334 THE MECHANICAL EQUIVALENT OF HEAT. 



pushing force, we must call in the aid of the higher mathe- 

 matics. 



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

 fective space is nought and the pressure constant, the New- 

 tonian conception of force is available only in the higher 

 branches of mechanics ; and it is plainly not advisable so to 

 choose our conception of " force " that it cannot be consist 

 ently employed in that branch (namely, the elementary parts 

 of the theory of motion) which of all others is chiefly con- 

 cerned 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 Gali- 

 leo's gravity thus inexactly and in some relations most incor- 

 rectly 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 ap- 

 proximately and under certain conditions ought never to be 

 employed to establish definitions. In a calculation, it is true, 

 we may correctly enough take an arc, which is sufficiently 

 small in comparison with the radius, as equal in size to the 

 sine or to the tangent ; but if we attempted to use such a rela- 

 tion in settling first principles, 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 element- 

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

 curve. Newton's force, or attraction, in specie gravity, #, is 

 equal to the differential quotient of the velocity by the time ; 



that is, g=-j* This expression is quite exact, but in order to 

 ctt 



understand and apply it a knowledge of the higher mathernat- 



