36 HISTORY OF MECHANICS'. 



upwards became equal to the accelerating force 

 downwards. Though the law of progress of a body 

 to this limiting velocity was not made out till the 

 Principia of Newton appeared, the views on which 

 Galileo made this assertion are perfectly sound, and 

 show that he had clearly conceived the. nature and 

 operation of accelerating and retarding force. 



When Uniform Accelerating Forces had once 

 been mastered, there remained only mathematical 

 difficulties in the treatment of Variable Forces. A 

 Variable Force was measured by the Limit of the 

 increment of the Velocity, compared with the incre- 

 ment of the Time ; just as a Variable Velocity was 

 measured by the Limit of the increment of the 

 Space compared with that of the Time. 



With this introduction of the Notion of Limits, 

 we are, of course, led to the Higher Geometry, 

 either in its geometrical or its analytical form. The 

 general laws of bodies falling by the action of any 

 Variable Forces, were given by Newton in the 

 Seventh Section of the Principia. The subject is 

 there, according to Newton's preference of geome- 

 trical methods, treated by means of the Quadrature 

 of Curves, the Doctrine of Limits being exhibited in 

 a peculiar manner in the First Section of the work, 

 in order to prepare the way for such applications of 

 it. Leibnitz, the Bernouillis, Euler, and since their 

 time, many other mathematicians, have treated such 

 questions by means of the analytical method of 

 limits, the Differential Calculus. The Rectilinear 



