ANALYTICAL AND GEOMETRICAL MECHANICS. XXX111 



abstraction and graphical construction. The power of abstraction alone 

 suffices, indeed, to comprehend in full generality metrical relations in ele- 

 mentary geometry and its simplest applications, but fails when the relations 

 sought must be attained step by step by the application of a number of 

 principles, or in the auxiliary figure by a number of constructions. If, 

 indeed, we take the relation sought directly from the auxiliary figure 

 itself, and even if it were possible to ta.ke out the required distances with 

 absolute accuracy, still this result obtained would stand to the general law 

 desired only in the same relation that the result of a particular numerical 

 computation does to the more general algebraic formula. 



Investigations by the aid of graphical figures can, however, make known 

 general relations of form and position, and have in this respect their special 

 advantage. So far also as by them metrical relations are sought, then, by 

 the exclusion of algebraic formulae, only the process of deduction the 

 routine of construction remains of general significance. Sciences, then, 

 which proceed in this manner, furnish indeed, with respect to metrical 

 relations, no general laws, but for the deduction of these relations do give 

 general methods. lu this category we may place descriptive geometry and 

 the more recent graphical statics. 



IL 



ANALYTICAL AND GEOMETRICAL MECHANICS. 



It is hardly necessary in these days to call attention to the advantages 

 of a geometrical treatment of mechanical problems. This, however, was 

 not always the case, and the most important developments of geometrical 

 mechanics belong to the present century. It is to Poinsot, Chasles, Mobius, 

 etc., that these developments are due. 



By the Calculus of Newton and Leibnitz (1646-1714), and its subsequent 

 development, analysis became such a powerful instrument that the activity 

 of mathematicians was for a long time solely directed towards analytical 

 investigations. The power of analysis was in mechanics carried to its 

 highest point by Lagrange (1736-1813), in his Mechanique analytique. He 

 undertook the problem of reducing mechanics to a series of analytical 

 operations : " On ne trouvera point de figures dans cet outrage. Les 

 methodes que fy expose ne demandent ni constructions ni raisonnement geo- 

 metrique ou mecanique, mais seulement des operations algebriques assujeties 

 d une marche reguliere et uniforme" (Mechanique analytique. Paris, 1788.) 

 The principle of virtual velocities formed his point of departure. A 

 number of text-books upon theoretical mechanics still follow the method 

 of Lagrange. 



The revival of pure geometrical investigations by Monge (1746-1818), 

 the creator of descriptive geometry, and his followers, could not well have 

 been without its influence upon mechanics. In the year 1804 appeared 

 the Elements de Statique, by Poinsot, in which, in contrast to Lagrange, 



