ARTICLE XV. 



On the Insufficiency of Taylor's Theorem as commonly investigated; roith Ob- 

 jections to the Demonstrations of Poisson and Cauchy, and the assumed Gene- 

 ralization of Mr. Peacock; to which are added a new Investigation and Remarks 

 on the Development and Continuity of Functions. By Charles Bomiycastle, 

 Professor of Mathematics in the University of Virginia. Read May 15, 1840. 



Section I. 



Development of Functions. 



In another place I have pointed out the errors and conflicting views result- 

 ing from the vague manner in which mathematical writers have usually con- 

 ceived the ultimate object of their peculiar logic. 



This discussion is only incidentally connected with the present paper, but 

 it will conduce to clearness if I state so much of the construction and object of 

 modern analysis as v/e shall have occasion to refer to in what follows. 



1. Numerical logic may be regarded as divided into two great branches — 

 the body of connected arrangements and rules, to which writers have given 

 that most awkward name '' the calculus," and the operations whereby all the 

 relations of quantity are reduced to its influence, and developed by its esta- 

 blished principles. 



2. The first of these branches, for which we have remarked a new name is 

 so much wanted,* consists, 1st, of arithmetic — which any good calculating ma- 



* See note 1 . 

 VII. — 3 E 



