688 LL. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 
tions afforded by the use of numerical tables. It would be equally 
possible to introduce, by means of these cards, the logarithms 
of numbers; but perhaps it might not be in this case either the 
shortest or the most appropriate method ; for the machine might 
be able to perform the same calculations by other more expedi- 
tious combinations, founded on the rapidity with which it exe- 
cutes the four first operations of arithmetic. To give an idea of 
this rapidity, we need only mention that Mr. Babbage believes 
he can, by his engine, form the product of two numbers, each 
containing twenty figures, in three minutes. 
Perhaps the immense number of cards required for the solu- 
tion of any rather complicated problem may appear to be an 
obstacle; but this does not seem to be the case. There is no 
limit to the number of cards that can be used. Certain stuffs 
require for their fabrication not less than ¢wenty thousand cards, 
and we may unquestionably far exceed even this quantity *. 
Resuming what we have explained concerning the Analytical 
Engine, we may conclude that it is based on two principles: 
the first, consisting in the fact that every arithmetical calculation 
ultimately depends on four principal operations—addition, sub- 
traction, multiplication, and division; the second, in the possi- 
bility of reducing every analytical calculation to that of the co- 
efficients for the several terms of a series. If this last principle 
be true, all the operations of analysis come within the domain of 
the engine. To take another point of view: the use of the cards 
offers a generality equal to that of algebraical formule, since 
such a formula simply indicates the nature and order of the 
operations requisite for arriving at a certain definite result, and 
similarly the cards merely command the engine to perform these 
game operations; but in order that the mechanisms may be able 
to act to any purpose, the numerical data of the problem must 
in every particular case be introduced. Thus the same series of 
cards will serve for all questions whose sameness of nature is 
such as to require nothing altered excepting the numerical data. 
In this light the cards are merely a translation of algebraical 
formule, or, to express it better, another form of analytical 
notation. 
Since the engine has a mode of acting peculiar to itself, it will 
in every particular case be necessary to arrange the series of 
calculations conformably to the means which the machine pos- 
* See Note F. 
