XV111 



THE PROGRESS OF 



it Thus he inferred from it, that two particles 

 of matter cannot possess exactly the same pro- 

 perties ; for if they did, the Supreme Being could 

 have no reason for employing the one more than 

 the other, and consequently both would he of 

 necessity rejected as if we were capable of 

 judging in what way motives act upon the mind 

 of the Deity, and as if position might not be 

 a sufficient motive for employing one particle 

 rather than another, supposing both possessed of 

 exactly the same properties. 



Another principle brought into view by Leib- 

 nitz, was the law of Continuity according to 

 which, nothing passes from one state to another, 

 without passing through all the intermediate 

 states. Though Leibnitz considers himself as 

 the first who pointed out this law, it is but fair 

 to state that it was distinctly laid down by 

 Galileo, who does not claim it as his own, but 

 ascribes its discovery to Plato. This principle 

 like the last, Leibnitz and his followers carried 

 to a blamable excess. Thus John Bernoulli 

 was induced by it to deny the existence of hard 

 bodies altogether; because in the collision of 

 such bodies a finite change must take place in 

 an instant, which, according to the principles of 

 the law of continuity is impossible. We can 

 obviate the objection of Bernoulli without re- 

 fusing, as 3Iaclaurin does, to admit the law of 

 continuity, by, admitting that the hard bodies 

 begin to act on each other before they come into 

 actual contact 



The last mechanical improvement of Leibnitz 

 introduced a controversy into mathematics, whicli 

 was discussed by the most eminent mathemati- 

 cians of Europe, for more than thirty years, with 

 great keenness, and not a little virulence ; though 

 neither side was able to produce any change of 

 opinion in their antagonists. Leibnitz, in 1686, 

 announced in the Leipsic Journal the demonstra- 

 tion of a great error, committed by Descartes and 

 others, in estimating the force of moving bodies. 

 In this paper he endeavours to show that the 

 force of a moving body is not proportional to its 

 velocity simply, but to the square of its velocity, 

 And he supported this new doctrine by very 

 plausible reasoning. A body projected upwards 

 against gravity, with a double velocity, ascends 

 four times the height ; with a triple velocity, to 

 nine times the height, and so on the height as- 

 cended being always as the square of the velocity. 

 Such was the reasoning, sufficiently simple and 

 satisfactory. 



The subject was soon taken up keenly, and the 

 world of science was divided into two parties. 

 The mathematicians of Germany, Holland, and 

 Italy, adopted the opinion of Leibnitz; those of 

 Great Britain the old opinion, that the force is 

 proportional simply to the velocity ; while those 



of France were divided between the two opinions, 



duria, Stirling, Desaguliers, Jurin, Clarke, 

 and Mairan, defended the old opinion ; while 

 Bernoulli, Hermann, Poleni, 'S Gravesende, and 

 Muschenbroek, supported the opinion of Leibnitz. 



\\ hat may appear at first sight singular in this 

 dispute, is, that the two parties who adopted 

 such different measures of force, when any 

 mechanical problem was proposed concerning 

 the action of bodies, whether at rest or in mo- 

 tion, resolved it in the same manner, and arrived 

 exactly at the same conclusions. It is clear from 

 this, that their ideas or opinions exactly coin- 

 cided. In reality, the two parties advanced 

 positions not inconsistent with each other ; and 

 both therefore were true. This was pointed out 

 by P'Alembert in his Dynamique, published in 

 1743. 



We may measure the force of one moving 

 body, by its effect upon another moving body. 

 Hence there is no doubt that the forces of such 

 bodies are as the quantities of matter multiplied 

 into the velocities ; because it is well known, that 

 the forces of bodies in which these products 

 are equal, if opposed, destroy each other. If we 

 employ this measure, it is evident that the forces 

 vary not as the squares, but simply as the 

 velocities. 



When a moving body is opposed by pressure, 

 or a resistance like that of gravity, the quantity 

 of such resistance required to extinguish the 

 motion must serve to measure the force of the 

 body. But there are two ways of computing the 

 amount of these retarding forces, which lead to 

 different results ; both of them just, and neither 

 of them to be assumed to the exclusion of {lie 

 other. Suppose a body to be projected perpen- 

 dicularly upwards, in a direction opposite to that 

 of gravity, we may either inquire into the retar- 

 dation which gravity produces during a given 

 time, or while the body is moving over a given 

 space. We may inquire how long the motion 

 will continue, or how far it will carry the body 

 before it be entirely exhausted. If we employ 

 the first of these for the measure of the force of 

 a body, that force must be proportional to the 

 velocity; for to this the time is manifestly pro- 

 portional. If we employ the second, namely the 

 length of the line which the moving body de- 

 scribes, as the measure of the force, then it must 

 be as the square of the velocity ; because to that 

 quantity the length of the line is known to be 

 proportional. 



Thus we obtain two values of the force ; the 

 one proportional to the velocity, the other to the 

 square of the velocity. Who does not perceive, 

 that the reason of this apparent inconsistency is 

 the different meaning applied to the term force 

 in the two cases? 





