29 



Horner, who arrived at Euler's results by aid of considerations 

 drawn from the theory of continued fractions,* 



So long, therefore, as the admitted operations of analysis 

 thus conduct to conclusions — and conclusions, too, mutually 

 confirmatory of one another, though arrived at by very difi"e- 

 rent paths — we are surely not authorized in summarily reject- 

 ing them as meaningless or absurd, merely on account of any 

 inherent difficulties involved in them. The only ground for 

 such rejection, that can generally be considered as sufficiently 

 cogent by analysts, must be errors in the reasoning by which 

 those conclusions are reached. In attempting, therefore, now 

 to point out the existence of these errors, it will be perceived 

 that I proceed on the assumption that nothing has as yet been 

 advanced, by the rejectors of diverging infinite series, against 

 the reasonings of Euler, Lacroix, and others, in reference to 

 this matter ; more especially that the method of definite inte- 

 grals, and that depending on the differential theorem, have 

 not as yet been shewn to be erroneous. I may be wrong in 

 this supposition ; if so, 1 should feel most anxious to withdraw 

 this Paper, rather than obtrude upon the attention of the 

 Academy the discussion of a topic already disposed of — and, 

 doubtless, in a more complete and satisfactory manner — else- 

 where. 



I As noticed above, the first step in the general theory 



of series occurs under the head of geometrical progression ; the 

 form of the series proposed for summation being 



a-\- ax -\- ax^ + ax^ -\- &c, (1) 



where it is to be observed that the " &c." implies the endless 

 progression of the terms beyond ax^, according to the law exr 

 hibited in the terms which precede ; excluding, however, every 

 thing in the form of supplement or correction. The general 

 expression for the sum of n terms of this series is known to be 



* Annals of Philosophy : July, 1826, p. 50. 



