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 diffe- 
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’ + axr?+ &e. (1) 
where it is to be observed that the ‘‘ &c.” implies the endless 
progression of the terms beyond az’, according to the law ex- 
hibited in the terms which precede ; excluding, however, every 
thing in the form of supplement or correction. The general 
expression for the sum of x terms of this series is known to be 
* Annals of Philosophy: July, 1826, p. 50. 
