ELEMENTS OF NATURAL PHILOSOPHY. 327 



which is the rate of displacement. And here we cannot but regret that the 

 authors, one of whom at least is an ardent disciple of Hamilton, have not at 

 once pointed out that every displacement is a vector, and taken the opportunity \ 

 of explaining the addition of vectors as a process, which, applied primarily to \ 

 displacements, is equally applicable to velocities, or the rates of change of \ 

 displacement, and to accelerations, or the rates of change of velocities. For it 

 is only in this way that the method of Newton, to which we are glad to see 

 that our authors have reverted, can be fully understood, and the "parallelogram 

 of forces" deduced from the "parallelogram of velocities." Another conception 

 of Hamilton's, however, that of the hodograph, is early introduced and employed 

 with great effect. The fundamental idea of the hodograph is the same as that 

 of vectors in general. The velocity of a body, being a vector, is defined by 

 its magnitude and direction, so that velocities may be represented by straight 

 lines, and these straight lines may be moved parallel to themselves into whatever 

 position is most suitable for exhibiting their geometrical relations, as for instance 

 in the hodograph they are all drawn from one point. The same idea is made 

 use of in the theorems of the " triangle " and the " polygon " of forces, and 

 in the more general method of "diagrams of stress," in which the Hues which 

 represent the stresses are drawn, not in the positions in which they actually 

 exist, but in those positions which most fully exhibit their geometrical relations. 

 We are sorry that a certain amount of slight is thrown on these methods in 

 411, where a different proposition is called the true triangle of forces. 



It is when a writer proceeds to set forth the first principle of dynamics 

 that his true character as a sound thinker or otherwise becomes conspicuous. 

 And here we are glad to see that the authors follow Newton, whose Leges 

 Motus, more perhaps than any other part of his great work, exhibit the 

 unimproveable completeness of that mind without a flaw. 



We would particularly recommend to writers on philosophy, first to deduce 

 from the best philosophical data at their command a definition of equal intervals 

 of time, and then to turn to 212, where such a definition is given as a 

 logical conversion of Newton's First Law. 



But it is in the exposition of the Third Law, which affirms that the actions 

 between bodies are mutual, that our authors have brought to light a doctrine, 

 which, though clearly stated by Newton, remained unknown to generations of 

 students and commentators, and even when acknowledged by the whole scientific 

 world was not known to be contained in a paragraph of the Princima till it 



