Re: Babbage/Menabrea: Analytical Engine

From: Button David (
Date: Wed Feb 28 2001 - 16:00:47 GMT

> Those labours which belong to the various branches of the mathematical
> sciences, although on first consideration they seem to be the
> exclusive province of intellect, may, nevertheless, be divided into
> two distinct sections; one of which may be called the mechanical,
> because it is subjected to precise and invariable laws, that are
> capable of being expressed by means of the operations of matter; while
> the other, demanding the intervention of reasoning, belongs more
> specially to the domain of the understanding.

Menabrea, in his opening sentences, attempts to classify the domain
into which Mr Babbage's Analytical Engine falls. He perceives the
possible two domains of mathematics as mechanical and intellectual - in
particular understanding. He then goes on to suggest that due to the
laws evident in the 'mechanical' domain, it would be possible to employ
machinery to execute mechanical calculations, and therefore ease the
workings of such problems.

He then explains previous attempts to realise such machines, and the
reasons for their considered failure due to need of constant
supervision and their lack of problem range:

> 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. The chief drawback hitherto on
> most of such machines is, that they require the continual intervention
> of a human agent to regulate their movements

With this in mind, it was Mr Babbage's 'dream' to devote himself to the
production of a machine that did not have these drawbacks:

> Struck with similar reflections, Mr. Babbage has devoted some years to
> the realization of a gigantic idea. He proposed to himself nothing
> less than the construction of a machine capable of executing not
> merely arithmetical calculations, but even all those of analysis, if
> their laws are known.

In the words 'if their laws are known' is the only drawback of Mr
Babbage's machine, but as Menabrea described earlier in the paper, the
problems that could be solved by such machines were limited to the
'mechanical' domain anyway, so essentially it was not a drawback.

Following this, it is made clear by Menabrea that the contents of the
paper are not a detailed or even brief description of Mr Babbage's
engine, but the ideas and thoughts perceived due to lectures given by
Mr Babbage.

> I must first premise that this engine is entirely different from that
> of which there is a notice in the 'Treatise on the Economy of
> Machinery,' by the same author. But as the latter gave rise to the
> idea of the engine in question, I consider it will be a useful
> preliminary briefly to recall what were Mr. Babbage's first essays,
> and also the circumstances in which they originated.

With these words, Menabrea introduces Mr Babbage's first engine, the
Difference Engine, on which his Analytical Engine was based. He
describes the principles of differences, and the way in which they can
be used to calculate values of a polynomial.

> To give some notion of this, it will suffice to consider the series of
> whole square numbers, 1, 4, 9, 16, 25, 36, 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 3, 5, 7, 9, 11, 13, 15, &c. On subtracting from each of these
> the preceding one, we obtain the Second Differences, which are all
> constant and equal to 2.

> From the mode in which the last two columns B and C have been formed,
> it is easy to see, that if, for instance, we desire to pass from the
> number 5 to the succeeding one 7, we must add to the former the
> constant difference 2; similarly, if from the square number 9 we would
> pass to the following one 16, we must add to the former the difference
> 7, which difference is in other words the preceding difference 5, plus
> the constant difference 2

With this example, Menabrea goes on to describe a more general theorem
for the machine that can cope with any polynomial of order m.

> The theorem on which is based the construction of the machine we have
> just been describing, is a particular case of the following more
> general theorem: that if in any polynomial whatever, the highest power
> of whose variable is m, this same variable be increased by equal
> degrees; the corresponding values of the polynomial then calculated,
> and the first, second, third, &c. differences of these be taken (as
> for the preceding series of squares); the mth differences will all be
> equal to each other. 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

The conclusion that is drawn from this is that the Difference Engine
can deal with any problem limited to simple additions and subtractions.
The problem is developed by the analysis of series with an infinite
number of terms:

> 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.

Clearly this is obvious, but the problem, as described by Menabrea can
be simplified as a large number of functions are convergent and can
therefore be simplified to series with a finite number of terms.
However, Menabrea highlights the problem of intermediate data - shown
in the square numbers example - where future values depend upon
previous values. These previous values are not necessarily convergent
to the same simplified equations and therefore the results must be
calculated differently for ranges of values.

With this invention, although fulfilling it's proposed purpose,
Babbage could see that there were huge areas of mathematical analysis
that were still not considered and as such he planned a new engine, the
Analytical Engine, which would solve any mathematical problem posed.

> It was while contemplating the vast field which yet remained to be
> traversed, that Mr. Babbage, renouncing his original essays, conceived
> the plan of another system of mechanism whose operations should
> themselves possess all the generality of algebraical notation, and
> which, on this account, he denominates the Analytical Engine.

> 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.

Menabrea then explains this principle with an example of the
multiplication of two binomials. In order to calculate the expression,
first the machine would need to be able to calculate the
multiplication and addition of the various coefficients, and then be
able to place them correctly within the powers of the equation.

> In order to reproduce these operations by means of a machine, the
> latter must therefore possess two distinct sets of powers: first, that
> of executing numerical calculations; secondly, that of rightly
> distributing the values so obtained.

Menabrea then considers the problem of human intervention and the
possible limitations of a machine that had the capability of executing
the two operations, but had to be marshalled by a human user. Therefore
it would have to be able to correctly separate the classes of operation
and also be able to act at the correct moments to employ both systems.

> 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.

> It is necessarily thus; for the machine is not a thinking being, but
> simply an automaton which acts according to the laws imposed upon it.
> This being fundamental, one of the earliest researches its author had
> to undertake, was that of finding means for effecting the division of
> one number by another without using the method of guessing indicated
> by the usual rules of arithmetic.

Menabrea does not detail the process of division used by the Analytical
Engine, instead it's success is assumed and therefore the first four
operations of arithmetic possible.

> 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.

This is rather confusing as in the previous parts of the paper Menabrea
claims that the machine of Pascal was limited by the fact that it could
only perform the first four operations of arithmetic which contradicts
the quotation above. It is possible that the meaning of this
contradiction is that Pascal's machine could only execute operations of
the first four operations, whereas Babbage's engine uses these
operations as an initial phase - the second is the distribution of the
intermediate variables within the powers of the equation.

Menabrea continues by describing the physical realisation of the engine
and then the issue of human control and intervention.

> It will now be inquired how the machine can of itself, and without
> having recourse to the hand of man, assume the successive dispositions
> suited to the operations.

Menabrea's next comments relate to the working of Babbage's engine by
the use of cards - an operation card and a variable card. The way in
which brocaded stuffs can be produced efficiently is given as an
example of the way in which these cards are meant.

> 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.

This example of combining two weaving processes is followed by an
example of resolving two first-degree equations with two unknowns. The
example shows how the operation and variable cards can be used to
indicate the way in which data is to be handled.

The initial part of the example deals with simple operation of the
engine which involves multiple copies of data and multiple copies of
operations cards.

> In the preceding table it will be remarked that the column for
> operations indicates four successive multiplications,
> two subtractions, and one division. Therefore, if desired, we need
> only use three operation-cards; to manage which, it is sufficient to
> introduce into the machine an apparatus which shall, after the first
> multiplication, for instance, retain the card which relates to this
> operation, and not allow it to advance so as to be replaced by another
> one, until after this same operation shall have been four times
> repeated.

The next part of the paper deals with a more optimised engine that
handles the above refinements. This new engine inscribes data back onto
columns that have previously been used as the data has been wiped off
the columns during transfer to the mill.

> But it is possible to simplify this process, and thus to diminish the
> chances of errors, which chances are greater, the larger the number of
> the quantities that have to be inscribed previous to setting the
> machine in action. To understand this simplification, we must remember
> that every number written on a column must, in order to be
> arithmetically combined with another number, be effaced from the
> column on which it is, and transferred to the mill.

Menabrea's focus then turns to the issue of human intervention, and it
is commented that to reduce the chance of errors, the operator need do
nothing except for inscribe the numerical data.

> 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.

This comment is quite relevant as it suggests that the only source of
error arises from the intervention of human hands. Due to the type of
mechanical problem that Babbage's engines were designed to consider,
this is very important. The only data that can be generated by the
engine is that enforced by the operations given by the cards and
therefore errors are only encountered due to incorrect human

Menabrea's next consideration is that of change of sign of the results
generated. The solution is that an extra disc is added to the columns
of the engine such that the digit presented on the disc makes the
number positive or negative.

> Hitherto no mention has been made of the signs in the results, and the
> machine would be far from perfect were it incapable of expressing and
> combining amongst each other positive and negative quantities. 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 next consideration of Menabrea is that of how to do arithmetic
operations on signed numerical data. The result is that using the
format of the sign disc, addition, subtraction, multiplication and
division can easily be implemented by manipulation of the sign disc.

Mr Babbage's original Difference Engine, on which the Analytical Engine
was based, was capable of performing simple arithmetic tasks, but the
purpose of the Analytical Engine was also to execute analytical

> 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.

> 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. When such cases
> present themselves, the machine is able, by means of a bell, to give
> notice that the passage through zero or infinity is taking place, and
> it then stops until the attendant has again set it in action for
> whatever process it may next be desired that it shall perform.

Such a condition of the engine suggests a certain amount of useful
'intelligence'. That is that it does not just blindly follow it's
instructions to calculate results from given data and operations, but
can also alert the operator if a problem is going to arise. However,
this is simply a reaction to particular data and cannot really be
considered as 'thought' even though fulfilling such a task allows the
resulting data to be error-free. In fact, in furtherance to this the
machine can be set to cope with such zero or infinity conditions and
can therefore continue to operate without necessarily involving the

> If this process has been foreseen, then the machine, instead of
> ringing, will so dispose itself as to present the new cards which have
> relation to the operation that is to succeed the passage through zero
> and infinity. These new cards may follow the first, but may only come
> into play contingently upon one or other of the two circumstances just
> mentioned taking place.

Menabrea concludes by first considering the principles of the engine -
that of arithmetic calculation, and that reducing analytic calculations
to that of their coefficients. He comments that the cards used within
the machine offer generality akin to algebraic formula, such that they
simply command the engine to perform operations.

> 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 formulae, or, to express it better, another form of
> analytical notation.

> 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

By this Menabrea suggests that the engine itself is not intelligent,
instead it aids intelligence.

As considered in the early parts of the paper, the purpose of such
machines as Babbage's and also the attempt of Pascal, was to aid in
computation. That is to execute whole calculations, or ease
calculations such that the scientist or mathematician could spend
his/her time on the intellectual contemplation of more 'serious'
matters. In this way, the concept of Babbage's Analytic Engine
fulfilled it's purpose, and although it itself could not interpret the
data produced, it could allow the operator to forgo the time consuming
task of calculation.

Although this can be interpreted to be unintelligent due to the simple
following of set rules, if this were the case of a considered
'intelligent' system (human for example), the following of such an
algorithm would be taken to be intelligent - not necessarily highly
intelligent (whatever that means) - but surely intelligent nonetheless.

David Button -

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