# Re: Lady Lovelace: Notes on Babbage/Menabrea

From: Patel Krupali (kp898@ecs.soton.ac.uk)
Date: Thu Mar 01 2001 - 15:41:20 GMT

Patel:
Lady Lovelace's Notes together with Menabrea's article are taken
together for us to understand the reasoning, behind the complete
demonstration of how "the developments and operations of analysis are
now capable of being executed by machinery". This is what we expect,
however when these notes are compared with the original article and with
the publication itself, a very different picture emerges. This picture
can be explained by Lovelace's background.

In keeping with the more general nature and status of the Analytical
Engine, Menabrea's article deals little with mechanical details. He
tends to describe the functional organisation and mathematical operation
of this more flexible and powerful engine. To illustrate its
capabilities, he presented several charts or tables of the steps which
the machine would have to go through when performing calculations and
finding numerical solutions to algebraic equations. These steps were the
instructions the engine's operator would punch in coded form on cards to
be fed into the machine. We can say these charts constituted the first
computer programs.

Menabrea's account of the Analytical Engine seems to focus on a more
general and abstract note, compared to Lardner's description of the
Difference Engine. While Lovelace's Notes seem to frequently move away
from the physical implementation and detail of the machine, she seems to
concentrate more on the metaphysical implications as well as latent
powers.

Lovelace's Notes are basically comments on specific points of Menabrea's
paper clarifying, elaborating, extending and occasionally correcting the
translated article.

> LOVELACE:
> Note A.
> The particular function whose integral the Difference Engine was constructed to tabulate, is
> D7uz= 0.
> The purpose which that engine has been specially intended and adapted to fulfil, is the computation
> of nautical and astronomical tables. The integral of D7uz= 0
> being uz = a + bx + cx2 + dx3 + ex4 + fx5 + gx6,
> the constants a, b, c, &c. are represented on the seven columns of discs, of which the engine consists.
> It can therefore tabulate accurately and to an unlimited extent, all series whose general term is
> comprised in the above formula; and it can also tabulate approximately between intervals of greater or
> less extent, all other series which are capable of tabulation by the Method of Differences.

Patel:
Computation is suppose to be determined by the symbol system it
implements and it is suppose to be implementation independent
(independent of the physical implementation). The integrals above are
to be used to implement the dynamical system. Therefore computation is
to be independent of these integrals and method of differences. However
implementing these equations cannot be the same as implementing a symbol
system. Are we performing the same computation here?
Does this mean that when we solve these integrals and equations with pen
and paper we are performing a computation? Surely if these tables are to
be implemented by a physical system we are executing our computations.
Therefore in a similar way we are implementing the mathematical laws
that the above formulae follow.

> LOVELACE:
> This granted, the two equations may be arranged according to the powers of y, and the coefficients of
> the powers of y may be arranged according to powers of x. The elimination of y will result from the
> successive multiplication's and subtractions of several such functions. In this, and in all other
> instances, as was explained above, the particular numerical data and the numerical results are
> determined by means and by portions of the mechanism which act quite independently of those that
> regulate the operations. In studying the action of the Analytical Engine, we find that the peculiar and
> independent nature of the considerations which in all mathematical analysis belong to operations, as
> distinguished from the objects operated upon and from the results of the operations performed upon
> those objects, is very strikingly defined and separated.

Patel:
Lovelace's account indicates results can not be produced unless the
system operated on the columns of decimally divided disks, with the
constant interval between successive divisions. This is the fixed
relationship, not the digits on the edges of the disks (which were
placed there for the benefit of observers only), that made the
Analytical Engine "numerical".

> LOVELACE:
> It were much to be desired, that when mathematical processes pass through the human brain
> instead of through the medium of inanimate mechanism, it were equally a necessity of things that the
> reasonings connected with operations should hold the same just place as a clear and well-defined
> branch of the subject of analysis, a fundamental but yet independent ingredient in the science, which
> they must do in studying the engine.

Patel:
She talks about the implications of the "mechanical generalisations" of
the Analytical Engine. Many of these are actually derived, or
metaphorically derived, from the physical separation between the
operating part of the engine and the numerical storage, between the
punched-card instruments for operations and the other. These separations
have both cognitive and cosmological significance. The "clear and
well-defined reasonings" to be applied to operations, taken from the
passage just quoted, refers to the rules for manipulations symbols. If
this is accepted, intelligence must be interpreted from the mathematical
derivation, along with the well- defined reasonings. And the
metaphorical reasoning is left for us as humans to find intelligent.
There doesn't seem to be any recollection of how the operations are to
be carried or any knowledge on Lovelace's part on the subject. This is
true for the most basic of operations too i.e. addition, subtraction.

> LOVELACE:
> The calculus of operations is likewise in itself a topic of so much interest, and has of late years been
> so much more written on and thought on than formerly, that any bearing which that engine, from its
> mode of constitution, may possess upon the illustration of this branch of mathematical science should
> not be overlooked. Whether the inventor of this engine had any such views in his mind while working
> out the invention, or whether he may subsequently ever have regarded it under this phase, we do not
> know; but it is one that forcibly occurred to ourselves on becoming acquainted with the means
> through which analytical combinations are actually attained by the mechanism. We cannot forbear
> suggesting one practical result which it appears to us must be greatly facilitated by the independent
> manner in which the engine orders and combines its operations: we allude to the attainment of those
> combinations into which imaginary quantities enter. This is a branch of its processes into which we
> have not had the opportunity of inquiring,

Patel:
The opportunity of inquiring was always there. Meanwhile Lovelace seems
to have passed from the practical to the speculative to the cosmological
in her notes. These are similar actions and objects which were reflected
in the design of the universe.

> LOVELACE:
> It seems to us obvious, however, that where operations are so independent in their mode of acting, it
> must be easy, by means of a few simple 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.) 2ndly,
> the symbolical results to be attached to those numerical results, which symbolical results are not less
> the necessary and logical consequences of operations performed upon symbolical data, than are
> numerical results when the data are numerical[2].

Patel:
There is nothing to indicate that Lady Lovelace was even aware of the
problems involved in making a machine to do algebra, a subject she seems
to have ignored several times. While Menabrea has cautiously interpreted
in his article the algebraic capability of calculating the coefficients
of power or functional series. Lady Lovelace made more ambitious claims
in most of her Notes, which seem to be unrestrained by Babbage as she
observed above. What is the difference between algebraic and numerical
results anyway? What makes a result numerical? The interpretation of
formulae reflects the concept of intelligence by using the numerical
results represented by symbols. These symbols are to be calculations
based on rules, which we assume, are correct. Thereby intelligence can
only be displayed here by reflection. We assume that the rule of
"attaching" the symbolic results would be for some human programmer to
have worked them out and arranged for them to be printed with the
corresponding numerical results. However, in Note E this claim is
repeated with a variation that makes it clear that Lady Lovelace had
symbolic processing by machine in mind.

> LOVELACE:

> Those who view mathematical science, not merely as a vast body of abstract and immutable truths,
> whose intrinsic beauty, symmetry and logical completeness, when regarded in their connexion
> together as a whole, entitle them to a prominent place in the interest of all profound and logical
> minds, but as possessing a yet deeper interest for the human race, when it is remembered that this
> science constitutes the language through which alone we can adequately express the great facts of the
> natural world, and those unceasing changes of mutual relationship which, visibly or invisibly,
> consciously or unconsciously to our immediate physical perceptions, are interminably going on in the
> agencies of the creation we live amidst : those who thus think on mathematical truth as the
> instrument through which the weak mind of man can most effectually read his Creator's works, will
> regard with especial interest all that can tend to facilitate the translation of its principles into explicit
> practical forms.

Patel:
The view that mathematical truths were a direct revelation of God's way
of thinking had been on the seen for well over a century. The rapid
technological and social changes that were taking place in the
nineteenth century meant that almost any scientific discovery could be
scrutinised for its implications for religious belief. In particular,
there was much interest in demonstrating that scientific activity could
lead to a strengthening rather than a weakening of faith.

> LOVELACE:
> In the case of the Analytical Engine we have undoubtedly to lay out a certain capital of analytical
> labour in one particular line; but this is in order that the engine may bring us in a much larger return
> in another line. It should be remembered also that the cards, when once made out for any formula,
> have all the generality of algebra, and include an infinite number of particular cases.

Patel:
The ideas set and the themes stressed, and the way they are presented by
Lovelace show that her Notes were inspired by views of the period in
which they were written. For her and Babbage the calculating machine
was a metaphor as well as a necessity of economic and scientific
progress. For example, the theme stressed in presenting the
"illustrations" or programs was that they demonstrate how certain
lengthy and complex calculations can be most efficiently executed -
from the point of view of the time and effort of the mathematician was
by organising the operations to be performed by the machine into
cyclical groups. The same set of operations can then be repeated over
and over by the engine, with only the starting and stopping places
indicated by the instructions. To describe this efficiency, Lady
Lovelace used an analogy taken from economic theory:

> LOVELACE:
> In the case of the Analytical Engine we have undoubtedly to lay out a certain capital of analytical
> labour in one particular line; but this is in order that the engine may bring us in a much larger return
> in another line. It should be remembered also that the cards, when once made out for any formula,
> have all the generality of algebra, and include an infinite number of particular cases.

Patel:
Where did this idea come from?
Lady Lovelace making her statement in the Notes, was comparing the
Analytical Engine with the Difference Engine. In most the Difference
machine had many disadvantages compared to the Analytical Engine.

> LOVELACE:
> To return to the explanation of the diagram: each circle at the top is intended to contain the
> algebraic sign + or -, either of which can be substituted[1] for the other, according as the number
> represented on the column below is positive or negative. In a similar manner any other purely
> symbolical results of algebraically processes might be made to appear in these circles. In Note A. the
> practicability of developing symbolical with no less ease than numerical results has been touched on.

Patel:
Touched on but not explained. In the passage just quoted, Lady Lovelace
seems to be confused between the planned engine and the representation
on her diagram. Although anything at all might be written in circles on
a diagram, it would be far more difficult to represent sines than signs
on the top disks of a metal column. This kind of confusion is inevitable
when the mechanism is based only on drawings. Much of her writing,
comes across as intuitive and mystical. She seems to be intrigued by
mathematical results, reasoning and processes, although she doesn't
understand symbols and techniques. Because of this she is never able to
turn her questions into answers. Again there is no evidence of
implementation of the machine. She could speculate that some
accomplishment might be possible, but without a firm understanding of
the subject matter she was unable to see that explanation was
necessary.

> LOVELACE:
> The engine can arrange and combine its numerical quantities exactly as if they were letters or any
> other general symbols; and in fact it might bring out its results in algebraically notation, were
> provisions made accordingly. It might develop three sets of results simultaneously, viz. Symbolic
> results (as already alluded to in Notes A. and B.); numerical results (its chief and primary object);
> and algebraical results in literal notation. This latter however has not been deemed a necessary or
> desirable addition to its powers,

Patel:
The distinction made here between "symbolic results" and the
"algebraicalresults in literal notation" might imply that the former
referred to the printing of formulae previously worked out by human
"analysts" and "attached" to the numerical results. If we look back to
Note B to confirm this there is confusion. Now Lady Lovelace makes the
claim for the algebraic capability of the engine with the aid of a
diagram, in which circles appear at the tops of the representations of
the columns of stored numbers. The circles represent the top disks on
which the signs of the numbers stored below are coded by means of odd
and even digits.

> LOVELACE:

> The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we
> know how to order it to perform. It can follow analysis; but it has no power of anticipating any
> analytical relations or truths. Its province is to assist us in making available what we are already
> acquainted with. This it is calculated to effect primarily and chiefly of course, through its executive
> faculties; but it is likely to exert an indirect and reciprocal influence on science itself in another
> manner. For, in so distributing and combining the truths and the formulae of analysis, that they may
> become most easily and rapidly amenable to the mechanical combinations of the engine, the relations
> and the nature of many subjects in that science are necessarily thrown into new lights, and more
> profoundly investigated. This is a decidedly indirect, and a somewhat speculative, consequence of
> such an invention.

Patel:
Lady Lovelace went further than Menabrea in suggesting what the
Analytical Engine could be made to achieve, making metaphysical
suggestions of the machines requirements.

> LOVELACE:
> 1. It performs the four operations of simple arithmetic upon any numbers whatever.
> 2. By means of certain artifices and arrangements (upon which we cannot enter within the restricted
> space which such a publication as the present may admit of), there is no limit either to the magnitude
> of the numbers used, or to the number of quantities (either variables or constants) that may be
> employed.
> 3. It can combine these numbers and these quantities either algebraically or arithmetically, in
> relations unlimited as to variety, extent, or complexity.
> 4. It uses algebraic signs according to their proper laws, and developes the logical consequences of
> these laws.

Patel:
This concerns the handling of the signs of the numbers in the process of
multiplication. She speculated, if the machine could be made
automatically to combine the plus and minus signs of the pairs of
numbers multiplied together, could it not be arranged to deal
appropriately with any other algebraic symbols that might accompany
numerical coefficients? Why could not symbols themselves be operated on?
It should be easy to make the engine do algebra.
The suggestion is reasonable as well as imaginative, not only of the
sign feature but also of the metaphysical status between the Difference
and the Analytical Engine. Surely algebraic symbols can represent more
than just numbers? We can not presume that they represent only numbers.
In this article they are based on rules for manipulating symbols.

There is another sense in which the engine was "numerical." There is
evidence that the number theory is applied (as well as algebra) to
simplify the mechanism or shorten the projected time required to execute
certain types of operations. The representation of the positive or
negative signs of the numbers as corresponding to odd or even digits on
the top wheel of each number column, which Lady Lovelace pointed to as
an instance of the symbolic powers of the engine, was actually one such
"arithmetical artifice." If a positive algebraic sign is represented by
an even digit and a negative sign by an odd digit, then the sign of the
result of multiplying (or diving) two numbers is represented by the
result of adding their sign digits. For example the sum of two even or
two odd digits is an even number; the sum of an even and an odd digits
is an even number; the sum of an even and an odd digit is an odd
number. Corresponding, the sign of the product of two positive or two
negative numbers is positive, and the sign of the product of a positive
and a negative number is negative.

> LOVELACE:
> Operation 7 will be unintelligible, unless it be remembered that if we were calculating for n = 1
> instead of n = 4, Operation 6 would have completed the computation of B1 itself; in which case the
> engine, instead of continuing its processes, would have to put B1 on V21 ; and then either to stop
> altogether, or to begin Operations 1, 2 . . . . 7 all over again for value of n (=2), in order to enter on
> the computation of B3; (having however taken care, previous to this recommencement, to make the
> number on V3 equal to two, by the addition of unity to the former n = 1 on that column). Now
> Operation 7 must either bring out a result equal to zero (if n = 1); or a result greater than zero, as in
> the present case; and the engine follows the one or the other of the two courses just explained,
> contingently on the one or the other result of Operation 7. In order fully to perceive the necessity of
> this experimental operation, it is important to keep in mind what was pointed out, that we are not
> treating a perfectly isolated and independent computation, but one of a series of antecedent and
> prospective computations.

Patel:
The machine is to perform these operations based on rules. If these
rules are not performed according to the rules the machine is
unintelligent. By saying this we assume that intelligence is only shown
by reflection. Therefore the machine only requires little intelligence
if all it has to do is remember operations and rules given to it
systematically. However we as humans are suppose to be the most
intelligent beings and we learn by many rules, which are the basis of
algorithms. For example we learn to recognise faces through sensorimotor
experiences. Many of our sensorimotor skills are learnt and are not
inborn, categorisation being the most important. Our capacities to learn
sort and label things. Similarly why should it be assumed that the
machine has inborn skills to do similar things. Surely operation 7 above
needs to learn that it has made a mistake and not considered
unintelligible straight away.

Throughout the notes there seems to be recurrent themes in a variety of
contexts and on different levels of abstraction. A frequently repeated
theme was the contrast between Babbage's two engines. Where Menabrea had
remarked that the Difference Engine "gave rise to the idea" of the
Analytical Engine ( which was both chronologically and logically true),
Lovelace's denies this relationship. She seems to insist that there was
no necessary temporal or conceptual relationship between the two
inventions, instead she stresses the metaphysical differences. The
Difference Engine, she went on to say , could do nothing but add and
tabulate therefore its entire significance in the numerical data it
processed. The Analytical Engine on the other hand, maintained a strict
separation between numerical data and operations, it produces only
numbers.

As a modern reader the important distinction between the two machines is
that the Difference Engine followed an unvarying computational path,
while the Analytical Engine was to be truly programmable and capable of
changing its path according to the results of calculations. However
Lovelace only seems to mention this in passing. Her notes show a series
of prototype programs, even they were written a century before the
actual construction of a programmable computer. Lovelace's focus is
quite different.

The ideas set and the themes stressed, and the way they are presented by
Lovelace show that her Notes were inspired by views of the period in
which they were written. For Babbage and Lovelace the calculating
machine was a metaphor as well an economic and scientific progress.

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