184 
MR. GRAVES ON A RECTIFICATION OF THE 
resulting series calculated to any number of terms, and the respective functions which they ought to 
represent. 
Such developments have been said to be analytically accurate, notwithstanding the numerical dis- 
crepancy in each particular case. “They serve,” it is argued, “ to represent their functions, and by 
performing algebraical operations upon them, correct conclusions are attained.” 
Now, it appeared to me that there was some confusion of expression in asserting universally that 
equations were analytically true, which, numerically considered, were, in particular instances, palpably 
false. In ascertaining the correctness of the conclusions deduced from them, and relied upon as 
evidence of the truth of their premises, I observed that the formerly rejected test of numerical identity 
was often appealed to. Nay further, I was induced to ascribe, in the absence of other visible causes, 
to the intervention of such equations the limited results which were occasionally elicited where pre- 
vious calculations would lead to the expectation of general ones, and even the conclusions absolutely 
and unlimitedly erroneous to which the mathematician was sometimes conducted by apparently un- 
deviating paths. 
To account for these difficulties, upon reverting to first principles, it will be found that the 
theorems of development (such as Taylor’s, Maclaurin’s, &c.) are based upon hypothetic rea- 
soning to this effect, viz. “ if the function be developable according to certain powers, it will be deve- 
loped in a certain form,” which is assigned. Now imagine a function of x, for instance, which for 
those values only of x that lie between certain limits, is capable of being developed according to the 
ascending integral powers of x, such a function, it would seem, evolved by Maclaurin’s theorem, 
would afford an expansion which, when x transgresses those limits, would be illusory. 
In the treatment of developments thus partially true, when more than one of them come in ques- 
tion, the extent of their compatibility should, in my opinion, be most carefully attended to ; for, if two 
such developments of a function were equated, whereof the one was applicable for values of the 
variable which would render the other illusory, the consequences derived from such equation might, 
in proportion to the extent of those values, be partly or entirely false. An instance of the limitation 
introduced by the caution here recommended is to be found in Appendix § 4. 
To learn how far a development was applicable, it might be useful to ascertain the error com- 
mitted upon calculating n terms of the series, and, then supposing n an infinitely great integer, to 
observe if there were any values of the variable which would prevent the expression for the error 
from vanishing. 
Should these reflections appear dubious or unfounded, I wish it to be fully understood that they 
may, in that case, be considered as operating on my results only, at most, by way of superfluous 
caution. Thus, if c be deemed unnecessary to the universal accuracy of the series [18], it has, at all 
events, the merit of ensuring its convergence. 
Since writing the above, I have been informed by Professor Hamilton that M. Poisson has lately 
given examples of the danger of using diverging series, even when the final development to which 
they conduct is converging. 
Note F. — This seems to prove that the logarithms of negative numbers are not in general the 
same as those of their positives, as Jean Bernouilli and D’Alembert thought. (See Lacroix, 
Traite,” &c. Introd. 82.) Hence also conversely by easy inference it seems to follow, that negative 
numbers have occasionally even real logarithms, contrary to the opinion that they have none whatever, 
