Babbage/Menabrea: Analytical Engine

From: Marinho Francis-Oladipo (
Date: Tue Feb 13 2001 - 19:23:34 GMT


This article by L.F. Menabrea of Turin, Italy, published in 1842
attempts to describe the vision that the inventor, Charles Babbage had
for the Analytical Engine. Based on lectures given by Mr Babbage a few
years previously, Menabrea wrote this paper to communicate his
understanding of the inventors dream rather than to give a blue-print
of the proposed machine. The machine in question was intended to be one
that would work to solve absolutely any problem using inputs, outputs
and punch cards.

> The illustrious inventor having been kind enough to communicate to
> me some of his views on this subject during a visit he made at
> Turin, I have, with his approbation, thrown together the
> impressions they have left on my mind. But the reader must not
> expect to find a description of Mr. Babbage's engine; the
> comprehension of this would entail studies of much length; and I
> shall endeavour merely to give an insight into the end proposed,
> and to develope the principles on which its attainment depends.

Menabrea, in order to distinguish or emphasise the difference between
the Analytic Engine and the Difference Engine (Babbage's predecessor
idea), first goes into detail about the origin of both ideas.
Commencing with the Difference Engine, he explains how an extension of
the decimal system via logarithmic and trigonometric tables undertaken
in England (following the French lead), prompted Mr Babbage to believe
that a part of the required operations could be performed by a machine
working on the principle of differences.


> It is well known that the French government, wishing to promote
> the extension of the decimal system, had ordered the construction
> of logarithmical and trigonometrical tables of enormous extent. M.
> de Prony, who had been entrusted with the direction of this
> undertaking, divided it into three sections, to each of which was
> appointed a special class of persons.

Menabrea then carries on, describing the process of the difference
principle using an example and explaining how the principle could be
implemented in a machine.

> To give some notion of this, it will suffice to consider the
> series of whole square numbers, 1, 4, 9, 16, 25, 6, 49, 64, &c.
> By subtracting each of these from the succeeding one, we obtain a
> new series, which we will name the Series of First Differences,
> consisting of the numbers , 5, 7, 9, 11, 1 , 15, &c. On
> subtracting from each of these the preceding one, we obtain the
> Second Differences, which are all constant and equal to 2.

> Now, to conceive how these operations may be reproduced by a
> machine, suppose the latter to have three dials, designated as A,
> B, C, on each of which are traced, say a thousand divisions, by
> way of example, over which a needle shall pass. The two dials, C,
> B, shall have in addition a registering hammer, which is to give a
> number of strokes equal to that of the divisions indicated by the
> needle. For each stroke of the registering hammer of the dial C,
> the needle B shall advance one division; similarly, the needle A
> shall advance one division for every stroke of the registering
> hammer of the dial B. Such is the general disposition of the
> mechanism.

The construction of the Difference Engine is then shown by Menabrea to
be a specific instance of a general theorem of polynomials. He also
talks about the requirements for such a machine to perform completely
(i.e. to handle subtraction as well as addition) and to handle
unbounded polynomial cases. Great care appears to have been taken to
cover every possible eventuality of what could be termed computation.
This is necessary if the machine is to live up to its name of being
able to perform any function or calculation.

> So that, in order to reproduce the series of values of the
> polynomial by means of a machine analogous to the one above
> described, it is sufficient that there be (m + 1) dials, having
> the mutual relations we have indicated. As the differences may be
> either positive or negative, the machine will have a contrivance
> for either advancing or retrograding each needle, according as the
> number to be algebraically added may have the sign plus or
> minus.

> If from a polynomial we pass to a series having an infinite number
> of terms, arranged according to the ascending powers of the
> variable, it would at first appear, that in order to apply the
> machine to the calculation of the function represented by such a
> series, the mechanism must include an infinite number of dials,
> which would in fact render the thing impossible. But in many 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
> approximation 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.

Menabrea concludes that the goal of the Difference Engine to meet the
specified requirements was satisfied even though its operations were
limited to addition and subtraction. It is clear that there are more
mathematical methods than this and he moves on to talk about the
follow-up idea of Charles Babbage, the Analytical Engine. This machine
was meant to solve the problem of dealing with absolutely any
mathematical problem. In order to do this, it is shown that the machine
has to be entirely automated without any intervention.

> When analysis is employed for the solution of any problem, there
> are usually two classes of operations to execute: first, the
> numerical calculation of the various coefficients; and secondly,
> their distribution in relation to the quantities affected by
> them.

> But if human intervention were necessary for directing each of
> these partial operations, nothing would be gained under the heads
> of correctness and economy of time; the machine must therefore
> have the additional requisite of executing by itself all the
> successive operations required for the solution of a problem
> proposed to it, when once the primitive numerical data for this
> same problem have been introduced.

Menabrea then without detail states that Charles Babbage did research
into mechanising the mathematical operation, division. It is not clear
why Menabrea could not explain Babbage's ideas on implementing division
but to get around that point he left it to assumption. He concludes
that given a machine that can implement the four main arithmetic
functions, i.e. addition, subtraction, multiplication and division,
any other problem could be solved. This slightly contrasted with his
earlier comment in the paper about Pascal's machine being less powerful
as it only implemented the four main arithmetic operations.

> For instance, the much-admired machine of Pascal is now simply an
> object of curiosity, which, whilst it displays the powerful
> intellect of its inventor, is yet of little utility in itself. Its
> powers extended no further than the execution of the first
> four[<]
> operations of arithmetic, and indeed were in reality confined to
> that of the first two, since multiplication and division were the
> result of a series of additions and subtractions.

> This granted, the machine is thence capable of performing every
> species of numerical calculation, for all such calculations
> ultimately resolve themselves into the four operations we have
> just named.

The next part of the paper deals with how the Analytic Engine could
carry out its functions by first tackling its number representation.
Menabrea then proceeds to show how a machine would eliminate its need
for human intervention, and as a final bid to reinforce the idea of
working of the Analytical Engine, he uses an example of two first
degree equations with two unknowns.

> Two species of threads are usually distinguished in woven stuffs;
> one is the warp or longitudinal thread, the other the woof or
> transverse thread, which is conveyed by the instrument called the
> shuttle, and which crosses the longitudinal thread or warp. When
> a brocaded stuff is required, it is necessary in turn to prevent
> certain threads from crossing the woof, and this according to a
> succession which is determined by the nature of the design that is
> to be reproduced.

> In order to diminish to the utmost the chances of error in
> inscribing the numerical data of the problem, they are
> successively placed on one of the columns of the mill; then, by
> means of cards arranged for this purpose, these same numbers are
> caused to arrange themselves on the requisite columns, without the
> operator having to give his attention to it; so that his undivided
> mind may be applied to the simple inscription of these same
> numbers.

Menabrea also takes into consideration the fact that the machine will
be functionally incomplete without handling signs. His description on
how to handle this was not however as a result of talking to Mr
Babbage, but more as a suggestion on how it could be dealt with.

> To accomplish this end, there is, above every column, both of the
> mill and of the store, a disc, similar to the discs of which the
> columns themselves consist. According as the digit on this disc
> is even or uneven, the number inscribed on the corresponding
> column below it will be considered as positive or negative.

The idea that a machine could perform any given calculatiion implied
that it would have to be able to cater for analytic problems as well as
numerical ones and Menabrea in this paper shows the possibility of this
given an assumption. He deals with the possibiliy of having a recurring
series and obstacles faced when simulating completely, analytic

> The machine is not only capable of executing those numerical
> calculations which depend on a given algebraical formula, but it
> is also fitted for analytical calculations in which there are one
> or several variables to be considered. It must be assumed that the
> analytical expression to be operated on can be developed according
> to powers of the variable, or according to determinate functions of
> this same variable, such as circular functions, for instance; and
> similarly for the result that is to be attained.

> Generally, since every analytical expression is susceptible of
> being expressed in a series ordered according to certain functions
> of the variable, we perceive that the machine will include all
> analytical calculations which can be definitively reduced to the
> formation of coefficients according to certain laws, and to the
> distribution of these with respect to the variables.

> We shall now further examine some of the difficulties which the
> machine must surmount, if its assimilation to analysis is to be
> complete. There are certain functions which necessarily change in
> nature when they pass through zero or infinity, or whose values
> cannot be admitted when they pass these limits.

Menabrea then begins to draw conclusions from his understanding of the
system proposed. He concluded that the two main principles involved in
such a machine would be numerical calculation using the four basic
mathematical operations, and reduction of analytic problems to
coefficients of terms of a series. He also talks about the
consideration of such a machine to be an intelligent one before
summarising the advantages of such a machine if built.

> Considered under the most general point of view, the essential
> object of the machine being to calculate, according to the laws
> dictated to it, the values of numerical coefficients which it is
> then to distribute appropriately on the columns which represent
> the variables, it follows that the interpretation of formulae and
> of results is beyond its province, unless indeed this very
> interpretation be itself susceptible of expression by means of the
> symbols which the machine employs. Thus, although it is not
> itself the being that reflects, it may yet be considered as the
> being which executes the conceptions of intelligence

> Now the engine, by the very nature of its mode of acting, which
> requires no human intervention during the course of its
> operations, presents every species of security under the head of
> correctness: besides, it carries with it its own check; for at the
> end of every operation it prints off, not only the results, but
> likewise the numerical data of the question; so that it is easy to
> verify whether the question has been correctly proposed. Secondly,
> economy of time: to convince ourselves of this, we need only
> recollect that the multiplication of two numbers, consisting each
> of twenty figures, requires at the very utmost three minutes.

> Thirdly, economy of intelligence: a simple arithmetical
> computation requires to be performed by a person possessing some
> capacity; and when we pass to more complicated calculations, and
> wish to use algebraical formulae in particular cases, knowledge
> must be possessed which presupposes preliminary mathematical
> studies of some extent. Now the engine, from its capability of
> performing by itself all these purely material operations, spares
> intellectual labour, which may be more profitably employed.

In my opinion, his conclusion about the machine that would perform
calculations based on rules it was given was correct to a degree. He
suggested that such a machine was not necessarily intelligent, on the
assumption that intelligence is displayed by reflection. He goes
further to say it executes conceptions of intelligence. Surely, if the
beliefs of Alan Turing and Alonzo Church are observed, then if a
machine can execute the concepts of intelligence, it should also be
able to be considered intelligent.

On the conclusion of economy of time, Menebrea does not give evidence
for this other than a statement made by Babbage with regards to the
speed of calculating the multiplication of two numbers each of twenty
figures. It is interesting to note that at the time Babbage would have
made such a claim, a working Analytical Engine had not been built. I
can only conclude that this statement came from examining his model and
own calculations but there does not seem to be any evidence to show
that such a machine could support the claim.

Touching again on the area of intelligence, Menabrea's opinion is that
little intelligence is required for such a machine as it is only
implementing operations that are given to it, in other words an
algorithm. So it speeds up calculations and does not require prior
training. Some would argue that humans implement algorithms too and as
they were learnt by humans who are considered intelligent, why should
the machine be considered to have lower intelligence since it also
learnt the algorithm.

Francis-Oladipo Marinho < >

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