176 CELESTIAL MECHANICS. I. iii, 14. 



is iiig-enious and elegant; but it is neither so general and 

 natural as one of Dr. Hamilton's, which is here adopted, 

 nor so simple and convincing as Maclaurin's, which it may 

 also be worth our while to notice. Supposing two equal 

 weights, of an ounce each, to be fixed at the ends of the 

 equal arms of a lever; in this case it is obvious that there 

 will be an equilibrium, since there is no reason why either 

 weight should preponderate. It is also evident, that the 

 fulcrum supports the whole weight of two ounces, neg- 

 lecting that of the lever; consequently we may substitute 

 for the fulcrum a force equivalent to two ounces, drawing 

 the lever upwards ; and instead of one of the weights, we 

 may place the end of the lever under a firm obstacle, and 

 this equilibrium will still remain, the lever being now of 

 the description which is called the second kind, the fixed 

 point being at one end. Here, therefore, the weight re- 

 maining at the other end of the lever counterbalances a 

 force of two ounces, acting at half the distance from the 

 new fulcrum ; and we may substitute for this force a 

 weight of two ounces, acting at an equal distance on the 

 other side of that fulcrum, supposing the lever to be suf- 

 ficiently lengthened ; and there will still be an equilibrium. 

 In this case the fulcrum will sustain a weight of three 

 ounces; and we may substitute for it a force of three 

 ounces, acting upwards, and proceed as before. In a 

 similar manner the demonstration may be extended to any 

 commensurable proportion of the arms ; and it is easy to 

 show that the same law must be true of all ratios whatever, 

 even if they happen to be incommensurable (120, Sch.); 

 the forces remaining always in equilibrium, when they are 

 to each other inversely as the distances at which they are 

 applied. Lagrange, in his Mecanique Analytique, has 



