ON BABBAGE’S ANALYTICAL ENGINE, 695 
provisions and additions in arranging the mechanism, to bring out a 
double set of results, viz.—1st, the numerical magnitudes which are 
the results of operations performed on numerical data. (These results 
are the primary object of the engine). Q2ndly, the symbolical results 
to be attached to those numerical results, which symbolical results are 
not less the necessary and logical consequences of operations per- 
formed upon symbolical data, than are numerical results when the data 
are numerical *. 
If we compare together the powers and the principles of construc- 
_ tion of the Difference and of the Analytical Engines, we shall perceive 
that the capabilities of the latter are immeasurably more extensive than 
those of the former, and that they in fact hold to each other the same 
relationship as that of analysis to arithmetic. The Difference En- 
_ gine can effect but one particular series of operations, viz. that re- 
quired for tabulating the integral of the special function 
Anu, =0; 
and as it can only do this for values of 2 up to 7+, it cannot be con- 
sidered as being the most general expression even of one particular 
function, much less as being the expression of any and all possible 
functions of all degrees of generality. The Difference Engine can in 
reality (as has been already partly explained) do nothing but add; and 
any other processes, not excepting those of simple subtraction, multi- 
plication and division, can be performed by it only just to that extent 
in which it is possible, by judicious mathematical arrangement and 
artifices, to reduce them to a series of additions. The method of differ- 
ences is, in fact, a method of additions; and as it includes within its 
means a larger number of results attainable by addition simply, than 
any other mathematical principle, it was very appropriately selected as 
the basis on which to construct an Adding Machine, so as to give to 
the powers of such a machine the widest possible range. The Ana- 
lytical Engine, on the contrary, can either add, subtract, multiply or 
divide with equal facility; and performs each of these four operations 
in a direct manner, without the aid of any of the other three. This 
one fact implies everything; and it is scarcely necessary to point out, 
for instance, that while the Difference Engine can merely tabulate, 
* In fact such an extension as we allude to, would merely constitute a 
rther and more perfected development of any system introduced for making 
e proper combinations of the signs plus and minus. How ably M. Menabrea 
has touched on this restricted case is pointed out in Note B. 
_ + The machine might have been constructed so as to tabulate for a higher 
_ value of z than seven. Since, however, every unit added to the value of z in- 
__ ereases the extent of the mechanism requisite, there would on this account be a 
limit beyond which it could not be practically carried. Seven is sufficiently 
high for the calculation of all ordinary tables. 
The fact that, in the Analytical Engine, the same extent of mechanism suf- 
ites for the solution of Av vz = 0, whether n= 7, n= 100,000, or n = any 
number whatever, at once suggests how entirely distinct must be the nature of 
the ee through whose application matter has been enabled to become the 
working agent of abstract mental operations in each of these engines respect- 
ively ; and it affords an equally obvious presumption, that in the case of the 
Analytical Engine, not only are those principles in themselves of a higher and 
more comprehensive description, but also such as must vastly extend the prac- 
tical value of the engine whose basis they constitute. 
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