LAW OF THE UNIVERSE. 247 



the lengths of similar curvilinear and concentric figures being 

 in some proportion, and that always the same, to the radii, 

 the lengths are to each other as those radii, and consequently 

 the velocity of the whole movement is increased in the same 

 proportion with the space moved through. Hence the times 

 taken for performing the whole motion must be the same. 

 Thus, if V and v are the velocities, E and r the radii, S and s 

 the lines described in the times T and , by two such bodies 

 round a common centre, V : v : : E : r, and S : s : : E : r ; and 



because V = 7=- and v = , -^ : : : E : r, and S : s : : T E : 

 -L r J. 6 



tr; or E : r : : T E : t r ; and therefore T = 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, com- 

 plete 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 what- 

 ever distance they fell, because the elliptic orbit being inde- 

 finitely 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. 



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 discovered first by 

 Galileo. 



