32 
—1=1424+4484164...—%%=a—~o, 
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 © — , 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 
