CAUCHY'S PROOF OF TAYLOR'S THEOREM.. 75 



97. The demonstration given of equation (4) in Art. 92, 

 which equation involves Taylor's Theorem, and may even 

 speaking loosely be called Taylor's Theorem, will probably 

 disappoint the reader. Though he may be unable to discover 

 any flaw in the reasoning, he will complain of the artificial 

 and tentative character of the whole, and he will urge the 

 same objection with respect to Cauchy's method of proof 

 which we shall presently give. Without denying the justice 

 of these objections, we may reply that the highly general 

 character of the theorem may be some excuse for the com- 

 plexity and indirect nature of the investigation. But with 

 respect particularly to the dissatisfaction felt in being com- 

 pelled to assent to a number of propositions without knowing 

 beforehand the general course which the demonstration might 

 be expected to take, we may remind the student that he must 

 not while engaged in the elements of a subject expect to be 

 able, as it were, to rediscover the theorems for himself. Instead 

 of asking, "what suggested this or that step?" he must 

 frequently be contented with the simpler question, " is the 

 reasoning correct ?" To this of course he has already, perhaps 

 unconsciously, been accustomed ; for example, if a complicated 

 construction occurred in Euclid, he merely confined himself, at 

 least for some time, to an examination of the consistency of 

 the construction, and the truth of the deductions from it, 

 without attempting to retrace the steps by which Euclid 

 arrived at his construction. 



98. On account of the importance of Taylor's Theorem 

 we shall add another demonstration ; this demonstration is 

 due in substance to Cauchy. 



Let F(x) and f(x) be two functions of x which remain 

 continuous, as also their differential coefficients, between the 

 values a and a + h of the variable x. Suppose also that be- 

 tween these same values the derived function f (x) does not 



vanish. Then the fraction -~ - ^ - ~~ shall be equal 



.+*_. 



to the value of .., ; : , when in the latter x has some value 



f-.w 



included between the specified values ; that is, 6 denoting 

 some proper fraction, we shall have 



