, PREFACE. 



opened a field entirely new, and showed that all the heavenly bodies, at 

 least those the motions of which are regulated by the attraction of the sun, 

 must necessarily, conform to the same laws, with a slight modification only, 

 by which KEPLER had found the five planets to be governed. KEPLER, rely 

 ing upon the evidence of observations, had announced that the orbit of every 

 planet is an ellipse, in which the areas are described uniformly about the 

 sun occupying one focus of the ellipse, and in such a manner that in differ 

 ent ellipses the times of revolution are in the sesquialteral ratio of the semi- 

 axes-major. On the other hand, NEWTON, starting from the principle of 

 universal gravitation, demonstrated d, priori that all bodies controlled by the 

 attractive force of the sun must move in conic sections, of which the planets 

 present one form to us, namely, ellipses, while the remaining forms, parabo 

 las and hyperbolas, must be regarded as being equally possible, provided 

 there may be bodies encountering the force of the sun with the requisite 

 velocity ; that the sun must always occupy one focus of the conic section ; 

 that the areas which the same body describes in different times about the 

 sun are proportional to those times; and finally, that the areas described 

 about the sun by different bodies, in equal times, are in the subduplicate 

 ratio of the semiparameters of the orbits: the latter of these laws, identical 

 in elliptic motion with the last law of KEPLER, extends to the parabolic and 

 hyperbolic motion, to which KEPLER S law cannot be applied, because the rev 

 olutions are wanting. The clue was now discovered by following which it 

 became possible to enter the hitherto inaccessible labyrinth of the motions of 

 the comets. And this was so successful that the single hypothesis, that their 

 orbits were parabolas, sufficed to explain the motions of all the comets which 

 had been accurately observed. Thus the system of universal gravitation had 



