NEWTON S PRINCIPIA. 55 



t r ; or R ; r :: T R : t r ; and therefore T = . Hence 

 if gravity were the same towards the sun that it is 

 between the surface and centre of each planet, or if the 

 sun were moved but a very little to one side, so as to be 

 in the centre of the ellipse, the whole planets would 

 revolve round him in the same time, and Saturn and Uranus 

 would, like Mercury, complete their vast courses in about 

 three of our lunar months instead of 30 and 80 years, a 

 velocity in the case of Uranus equal to 75,000 miles in a 

 second, or nearly one-third that of light. 



It also follows from this proposition that, if such a law 

 of attraction prevailed, all bodies descending in a straight 

 line to the centre would reach it in the same time from 

 whatever distance they fell, because the elliptic orbit 

 being indefinitely stretched out in length and narrowed 

 till it became a straight line, bodies would move or vibrate 

 in equal times . through that line. This is the law of 

 gravity at all points within the earth s surface, and Sir 

 I. Newton has adapted one of his investigations to it, 

 when treating of the pendulum. 



Another consequence of this proposition is, that if the 

 centre of the ellipse be supposed to be removed to an 

 infinite distance, and the figure to become a parabola, 

 the centripetal force being directed to a point infinitely 

 remote, becomes constant and equable ; a proposition dis 

 covered first by Galileo. 



Sir Isaac Newton having treated of the centripetal force 

 in conic sections, where the centre of forces is the centre 

 of the figure (and generally whatever be the centre in 

 the case of the circle), proceeds to treat of that force where 

 it is directed towards the focus of one or other of those 

 curves, and not to the centre. It is easy to demonstrate 

 a compendious theorem, that which forms the subject of 

 his three first propositions, in which he determines the 



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