506 Dr. J. R. Mayer on the Mechanical Equivalent of Heat. 



and pleasant to us by long usage, we cannot long hesitate in the 

 choice we have to make between the conceptions I. and II. 



Let us consider the elementary case of a mass, originally at 

 rest, which receives motion : this happens, as has been already 

 said, by the mass being subjected to a certain push or pull under 

 the influence of which it traverses a certain space, the effective 

 space. Now, however, both the velocity and also the intensity 

 of the push (Newton's force) always vary at every point of the 

 effective space ; and in order to multiply these variable magni- 

 tudes into the effective space, that is, to deduce the quantity of 

 motion from the intensity of the pushing force, we must call in 

 the aid of the higher mathematics. 



But hence it follows that, except in statics, where the effec- 

 tive space is nought and the pressure constant, the Newtonian 

 conception of force is available only in the higher branches of 

 mechanics ; and it is plainly not advisable so to choose our con- 

 ception of " force " that it cannot be consistently employed in 

 that branch (namely the elementary parts of the theory of motion) 

 which of all others is chiefly concerned with fundamental notions. 



It is, however, a totally mistaken method to try to adapt the 

 idea of a force, such as gravity, conceived in Newton's sense, to 

 the elementary parts of science, by leaving out of consideration 

 one of its most important properties, namely its dependence on 

 distance, and to make a "force" out of Galileo's gravity thus 

 inexactly and in some relations most incorrectly conceived. Some 

 such ideal force (No. III.) seems to hover before the minds of 

 most writers on natural science as the original type of a " force of 

 nature." 



Such quantitative determinations as hold good only approxi- 

 mately and under certain conditions ought never to be employed 

 to establish definitions. In a calculation, it is true, we may cor- 

 rectly enough take an arc, which is sufficiently small in com- 

 parison with the radius, as equal to the sine or to the tangent ; 

 but if we attempted to use such a relation in settling first prin- 

 ciples, we should lay a foundation for fallacies and errors. 



The Newtonian idea of force, however, transplanted in the 



manner that is commonly done into the region of elementary 



science, is no whit better than the notion of a straight curve. 



Newton's force, or attraction, in specie gravity, g, is equal to the 



dc 

 differential quotient of the velocity by the time; that is, g= j. 



This expression is quite exact, but in order to understand and 

 apply it a knowledge of the higher mathematics is required. On 

 the other hand, it is quite true that, so long as we have to do 

 only with cases in which the space fallen through is so small in 

 comparison with the earth's semidiameter that it may be disre- 



