66 NEWTON S PRINCIPIA. 



describe from each of the given points a circle, with the dis 

 tance of that point from the given focus as a radius, and the 

 straight line touching these two circles will be the directrix 

 of the parabola, and the perpendicular to it from the focus, 

 its axis ; the principal vertex being the middle point of 

 that perpendicular. The coincidence of the very eccentric 

 elliptical orbits of the comets with the parabola, makes this 

 parabolic hypothesis answer for determining their places 

 and times in the general case. 



The correction of the orbit thus found is reduced to 

 finding the orbit of an ellipse which shall pass through 

 three given points, and this is done by the 2 1 stpropo- 

 sition of Book I., or rather by the 16th lemma, to which 

 it is a corollary, for inflecting three straight lines from three 

 given points, the differences, if any, between the lines, being 

 given. 



Sir Isaac Newton tried the accuracy of the methods 

 thus found upon several comets, and particularly on the 

 celebrated one of 1680, called Halley s comet, from the 

 great labour which that mathematician, in aid of his illus 

 trious friend and master, bestowed upon the calculation of 

 its orbit. The following is a short statement of the general 

 result of a comparison between the places computed from 

 the theory, and the places found by actual observation, 

 in the cases tried. 



of facilitating the solution of this difficult problem by another method ; 

 but the author informs us that he subsequently fell upon the method which 

 he has given in the ^third^book, and which he prefers for its greater sim 

 plicity. It is, however, very Remarkable that he overlooked the important 

 circumstance of there being a porism connected with his solution, or a 

 case in which the problem becomes indeterminate and has an infinite number 

 of solutions ; and what is still more singular that the case of the comet is 

 that of the porism, so that the solution is wholly inapplicable. This 

 was first discovered by F. Boscovich in 1749; it being found that the solution 

 had thrown the comet upon the wrong side of the sun. (See Life of Sinison, 

 ~~&quot; 



