
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ut 2 VI+II, adeoque
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(
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DFGE/DE
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) ut (2VI+II/
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DE
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), id eſt, ſi primæ quantitatum naſcentium
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rationes ſumantur, longitudo
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DF
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ut quantitas (2VI/
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DE
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), adeoque e
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tiam ut quantitatis hujus dimidium (IXV/
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DE
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). Eſt autem tempus quo </
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