DISCOVERY OF THE LAWS OF MOTION. 333 



Weights and Powers hypothetical motions, arising from some other 

 cause ; and then, by the construction of the machine, the velocities of 

 the Weights and Powers must have certain definite ratios. These 

 velocities, being thus hypothetically supposed and not actually pro- 

 duced, are called Virtual Velocities. And the general law of equilib- 

 rium is, that in any machine, the Weights which balance each other, 

 are reciprocally to each other as their Virtual Velocities. This is 

 called the Principle of Virtual Velocities. 



This Principle (which was afterwards still further generalized) is, by 

 some of the admirers of Galileo, dwelt upon as one of his great services 

 to Mechanics. But if we examine it more nearly, we shall see that it 

 has not much importance in our history. It is a generalization, but a 

 generalization established rather by enumeration of cases, than by any 

 induction proceeding upon one distinct Idea, like those generalizations 

 of Facts by which Laws are primarily established. It rather serves 

 verbally to conjoin Laws previously known, than to exhibit a connection 

 in them : it is rather a help for the memory than a proof for the 

 reason. 



The Principle of Virtual Velocities is so far from implying any 

 clear possession of mechanical ideas, that any one who knows the pro- 

 perty of the Lever, whether he is capable of seeing the reason for it 

 or not, can see that the greater weight moves slower in the exact pro- 

 portion of its greater magnitude. Accordingly, Aristotle, whose en- 

 tire want of sound mechanical views we have shown, has yet noticed 

 this truth. When Galileo treats of it, instead of offering any reasons 

 which could independently establish this principle, he gives his readers 

 a number of analogies and illustrations, many of them very loose 

 ones. Thus the raising a great weight by a small force, he illustrates 

 by supposing the weight broken into many small parts, and conceiving 

 those parts raised one by one. By other persons, the analogy, already 

 intimated, of gain and loss is referred to as an argument for the prin- 

 ciple in question. Such images may please the fancy, but they cannot 

 be accepted as mechanical reasons. 



Since Galileo neither first enunciated this rule, nor ever proved it 

 as an independent principle of Mechanics, we cannot consider the dis- 

 covery of it as one of his mechanical achievements. Still less can we 

 compare his reference to this principle with Stevinus's proof of the 

 Inclined Plane ; which, as we have seen, was rigorously inferred from 

 1 the sound axiom, that a body cannot put itself in motion. If we were 

 to assent to the really self-evident axioms of Stevinus, only in virtue 



