SEQUEL TO THE GENERALIZATION. 363 



of Mechanics may assume the various aspects which belong to the 

 different modes of dealing with mathematical quantities. Mechanics, 

 like pure mathematics, may be geometrical or may be analytical ; that 

 is, it may treat space either by a direct consideration of its properties, 

 or by a symbolical representation of tbem: Mechanics, like pure 

 mathematics, may proceed from special cases, to problems and methods 

 of extreme generality ; — may summon to its aid the curious and refined 

 relations of symmetry, 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 further expanded, inasmuch as symbols are a more general 

 language than words. "We shall very briefly notice a series of mod- 

 ifications of this kind. 



1. Geometrical Mechanics, Newton, dx. — 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 symbol- 

 ically, or by reference to number ; but numbers, as, for instance, those 

 which measure time and force, are represented 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 theorems, even where he had made the 

 discovery of them by analytical calculations. The intuitions of space 

 appeared to him, as they have appeared to many of his followers, to 

 be a more clear and satisfactory road to knowledge, than the operations 

 of symbolical language. Hermann, whose Phoronomia was the next 

 great work on this subject, pursued a like course ; employing curves, 

 which he calls " the scale of velocities," " of forces," &c. Methods 

 nearly similar were employed by the two first Bernoullis, and other 

 mathematicians of that period ; and were, indeed, so long familiar, that 

 the influence of them may still be traced in some of the terms which 

 are used on such subjects ; as, for instance, when we talk of " reducing 

 a problem to quadratures," that is, to the finding the area of the curves 

 employed in these methods. 



2. Analytical Mechanics. Euler. — As analysis was more cultivated, 

 it gained a predominancy over geometry ; being found to be a far 

 more powerful instrument for obtaining results ; and possessing a 

 beauty and an evidence, which, though different from those of geom- 

 etry, had great attractions for minds to which they became familiar. 

 The person who did most to give to analysis the generality and sym- 



