220 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



Supposing, then, a numerical function P developing in a known series 



P, P, P . . . P . . &C., (4) 



and a modal function Q developing, modally, in a similar series 



Q, Q, Q . . . Q . . &c.; (5) 



it is evident that if we can so determine P that any corresponding terms in 



1 

 these developments shall agree — as, for example, that P shall be the numerical 



1 

 representative of Q, the relation which exists among the terms, and whereby 



we pass from term to term, in either direction, along the series, will immedi- 







ately determine the connexion of P and Q, of P and Q, or of any other of these 



related functions. A single example, the method of determining areas, will 



render this abundantly clear, and as the subject is elementary, and well known, 



we may dismiss it in a few words: — The modal function Q is, in this case, the 



012 

 increased area; the related, or derived functions Q, Q, Q, &c., are the unin- 



creased area — the parallelogram that composes the largest part of the increase 

 — the right lined triangle that stands on this parallelogram, and so on. Now 

 any one of these derived terms, the first excepted, is easily expressed algebra- 

 ically, and thus, by the converse of the process that derives a term from that 



which precedes it, we are enabled to repass, backward, along the developments, 







and to determine the relation of P and Q. 



It is very remarkable that mathematicians employing extensively the theory 

 described above, and which depends on the use made of that remainder R, con- 

 cerning which so much has been said, should have entertained notions in re- 

 gard to this last that were far from correct. The nature and magnitude of R 

 is manifestly an inquiry essential to the proofs, both that the development is 

 possible and that it is prime, and which is again referred to in that part of the 

 application wherein any term of the series is said to exceed all the rest of the 

 development. D'Alembert and Lagrange were thus led to compute the 

 value of a term on which so much depended; but, proceeding in a vicious cir- 

 cle, their analysis assumed the development which it was meant to demon- 

 strate, as we shall presently show. 



Cauchy, perceiving this error, but not distinguishing between developments 



