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XXXII. Sojne Olservaliojis o?i Dr. Taylor's Theorem for 

 the Development of the Function <p {a + x).* 



VV HEN the Melhodns Incrementorum, &c. was first pub- 

 lished, some of the continental analysts seemed to think 



V d <3 ( (l) x^ 



that the theorem <p (a + x) = v' («) + - — V^— ^ + 



^ ^ ^ r V / 1 J ^^ ^ ■1-2 



— -J ^ 1-, &c. which doctor Taylor had given as a disco- 

 very of his own, was really the same with another theorem 

 which John Bernoulli had previously investigated, and by 

 which he had been led to the value of S y dx^ when y is a 

 function of x alone. Bernoulli, however, was himself too 

 well informed to support this opinion openly; but under 

 his patronage the question was agitated ; and he tacitly, at least, 

 permitted the republication of the papers against Taylor in 

 the collection of his works, which was edited at Lausanne 

 in 1745. But, whatever may have been the opinion of 

 Bernoulli and his supporters, the after-judgment of suc- 

 ceeding analysts has settled the debate; and it is now uni- 

 versally admitted that the two theorems are totally different 

 in the objects and purposes to which they can be applied. 

 That which was discovered by Taylor has, under the hand* 

 of La Grange, been made the basis of a calculus similar in 

 its uses and extent to the differential calculus; and the other, 

 which was given by Bernoulli, performs the same office in 

 the integral calculus to which the former is adapted in the 

 differential. 



Betwixt the two theorems, however, there is not that dif- 

 ference which might be expected to arise from the diversity 

 of their objects of application, and the opposite principle 

 upon which their demonstrations have been made to rest. 

 The one may be very easily deduced from the other: but, 

 although this is the case, there is not the least reason for 

 confounding the theorems together; nor docs this circum- 

 stance afford even the shadow of a reason for alleging any 

 similarity between them. The process on which we are 



* Communicated by the Autlior. 



O 2 about 



