352 HISTORY OF MECHANICS. 



CHAPTER V. 



GENERALIZATION OF THE PRINCIPLES OF MECHANICS. 



ec t. i. Generalization of the Second Law of Motion. Central forces. 



THE Second Law of Motion being proved for constant Forces which 

 act in parallel lines, and the Third Law for the Direct Action of 

 bodies, it still required great mathematical talent, and some inductive 

 power, to see clearly the laws which govern the motion of any number 

 of bodies, acted upon by each other, and by any forces, anyhow vary- 

 ing in magnitude and direction. This was the task of the generaliza- 

 tion of the laws of motion. 



Galileo had convinced himself that the velocity of projection, and 

 that which gravity alone would produce, are "both maintained, with- 

 out being altered, perturbed, or impeded in their mixture." It is to be 

 observed, however, that the truth of this result depends upon a par- 

 ticular circumstance, namely, that gravity, at all points, acts in lines, 

 which, ns to sense, are parallel. When we have to consider cases in 

 which this is not true, as when the force tends to the centre of a circle, 

 the law of composition cannot be applied in the same way ; and, in 

 this case, mathematicians were met by some peculiar difficulties. 



One of these difficulties arises from the apparent inconsistency of the 

 statical and dynamical measures of force. When a body moves in a 

 circle, the force which urges the body to the centre is only a tendency 

 to motion ; for the body does not, in fact, approach to the centre ; and 

 this mere tendency to motion is combined with an actual motion, which 

 takes place in the circumference. We appear to have to compare two 

 things which are heterogeneous. Descartes had noticed this difficulty, 

 but without giving any satisfactory solution of it. 1 If we combine the 

 actual motion to or from the centre with the traverse motion about 

 the centre, we obtain a result which is false on mechanical principles. 

 Galileo endeavored in this way to find the curve described by a body 

 which falls towards the earth's centre, and is, at the same time, carried 



Princip. P. iii. 59. 



