NXMV ANALYTICAL AND GEOMETRICAL MECHANICS. 



we find : " qus tony Is* the.oreme de la Statique rationale ne sont plus au 

 /"/a/ 'iii,-, lot th-"r,'ni'x //- (r,'i>m,'trif.." This work was the beginning of a 

 scries of treatises in which the advantages of the synthetic development 

 and geometrical treatment of mechanics were defended and, by most 

 important results, strikingly demonstrated. 



At this time the views as to the best method of treating mathematical 

 problems were sharply opposed. Carnot (1753-1823), to whom, however, 

 the modern geometry itself owes no slight impulse, gives the preference 

 to analysis. For synthesis "est restreinte par la nature de ces precedes; 

 eUe ne pent jamai* perdre de vue son objet, ilfaut que cet objet s'offre tou- 

 jmirs A Vesprit, reel et net, ainsi que tous les rapprochements el combinaisons 

 qu'on enfaiV (Oeometrie de position. Paris, 1803.) That which here Car- 

 not considers as a defect in the synthetic and geometrical method, Poinsot 

 claims as its special advantage : " On pent bien par ces calctds plus ou moins 

 longs et compliques narvenir d determiner le lieu ou se trouvera le corps au 

 bout (Fun temps donne, metis on le perd enticement de vue, tandis qu'on vou- 

 drait Fobsereer et le suivre, pour ainsi dire, des yeux dans tout le cours de sa 

 rotation " (Theon'e nouv. d. I. rot. d. corps). 



The example of Poinsot found numerous followers. In Germany, Mo- 

 bius followed with his " LeJirbuch der Statik." Mechanics as well as 

 geometry thus received enrichment. Mobius gives the preference always 

 to the synthetic method, and also endeavors' to interpret geometrically, 

 analytically deduced formulae " because in investigations concerning 

 bodies in space the geometrical method is a treatment of the subject itself, 

 and is therefore the most natural, while by the analytical method the sub- 

 ject is concealed and more or less lost sight of under extraneous signs " 

 (Lehrb. d. Statik. Leipzig, 1837.) 



Even in analytical operations, geometrical considerations came more and 

 more in the foreground. On all sides the development of Kinematics, the 

 theory of motion without reference to its cause, was prosecuted. But, 

 neglecting the cause of motion, there remains only its path ; that is, geo- 

 metry proper (Kinematical geometry, or the geometry of motion}. The in- 

 vestigations of Chasles, Mobius, Rodrigues, Jouquiere. and others, may yet 

 be still further pursued ; and when by the aid of geometry a certain com- 

 pleteness has been given to the theory of the motion of invariable systems, 

 the geometrical theory of regular variable systems (to which the flexible 

 and elastic belong; will be possible. For the discussion of such branches 

 of mathematics, the synthetic geometry is necessary ; for their foundation 

 lies in a theory of the relationship of systems. 



The advantage of the synthetic method in mechanics is denied by no 

 one. Wherever it is possible, we obtain more comprehensive conclusions 

 as to the nature of the phenomena, while all the properties of the same fol- 

 low directly from the simple and known truths premised. In analytical 

 investigations it is necessary, even when definite equations are obtained, to 

 deduce the actual laws singly and in a supplementary manner, although 

 they are indeed all contained in the equations themselves. 



It is not, however, always possible to preserve the synthetic process 

 throughout. From the first truth the ways diverge hi all directions, and 



