674 LL. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 
cases the difficulty will disappear, if we observe that for a great 
number of functions the series which represent them may be 
rendered convergent; so that, according to the degree of ap- 
proximation desired, we may limit ourselves to the calculation 
of a certain number of terms of the series, neglecting the rest. 
By this method the question is reduced to the primitive case of 
a finite polynomial. It is thus that we can calculate the suc- 
cession of the logarithms of numbers. But since, in this particular 
instance, the terms which had been originally neglected receive 
increments in a ratio so continually increasing for equal incre- 
ments of the variable, that the degree of approximation required 
would ultimately be affected, it is necessary, at certain intervals, 
to calculate the value of the function by different methods, and 
then respectively to use the results thus obtained, as data whence 
to deduce, by means of the machine, the other intermediate 
values. We see that the machine here performs the office of 
the third section of calculators mentioned in describing the tables 
computed by order of the French government, and that the end 
originally proposed is thus fulfilled by it. 
Such is the nature of the first machine which Mr. Babbage 
conceived. We see that its use is confined to cases where the 
numbers required are such as can be obtained by means of 
simple additions or subtractions; that the machine is, so to 
speak, merely the expression of one* particular theorem of ana- 
lysis; and that, in short, its operations cannot be extended so 
as to embrace the solution of an infinity of other questions in- 
cluded within the domain of mathematical analysis. It was 
while contemplating the vast field which yet remained to be tra- 
versed, that Mr. Babbage, renouncing his original essays, con- 
ceived the plan of another system of mechanism whose opera- 
tions should themselves possess all the generality of algebraical 
notation, and which, on this account, he denominates the dna- 
lytical Engine. 
Having now explained the state of the question, it is time for 
me to develope the principle on which is based the construction 
of this latter machine. When analysis is employed for the solu- 
tion of any problem, there are usually two classes of operations 
to execute: firstly, the numerical calculation of the various co- 
efficients; and secondly, their distribution in relation to the 
quantities affected by them. If, for example, we have to obtain 
* See Note A. 
