VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 381 



I considered that the weight of the axis and arms of the machine was incon- 

 siderable, compared with the weight of the 1 cylinders of lead, and also that the 

 quantity a bore a very small proportion to the length of the cylindrical arms of 

 fir. And since the acceleratincr force is always as — ^^-7-, or as ■ "^ , , and the 



° •' ap + tip ay -\- abv 



quantity ab? or e expresses the sum of all the particles multiplied by the squares of 

 their distances from the axis of motion, it is plain that e must far exceed a^p; 

 and lastly, since the quantity e is the same both in the 1st and 2d experiment, 

 it follows, that the forces are very nearly to each other as a^p to -^, or as 4 to 1 : 

 and in the same way the other experiments are shown to be consistent with the 

 theory. 



I chose to premise a short account of the opinions which the philosophers 

 before Galileo entertained concerning the motions of bodies; because their 

 mistaken ideas of the effects of gravity are analogous to some opinions of 

 a later date, whicli indeed suggested the necessity of resuming these inquiries. 

 And as nothing in controversial matters so completely satisfies the mind as an 

 exact knowledge of that particular which produces the dispute; I have shown, 

 that the terms made use of to express the 3d law of motion were taken in 2 very 

 different senses: that Sir Isaac Newton's explication of them is at best ambiguous, 

 and Maclaurin's absolutely false. 1st. In the demonstration of the first case, 

 we see that the assertion of Leibnitz is true in one particular instance. When 

 2 elastic balls move in the same straight line, the sum of their forces is not 

 altered by collision; and it is more than probable, that this single circumstance 

 was the cause of affixing new ideas to the terms action and re-action. For, 



2d. In the 2d case, the same principle is taken for granted by J. Bernoulli. 

 We have examined into the consequences of this author's solution, and shown 

 that his hypothesis will prove all bodies to be perfectly elastic. As the steps by 

 which he deceived himself are here exposed, whoever carefully attends to these 

 2 examples cannot easily mistake in any case that may occur. It is plain, that 

 if any one contends for the equality of action and re-action, and explains those 

 terms by the changes produced in the absolute forces of the bodies, the dispute 

 is not merely verbal. 



3d. When a conclusion, agreeable to experience, is deduced from any hypo- 

 thesis, it does not therefore necessarily follow, that the hypothesis is universally 

 true, not even supposing the converse of the proposition to hold. In this 3d 

 case it is shown, what kind of answer we are to give such reasoning. The con- 

 servatio virium vivarium is never to be admitted, unless its connection with 

 simple facts, which are incontestable, be first made out. The solution of this 

 problem depends on this, that the motion lost is equal to the motion communi- 

 cated in a contrary direction, after the property of the lever is taken into the 



