Datasets:

lgrobol

/

openminuscule

like 0

Thus not only the mental and the material, but the theoretical and the practical in the mathematical world, are brought into more intimate and effective connexion with each other.

We are not aware of its being on record that anything partaking in the nature of what is so well designated the Analytical Engine has been hitherto proposed, or even thought of, as a practical possibility, any more than the idea of a thinking or of a reasoning machine.

We will touch on another point which constitutes an important distinction in the modes of operating of the Difference and Analytical Engines.

In order to enable the former to do its business, it is necessary to put into its columns the series of numbers constituting the first terms of the several orders of differences for whatever is the particular table under consideration.

The machine then works upon these as its data.

But these data must themselves have been already computed through a series of calculations by a human head.

Therefore that engine can only produce results depending on data which have been arrived at by the explicit and actual working out of processes that are in their nature different from any that come within the sphere of its own powers.

In other words, an analysing process must have been gone through by a human mind in order to obtain the data upon which the engine then synthetically builds its results.

The Difference Engine is in its character exclusively synthetical, while the Analytical Engine is equally capable of analysis or of synthesis.

It is true that the Difference Engine can calculate to a much greater extent with these few preliminary data, than the data themselves required for their own determination.

The table of squares, for instance, can be calculated to any extent whatever, when the numbers one and two are furnished; and a very few differences computed at any part of a table of logarithms would enable the engine to calculate many hundreds or even thousands of logarithms.

Still the circumstance of its requiring, as a previous condition, that any function whatever shall have been numerically worked out, makes it very inferior in its nature and advantages to an engine which, like the Analytical Engine, requires merely that we should know the succession and distribution of the operations to be performed; without there being any occasion, in order to obtain data on which it can work, for our ever having gone through either the same particular operations which it is itself to effect, or any others.

Numerical data must of course be given it, but they are mere arbitrary ones; not data that could only be arrived at through a systematic and necessary series of previous numerical calculations, which is quite a different thing.

To this it may be replied, that an analysing process must equally have been performed in order to furnish the Analytical Engine with the necessary operative data; and that herein may also lie a possible source of error.

Granted that the actual mechanism is unerring in its processes, the cards may give it wrong orders.

This is unquestionably the case; but there is much less chance of error, and likewise far less expenditure of time and labour, where operations only, and the distribution of these operations, have to be made out, than where explicit numerical results are to be attained.

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.

We have dwelt considerably on the distinctive peculiarities of each of these engines, because we think it essential to place their respective attributes in strong relief before the apprehension of the public; and to define with clearness and accuracy the wholly different nature of the principles on which each is based, so as to make it self-evident to the reader (the mathematical reader at least) in what manner and degree the powers of the Analytical Engine transcend those of an engine, which, like the Difference Engine, can only work out such results as may be derived from one restricted and particular series of processes, such as those included in \Delta^nu_z=0.

We think this of importance, because we know that there exists considerable vagueness and inaccuracy in the mind of persons in general on the subject.

There is a misty notion amongst most of those who have attended at all to it, that two “calculating machines” have been successively invented by the same person within the last few years; while others again have never heard but of the one original “calculating machine,” and are not aware of there being any extension upon this.

For either of these two classes of persons the above considerations are appropriate.

While the latter require a knowledge of the fact that there are two such inventions, the former are not less in want of accurate and well-defined information on the subject.

No very clear or correct ideas prevail as to the characteristics of each engine, or their respective advantages or disadvantages; and in meeting with those incidental allusions, of a more or less direct kind, which occur in so many publications of the day, to these machines, it must frequently be matter of doubt which “calculating machine” is referred to, or whether both are included in the general allusion.

We are desirous likewise of removing two misapprehensions which we know obtain, to some extent, respecting these engines.

In the first place it is very generally supposed that the Difference Engine, after it had been completed up to a certain point, suggested the idea of the Analytical Engine; and that the second is in fact the improved offspring of the first, and grew out of the existence of its predecessor, through some natural or else accidental combination of ideas suggested by this one.

Such a supposition is in this instance contrary to the facts; although it seems to be almost an obvious inference, wherever two inventions, similar in their nature and objects, succeed each other closely in order of time, and strikingly in order of value; more especially when the same individual is the author of both.

Nevertheless the ideas which led to the Analytical Engine occurred in a manner wholly independent of any that were connected with the Difference Engine.

These ideas are indeed in their own intrinsic nature independent of the latter engine, and might equally have occurred had it never existed nor been even thought of at all.

The second of the misapprehensions above alluded to relates to the well-known suspension, during some years past, of all progress in the construction of the Difference Engine.

Respecting the circumstances which have interfered with the actual completion of either invention, we offer no opinion; and in fact are not possessed of the data for doing so, had we the inclination.

But we know that some persons suppose these obstacles (be they what they may) to have arisen in consequence of the subsequent invention of the Analytical Engine while the former was in progress.

We have ourselves heard it even lamented that an idea should ever have occurred at all, which had turned out to be merely the means of arresting what was already in a course of successful execution, without substituting the superior invention in its stead.

This notion we can contradict in the most unqualified manner.

The progress of the Difference Engine had long been suspended, before there were even the least crude glimmerings of any invention superior to it.

Such glimmerings, therefore, and their subsequent development, were in no way the original cause of that suspension; although, where difficulties of some kind or other evidently already existed, it was not perhaps calculated to remove or lessen them that an invention should have been meanwhile thought of, which, while including all that the first was capable of, possesses powers so extended as to eclipse it altogether.

We leave it for the decision of each individual (after he has possessed himself of competent information as to the characteristics of each engine) to determine how far it ought to be matter of regret that such an accession has been made to the powers of human science, even if it has (which we greatly doubt) increased to a certain limited extent some already existing difficulties that had arisen in the way of completing a valuable but lesser work.

We leave it for each to satisfy himself as to the wisdom of desiring the obliteration (were that now possible) of all records of the more perfect invention, in order that the comparatively limited one might be finished.

The Difference Engine would doubtless fulfil all those practical objects which it was originally destined for.

It would certainly calculate all the tables that are more directly necessary for the physical purposes of life, such as nautical and other computations.

Those who incline to very strictly utilitarian views may perhaps feel that the peculiar powers of the Analytical Engine bear upon questions of abstract and speculative science, rather than upon those involving every-day and ordinary human interests.

These persons being likely to possess but little sympathy, or possibly acquaintance, with any branches of science which they do not find to be useful (according to their definition of that word), may conceive that the undertaking of that engine, now that the other one is already in progress, would be a barren and unproductive laying out of yet more money and labour; in fact, a work of supererogation.

Even in the utilitarian aspect, however, we do not doubt that very valuable practical results would be developed by the extended faculties of the Analytical Engine; some of which results we think we could now hint at, had we the space; and others, which it may not yet be possible to foresee, but which would be brought forth by the daily increasing requirements of science, and by a more intimate practical acquaintance with the powers of the engine, were it in actual existence.

On general grounds, both of an a priori description as well as those founded on the scientific history and experience of mankind, we see strong presumptions that such would be the case.

Nevertheless all will probably concur in feeling that the completion of the Difference Engine would be far preferable to the non-completion of any calculating engine at all.

With whomsoever or wheresoever may rest the present causes of difficulty that apparently exist towards either the completion of the old engine, or the commencement of the new one, we trust they will not ultimately result in this generation's being acquainted with these inventions through the medium of pen, ink and paper merely; and still more do we hope, that for the honour of our country's reputation in the future pages of history, these causes will not lead to the completion of the undertaking by some other nation or government.

This could not but be matter of just regret; and equally so, whether the obstacles may have originated in private interests and feelings, in considerations of a more public description, or in causes combining the nature of both such solutions.

We refer the reader to the ‘Edinburgh Review’ of July 1834, for a very able account of the Difference Engine.

The writer of the article we allude to has selected as his prominent matter for exposition, a wholly different view of the subject from that which M. Menabrea has chosen.

The former chiefly treats it under its mechanical aspect, entering but slightly into the mathematical principles of which that engine is the representative, but giving, in considerable length, many details of the mechanism and contrivances by means of which it tabulates the various orders of differences.

M. Menabrea, on the contrary, exclusively developes the analytical view; taking it for granted that mechanism is able to perform certain processes, but without attempting to explain how; and devoting his whole attention to explanations and illustrations of the manner in which analytical laws can be so arranged and combined as to bring every branch of that vast subject within the grasp of the assumed powers of mechanism.

It is obvious that, in the invention of a calculating engine, these two branches of the subject are equally essential fields of investigation, and that on their mutual adjustment, one to the other, must depend all success.

They must be made to meet each other, so that the weak points in the powers of either department may be compensated by the strong points in those of the other.

They are indissolubly connected, though so different in their intrinsic nature, that perhaps the same mind might not be likely to prove equally profound or successful in both.

We know those who doubt whether the powers of mechanism will in practice prove adequate in all respects to the demands made upon them in the working of such complicated trains of machinery as those of the above engines, and who apprehend that unforeseen practical difficulties and disturbances will arise in the way of accuracy and of facility of operation.

The Difference Engine, however, appears to us to be in a great measure an answer to these doubts.

It is complete as far as it goes, and it does work with all the anticipated success.

The Analytical Engine, far from being more complicated, will in many respects be of simpler construction; and it is a remarkable circumstance attending it, that with very simplified means it is so much more powerful.

The article in the ‘Edinburgh Review’ was written some time previous to the occurrence of any ideas such as afterwards led to the invention of the Analytical Engine; and in the nature of the Difference Engine there is much less that would invite a writer to take exclusively, or even prominently, the mathematical view of it, than in that of the Analytical Engine; although mechanism has undoubtedly gone much further to meet mathematics, in the case of this engine, than of the former one.

Some publication embracing the mechanical view of the Analytical Engine is a desideratum which we trust will be supplied before long.

Those who may have the patience to study a moderate quantity of rather dry details will find ample compensation, after perusing the article of 1834, in the clearness with which a succinct view will have been attained of the various practical steps through which mechanism can accomplish certain processes; and they will also find themselves still further capable of appreciating M. Menabrea's more comprehensive and generalized memoir.

The very difference in the style and object of these two articles makes them peculiarly valuable to each other; at least for the purposes of those who really desire something more than a merely superficial and popular comprehension of the subject of calculating engines.

A. A. L.

Note B

That portion of the Analytical Engine here alluded to is called the storehouse.

It contains an indefinite number of the columns of discs described by M. Menabrea.

The reader may picture to himself a pile of rather large draughtsmen heaped perpendicularly one above another to a considerable height, each counter having the digits from 0 to 9 inscribed on its edge at equal intervals; and if he then conceives that the counters do not actually lie one upon another so as to be in contact, but are fixed at small intervals of vertical distance on a common axis which passes perpendicularly through their centres, and around which each disc can revolve horizontally so that any required digit amongst those inscribed on its margin can be brought into view, he will have a good idea of one of these columns.

The lowest of the discs on any column belongs to the units, the next above to the tens, the next above this to the hundreds, and so on.

Thus, if we wished to inscribe 1345 on a column of the engine, it would stand thus:—

1

3

4

5

In the Difference Engine there are seven of these columns placed side by side in a row, and the working mechanism extends behind them: the general form of the whole mass of machinery is that of a quadrangular prism (more or less approaching to the cube); the results always appearing on that perpendicular face of the engine which contains the columns of discs, opposite to which face a spectator may place himself.

In the Analytical Engine there would be many more of these columns, probably at least two hundred.

The precise form and arrangement which the whole mass of its mechanism will assume is not yet finally determined.

We may conveniently represent the columns of discs on paper in a diagram like the following:—

The V's are for the purpose of convenient reference to any column, either in writing or speaking, and are consequently numbered.

The reason why the letter V is chosen for the purpose in preference to any other letter, is because these columns are designated (as the reader will find in proceeding with the Memoir) the Variables, and sometimes the Variable columns, or the columns of Variables.

The origin of this appellation is, that the values on the columns are destined to change, that is to vary, in every conceivable manner.

But it is necessary to guard against the natural misapprehension that the columns are only intended to receive the values of the variables in an analytical formula, and not of the constants.

The columns are called Variables on a ground wholly unconnected with the analytical distinction between constants and variables.

In order to prevent the possibility of confusion, we have, both in the translation and in the notes, written Variable with a capital letter when we use the word to signify a column of the engine, and variable with a small letter when we mean the variable of a formula.

Similarly, Variable-cards signify any cards that belong to a column of the engine.

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 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 algebraical 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.

The zeros beneath the symbolic circles represent each of them a disc, supposed to have the digit 0 presented in front.

Only four tiers of zeros have been figured in the diagram, but these may be considered as representing thirty or forty, or any number of tiers of discs that may be required.

Since each disc can present any digit, and each circle any sign, the discs of every column may be so adjusted as to express any positive or negative number whatever within the limits of the machine; which limits depend on the perpendicular extent of the mechanism, that is, on the number of discs to a column.

Each of the squares below the zeros is intended for the inscription of any general symbol or combination of symbols we please; it being understood that the number represented on the column immediately above is the numerical value of that symbol, or combination of symbols.

Let us, for instance, represent the three quantities a, n, x, and let us further suppose that a = 5, n = 7, x = 98.

We should have—

Read footnote.

We may now combine these symbols in a variety of ways, so as to form any required function or functions of them, and we may then inscribe each such function below brackets, every bracket uniting together those quantities (and those only) which enter into the function inscribed below it.

We must also, when we have decided on the particular function whose numerical value we desire to calculate, assign another column to the right-hand for receiving the results, and must inscribe the function in the square below this column.

In the above instance we might have any one of the following functions:—

$ax^n$, $x^{an}$, $a\cdot n\cdot x$, $\frac{a}{n}x$, $a+n+x$, \&c.

\&.c

Let us select the first.