384 HISTOEY OF MECHANICS. 



of the unproved verbal generalization of Galileo, we should be in great 

 danger of allowing ourselves to be referred successively from one truth 

 to another, without any reasonable hope of ever arriving at any thing 

 ultimate and fundamental. 



But though this Principle of Virtual Velocity cannot be looked 

 upon as a great discovery of Galileo, it is a highly useful rule ; and 

 the various forms under which he and his successors urged it, tended 

 much to dissipate the vague wonder with which the effects of machines 

 had been looked upon ; and thus to diffuse sounder and clearer notions 

 on such subjects. 



The Principle of Virtual Velocities also affected the progress of 

 mechanical science in another way : it suggested some of the analogies 

 by the aid of which the Third Law of Motion was made out ; leading 

 to the adoption of the notion of Momentum as the arithmetical pro- 

 duct of weight and velocity. Since on a machine on which a weight 

 of two pounds at one part balances three pounds at another part, the 

 former weight would move through three inches while the latter would 

 move through two inches ; we see (since three multiplied into two is 

 equal to two multiplied into three) that the Product of the weight 

 and the velocity is the same for the two balancing weights ; and if we 

 call this Product Momentum, the Law of Equilibrium is, that when 

 two weights balance on a machine, the Momentum of the two would 

 be the same, if they were put in motion. 



The Notion of Momentum was here employed in connection with 

 Virtual Velocities ; but it also came under consideration in treating of 

 Actual Velocities, as we shall soon see. 



Sect. 5. — Attempts at the Third Law of Motion. — Notion of 



Momentum. 



In the questions we have hitherto had to consider respecting Motion, 

 no regard is had to the Size of the body moved, but only to the 

 Velocity and Direction of the motion. We must now trace the pro- 

 gress of knowledge respecting the mode in which the Mass of the 

 body influences the effect of Force. This is a more difficult and com- 

 plex branch of the subject; but it is one which requires to be noticed, 

 as obviously as the former. Questions belonging to this department 

 of Mechanics, as well as to the others, occur in Aristotle's Mechanical 

 Problems. " Why," says he, " is it, that neither very small nor very 

 large bodies go far when we throw them ; but, in order that this may 



