NOTES. 301 



c' fl" C 2 ~ Q* 



obtain for the given chord = d x 1- n, and 



c cm 



therefore m = - ; and thus we have the chord which 



c* a n c 



it was required to find. But if the given chord be the line 

 joining the points of contact of the tangents drawn to the 



a 9 



ellipse from the given point; then x = c (= my\ and 



c 



c* a* 



therefore d = 0, and n = ; so that m = , and the 



c 



problem becoming indeterminate, any chord answers the con- 

 ditions of harmonic division. 



It is remarkable with how great earnestness M. Chasles 

 inculcates the advantage of studying the ancient writers, and 

 how much he extols the preference which Sir I. Newton gives 

 to synthetical demonstration conducted geometrically. We 

 may be allowed, however, to question the degree in which he 

 regards the Newtonian investigations as purely geometrical, 

 and still more the assertion that they were conducted by the 

 resources of the ancient geometry. The saying of Machin is 

 well known that the Principia was algebra in disguise ; and 

 no one can doubt that the investigations were carried on by 

 the resources of the calculus ; the analysis being algebraical, 

 and the composition or synthesis geometrical, at least in most 

 instances. 



NOTE III., p. 38. 



Professor Playfair's enunciation of the principle is not quite 

 satisfactory. "If," he says, " the motion which the particles 

 " of a moving or a system of moving bodies, have at any 

 " instant, be resolved into each two, one of which is the 

 " motion which the particles had in the preceding instant, 

 " then the sum of all these third motions must be such that 

 " they are in equilibrium with one another." (Ed. JRev. 

 xi. 253.) 



