32 



-1 = 1+2 + 4+8 + 16 + .. .-2" = 00 — 00, 

 which involves no error or contradiction. 



It hence appears that when the geometrical development 

 is a converging series, for an arithmetical value of the common 

 ratio, no error can arise from the omission of the supplemen- 

 tary correction, which is always necessary for the completion 

 of the algebraic form of that development; but that when the 

 arithmetical value of the ratio is such as to render the series 

 divergent, the algebraic error necessarily introduces an arith- 

 metical error infinitely great : the correction of the algebraic 

 form furnishes, in such a case, the expression co — oo, that is 

 the difference of two infinites, for the finite undeveloped nume- 

 rical value : and in this there is nothing inexplicable or pecu- 

 liar. 



We see, therefore, that in passing from the convergent to 

 the divergent state of a geometrical series, we have no oc- 

 casion for any new principle, such, for instance, as the sign of 

 transition, introduced by Dr. Peacock, in the discussion of 

 this subject, in his very valuable and instructive Report on 

 Analysis, presented at the third meeting of the British Asso- 

 ciation. If there only be strict algebraic accuracy between 

 the finite expression and its developed form, there will neces- 

 sarily be equally strict numerical accuracy, whatever arithme- 

 tical values be given to the arbitrary symbols : a truth which 

 must indeed universally hold in all the results of analysis. 



II- — The developments of the binomial theorem, as well 

 as those considered above, have also been the source of much 

 perplexity and misinterpretation, when they have assumed a 

 divergent form. In contemplating these developments, the 

 fact has been overlooked, that although, when interminable, 

 they each involve an infinite series, whose terms succeed one 

 another, according to a certain uniform law, yet that series 

 alone is not the complete algebraical equivalent of the unde- 

 veloped expression : a supplementary function of the symbols 



