310 NEWTON S PRINCIPIA. . 



Both these methods, it will be observed, require the dis 

 tance of the comet at the time of the observation to be 

 known at least approximately. The expedient* suggested 

 was to consider a small portion of each orbit as rectilinear 

 and described with uniform motion. Then four observa 

 tions being made at moderate intervals of time, four straight 

 lines are given, across which a straight line is to be arawn 

 so as to be cut in three parts in the same ratios of the 

 intervals of time. This geometrical problem had been al 

 ready solved by Wallis, Wren, and Newton, but its applica 

 tion to comets had never led to any satisfactory result. A 

 little consideration will enable us to understand why. If 

 the four straight lines pass through one point, the problem 

 admits of more than one solution. Geometers call this a 

 porismatic case. In fact, a whole family of straight lines 

 can be drawn cutting the four straight lines in the re 

 quired ratios. When the straight lines do not exactly 

 meet in a point, only one straight line can be drawn ; 

 but, as might be expected, the construction to draw this 

 line is such that any small error in the data will make a 

 very great change in the cutting line. This, Boscovich 

 remarks f, is exactly what occurs in the application to 

 comets ; the arcs of the two orbits, which are considered as 

 straight lines, are necessarily small, and in this case the 

 four straight lines can be shown to meet very nearly in 

 one point. 



Newton was not a man to be content with merely a 

 general theory of comets ; he at once reduced it to practice. 

 He proceeded to try his method on the comet of 1680. 

 By scale and compass, in a figure in which the radius of 

 the earth was 16^ inches, he determined the elements of 

 the orbit. Halley afterwards again calculated these with 



* Edin. Trans., vol. iii. 



f Phil. Keceiit. a Bcncdicto Stay cum adnot, Boscovich, lib. ii. p. 345. 



