THE SCOPE OF PHILOSOPHY 



forces, D*Alembert established for the first 

 time the equations of equilibrium of any system 

 of forces applied to the different points of a 

 solid body — equations which include all cases 

 of levers and an infinity of cases besides." 

 But, as Comte observes, " before hydrostatics 

 could be comprehended under statics, it was 

 necessary that the abstract theory of equilibrium 

 should be made so general as to apply directly 

 to fluids as well as solids. This was accom- 

 plished when Lagrange supplied, as the basis 

 of the whole of mechanics, the single principle 

 of virtual velocities,'* — or the principle that 

 whenever weights balance each other, " the re- 

 lation of one set of weights to their velocities 

 equals the relation of the other set of velocities 

 to their weights." ^ So geometry in ancient times 

 treated of questions relating to particular fig- 

 ures ; but since the great discovery of Descartes, 

 it has dealt with questions relating to any fig- 

 ure whatever. So, in the progress of analytical 

 mathematics, we have first arithmetic, which 

 " can express in one formula the value of a par- 

 ticular tangent to a particular curve ; " and, at 

 a later date, algebra, which can express in one 

 formula the values of all possible tangents to a 

 particular curve ; and, at a still later date, the 

 calculus, which can express in one formula the 



^ [Spencer's Essay on the ** Genesis of Science," Library 

 Edition of the Essays (Appleton, 1891), vol. ii. p. 59.] 



53 



