DISCOVERY OF THE LAWS OF MOTION". 329 



in his works, and in those of his scholars and successors. The motioc 

 of bodies falling freely was, however, in such treatises, generally com- 

 bined with the motion of bodies Falling along Inclined Planes; a part 

 of the theory of which we have still to speak. 



The Notion 'of Accelerating Force and of its operation, once formed, 

 was naturally applied in other cases than that of bodies falling freely, 

 The different velocities with which heavy and light bodies fall were 

 explained by the different resistance of the air, which, diminishes the 

 accelerating force ; 9 and it was boldly asserted, that in a vacuum a lock 

 of wool and a piece of lead would fall equally quickly. It was also 

 maintained 10 that any falling body, however large and heavy, would 

 always have its velocity in some degree diminished by the air in which 



V I/O 



it falls, and would at last be reduced to a state of uniform motion, as 

 soon as the resistance 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 accel- 

 erating and retarding force. 



O o 



"When Uniform Accelerating Forces had once been mastered, there 



o ' 



remained only mathematical difficulties in the treatment of Variable 

 Forces. A Variable Force was measured by the Limit of the incre- 

 ment of the Velocity, compared with the increment 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 geometrical meth- 

 ods, 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, Enler, and since their time, many other mathemati- 

 cians, have treated such questions by means of the analytical method 

 of limits, the Differential Calculus. The Rectilinear Motion of bodies 

 acted upon by variable forces is, of course, a simpler problem than 

 Uieir Curvilinear Motion, to which we have now to proceed. But il 



8 Galileo, iii. 43. 10 iii. 54. 



