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CHAPTER XXVII. 



ON DIFFERENTIALS. 



365. IN the preceding pages we have given the proposi- 

 tions commonly found in works on the Differential Calculus, 

 and have used the method of limits in all the demonstrations. 

 We now offer a few remarks on another method of treating 

 the subject. 



In the expansion of f(os + h) by Taylor's Theorem, the 

 coefficient of h was shewn to be that function of a; which we 

 had called the differential coefficient off(x) with respect to x. 

 Lagrange proposed to define the differential coefficient of f(x) 

 with respect to x as the coefficient of h in the expansion of 

 f(x + h), and thus to avoid all reference to the theory of 

 limits. Lagrange's views were propounded towards the close 

 of the last century and were generally adopted by elementary 

 writers. 



One objection to this method is its use of infinite series 

 without ascertaining that those series are convergent, and the 

 proof that / (x + h) can always be expanded in a series of 

 ascending powers of h, which is made the foundation of the 

 Differential Calculus, labours under serious defects. Another 

 objection is that it is impossible to avoid introducing the 

 notion of a limit in the applications of the subject to geometry 

 and mechanics ; the definition of the tangent line to a curve 

 may be given as an example. 



366. Nearly all the recent treatises on the Differential 

 Calculus have followed the method of limits, and the only 

 point of importance in which a difference exists among them 

 is with respect to the use of differentials. In the present 



dii 

 work has been defined as one symbol, thus : if y = (j> (-z) 



