SEQUEL OF THE GENERALIZATION. 93 



aspects which belong to the different modes of deal- 

 ing with mathematical quantities. Mechanics may, 

 like pure mathematics, be geometrical or analyti- 

 cal; that is, it may treat space by a direct consi- 

 deration of its properties, or by a symbolical repre- 

 sentation of them : mechanics, like pure mathema- 

 tics, may proceed from special cases, to problems 

 and methods of extreme generality ; may summon 

 to its aid the curious and refined relations of sym- 

 metry, by which general and complex conditions 

 are simplified ; may become more powerful by the 

 discovery of more powerful analytical artifices; 

 may even have the generality of its principles fur- 

 ther expanded, inasmuch as symbols are a more 

 general language than words. We shall very briefly 

 notice a series of modifications of this kind. 



1. Geometrical Mechanics. Newton, 8yc. The 

 first great systematical Treatise on Mechanics, in 

 the most general sense, is the two first Books of the 

 Principia of Newton. In this work, the method 

 employed is predominantly geometrical : not only 

 space is not represented symbolically, or by refer- 

 ence to number; but numbers, as, for instance, 

 those which measure time and force, are repre- 

 sented by spaces ; and the laws of their changes are 

 indicated by the properties of curve lines. It is 

 well known that Newton employed, by preference, 

 methods of this kind in the exposition of his theo- 

 rems, even where he had made the discovery of 

 them by analytical calculations. The intuitions of 



