NOTES BY THE TRANSLATOR.
The particular function whose integral the Difference Engine was constructed to tabulate, isDelta^7 u_z=0The purpose which that engine has been specially intended and adapted to fulfil, is the computation of nautical and astronomical tables. The integral ofDelta^7 u_z=0being u_z=a+bx+cx^{2}+dx^{3}+ex^{4}+fx^{5}+g x^{6}the constantsa,b,c,&c. are represented on the seven columns of discs, of which the engine consists. It can therefore tabulateaccuratelyand to anunlimited extent, all series whose general term is comprised in the above formula; and it can also tabulateapproximativelybetweenintervals of greater or less extent, all other series which are capable of tabulation by the Method of Differences.
The Analytical Engine, on the contrary, is not merely adapted fortabulatingthe results of one particular function and of no other, but fordeveloping and tabulatingany function whatever. In fact the engine may be described as being the material expression of any indefinite function of any degree of generality and complexity, such as for instance,F(x,y,z,log x,sin y,x^{p}, &c.)which is, it will be observed, a function of all other possible functions of any number of quantities.
In this, which we may call theneutralorzerostate of the engine, it is ready to receive at any moment, by means of cards constituting a portion of its mechanism (and applied on the principle of those used in the Jacquard-loom), the impress of whateverspecialfunction we may desire to develope or to tabulate. These cards contain within themselves (in a manner explained in the Memoir itself, pages 12 and 13) the law of development of the particular function that may be under consideration, and they compel the mechanism to act accordingly in a certain corresponding order. One of the simplest cases would be, for example, to suppose thatF(x,y,z, &c., &c.)is the particular functionDelta^{n} u_z = 0which the Difference Engine tabulates for values ofnonly up to 7. In this case the cards would order the mechanism to go through that succession of operations which would tabulateu_z = a + bx + cx^{2} + ... mx^{n-1}wherenmight be any number whatever.
These cards, however, have nothing to do with the regulation of the particularnumericaldata. They merely determine theoperations[16]to be effected, which operations may of course be performed on an infinite variety of particular numerical values, and do not bring out any definite numerical results unless the numerical data of the problem have been impressed on the requisite portions of the train of mechanism. In the above example, the first essential step towards an arithmetical result, would be the substitution of specific numbers forn,and for the other primitive quantities which enter into the function.
Again, let us suppose that forFwe put two complete equations of the fourth degree betweenxandy.We must then express on the cards the law of elimination for such equations. The engine would follow out those laws, and would ultimately give the equation of one variable which results from such elimination. Variousmodesof elimination might be selected; and of course the cards must be made out accordingly. The following is one mode that might be adopted. The engine is able to multiply together any two functions of the forma+bx+cx^{2}+ ... px^{n}This granted, the two equations may be arranged according to the powers ofy,and the coefficients of the powers ofymay be arranged according to powers ofx.The elimination ofywill result from the successive multiplications and subtractions of several such functions. In this, and in all other instances, as was explained above, the particularnumericaldata and thenumericalresults are determined by means and by portions of the mechanism which act quite independently of those that regulate theoperations.
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 tooperations, as distinguished fromthe objects operated uponand from theresultsof the operations performed upon those objects, is very strikingly defined and separated.
It is well to draw attention to this point, not only because its full appreciation is essential to the attainment of any very just and adequate general comprehension of the powers and mode of action of the Analytical Engine, but also because it is one which is perhaps too little kept in view in the study of mathematical science in general. It is, however, impossible to confound it with other considerations, either when we trace the manner in which that engine attains its results, or when we prepare the data for its attainment of those results. 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 withoperationsshould hold the same just place as a clear and well-defined branch of the subject of analysis, a fundamental but yet independent ingredient in thescience, which they must do in studying the engine. The confusion, the difficulties, the contradictions which, in consequence of a want of accurate distinctions in this particular, have up to even a recent period encumbered mathematics in all those branches involving the consideration of negative and impossible quantities, will at once occur to the reader who is at all versed in this science, and would alone suffice to justify dwelling somewhat on the point, in connexion with any subject so peculiarly fitted to give forcible illustration of it, as the Analytical Engine. It may be desirable to explain, that by the wordoperation, we meanany process which alters the mutual relation of two or more things, be this relation of what kind it may. This is the most general definition, and would include all subjects in the universe. In abstract mathematics, of course operations alter those particular relations which are involved in the considerations of number and space, and theresultsof operations are those peculiar results which correspond to the nature of the subjects of operation. But the science of operations, as derived from mathematics more especially, is a science of itself, and has its own abstract truth and value; just as logic has its own peculiar truth and value, independently of the subjects to which we may apply its reasonings and processes. Those who are accustomed to some of the more modern views of the above subject, will know that a few fundamental relations being true, certain other combinations of relations must of necessity follow; combinations unlimited in variety and extent if the deductions from the primary relations be carried on far enough. They will also be aware that one main reason why the separate nature of the science of operations has been little felt, and in general little dwelt on, is theshiftingmeaning of many of the symbols used in mathematical notation. First, the symbols ofoperationare frequentlyalsothe symbols of theresultsof operations. We may say that these symbols are apt to have both aretrospectiveand aprospectivesignification. They may signify either relations that are the consequence of a series of processes already performed, or relations that are yet to be effected through certain processes. Secondly, figures, the symbols ofnumerical magnitude, are frequentlyalsothe symbols ofoperations, as when they are the indices of powers. Wherever terms have a shifting meaning, independent sets of considerations are liable to become complicated together, and reasonings and results are frequently falsified. Now in the Analytical Engine the operations which come under the first of the above heads, are ordered and combined by means of a notation and of a train of mechanism which belong exclusively to themselves; and with respect to the second head, whenever numbers meaningoperationsand notquantities(such as the indices of powers), are inscribed on any column or set of columns, those columns immediately act in a wholly separate and independent manner, becoming connected with theoperating mechanismexclusively, and re-acting upon this. They never come into combination with numbers upon any other columns meaningquantities; though, of course, if there are numbers meaningoperationsuponncolumns, these maycombine amongst each other, and will often be required to do so, just as numbers meaningquantitiescombine with each other in any variety. It might have been arranged that all numbers meaningoperationsshould have appeared on some separate portion of the engine from that which presents numericalquantities; but the present mode is in some cases more simple, and offers in reality quite as much distinctness when understood.
The operating mechanism can even be thrown into action independently of any object to operate upon (although of course noresultcould then be developed). Again, it might act upon other things besidesnumber, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine. Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.
The Analytical Engine is anembodying of the science of operations, constructed with peculiar reference to abstract number as the subject of those operations. The Difference Engine is the embodying ofone particular and very limited set of operations, which (see the notation used in note B) may be expressed thus,(+, +, +, +, +, +), or thus, 6(+). Six repetitions of the one operation, +, is, in fact, the whole sum and object of that engine. It has seven columns, and a number on any column can add itself to a number on the next column to itsright-hand. So that, beginning with the column furthest to the left, six additions can be effected, and the result appears on the seventh column, which is the last on the right-hand. Theoperatingmechanism of this engine acts in as separate and independent a manner as that of the Analytical Engine; but being susceptible of only one unvarying and restricted combination, it has little force or interest in illustration of the distinct nature of thescience of operations. The importance of regarding the Analytical Engine under this point of view will, we think, become more and more obvious, as the reader proceeds with M. Menabrea’s clear and masterly article. 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 itsoperations: we allude to the attainment of those combinations into whichimaginary quantitiesenter. This is a branch of its processes into which we have not had the opportunity of inquiring, and our conjecture therefore as to the principle on which we conceive the accomplishment of such results may have been made to depend, is very probably not in accordance with the fact, and less subservient for the purpose than some other principles, or at least requiring the cooperation of others. 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 themechanism, to bring out adoubleset ofresults, viz.—1st, thenumerical magnitudeswhich are the results of operations performed onnumerical data. (These results are the primary object of the engine). 2ndly, thesymbolical resultsto be attached to those numerical results, which symbolical results are not less the necessary and logical consequences of operations performed uponsymbolical data, than are numerical results when the data are numerical[17].
If we compare together the powers and the principles of construction of the Difference and of the Analytical Engines, we shall perceive that the capabilities of the latter are immeasurably more extensive than those of the former, and that they in fact hold to each other the same relationship as that of analysis to arithmetic. The Difference Engine can effect but one particular series of operations, viz. that required for tabulating the integral of the special functionDelta^{n} u_z = 0and as it can only do this for values ofnup to 7[18], it cannot be considered as being the mostgeneralexpression even ofone particularfunction, much less as being the expression of any and all possible functions of all degrees of generality. The Difference Engine can in reality (as has been already partly explained) do nothing butadd; and any other processes, not excepting those of simple subtraction, multiplication and division, can be performed by it only just to that extent in which it is possible, by judicious mathematical arrangement and artifices, to reduce them to aseries of additions. The method of differences is, in fact, a method of additions; and as it includes within its means a larger number of results attainable byadditionsimply, than any other mathematical principle, it was very appropriately selected as the basis on which to constructan Adding Machine, so as to give to the powers of such a machine the widest possible range. The Analytical Engine, on the contrary, can either add, subtract, multiply or divide with equal facility; and performs each of these four operations in a direct manner, without the aid of any of the other three. This one fact implies everything; and it is scarcely necessary to point out, for instance, that while the Difference Engine can merelytabulate,and is incapable ofdeveloping, the Analytical Engine caneither tabulate or develope.
The former engine is in its nature strictlyarithmetical, and the results it can arrive at lie within a very clearly defined and restricted range, while there is no finite line of demarcation which limits the powers of the Analytical Engine. These powers are co-extensive with our knowledge of the laws of analysis itself, and need be bounded only by our acquaintance with the latter. Indeed we may consider the engine as thematerial and mechanical representativeof analysis, and that our actual working powers in this department of human study will be enabled more effectually than heretofore to keep pace with our theoretical knowledge of its principles and laws, through the complete control which the engine gives us over theexecutive manipulationof algebraical and numerical symbols.
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.
The distinctive characteristic of the Analytical Engine, and that which has rendered it possible to endow mechanism with such extensive faculties as bid fair to make this engine the executive right-hand of abstract algebra, is the introduction into it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in the fabrication of brocaded stuffs. It is in this that the distinction between the two engines lies. Nothing of the sort exists in the Difference Engine. We may say most aptly that the Analytical Engineweaves algebraical patternsjust as the Jacquard-loom weaves flowers and leaves. Here, it seems to us, resides much more of originality than the Difference Engine can be fairly entitled to claim. We do not wish to deny to this latter all such claims. We believe that it is the only proposal or attempt ever made to construct a calculating machinefounded on the principle of successive orders of differences, and capable ofprinting off its own results; and that this engine surpasses its predecessors, both in the extent of the calculations which it can perform, in the facility, certainty and accuracy with which it can effect them, and in the absence of all necessity for the intervention of human intelligenceduring the performance of its calculations. Its nature is, however, limited to the strictly arithmetical, and it is far from being the first or only scheme for constructingarithmeticalcalculating machines with more or less of success.
The bounds ofarithmeticwere however outstepped the moment theidea of applying the cards had occurred; and the Analytical Engine does not occupy common ground with mere “calculating machines.” It holds a position wholly its own; and the considerations it suggests ate most interesting in their nature. In enabling mechanism to combine togethergeneralsymbols, in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of themost abstractbranch of mathematical science. A new, a vast, and a powerful language is developed for the future use of analysis, in which to wield its truths so that these may become of more speedy and accurate practical application for the purposes of mankind than the means hitherto in our possession have rendered possible. 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 theAnalyticalEngine 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 worksuponthese 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, ananalysingprocess must have been gone through by a human mind in order to obtain the data upon which the engine thensyntheticallybuilds its results. The Difference Engine is in its character exclusivelysynthetical, 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 numbersoneandtwoare 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 thesuccession and distribution of the operationsto be performed; without there being any occasion[19], 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 necessaryoperativedata; 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 fromone restricted and particular series of processes, such as those included inDelta^{n} u_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, thattwo“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 thereare twosuch 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 doubtwhich“calculating machine” is referred to, or whetherbothare 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,suggestedthe idea of the Analytical Engine; and that the second is in fact the improved offspring of the first, andgrew outof 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 oftime, and strikingly in order ofvalue; more especially when the sameindividual 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 arisenin consequenceof the subsequent invention of the Analytical Engine while the former was in progress. We have ourselves heard it evenlamentedthat 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 originalcauseof 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 hist 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 himselfof 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 ithas(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 tallies 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 even-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 beuseful(according totheirdefinition 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 hintat, 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 anà prioridescription 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 someothernation 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 explainhow; 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 beingmore complicated, will in many respects be of simpler construction; and it is a remarkable circumstance attending it, that with verysimplifiedmeans 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 themechanicalview 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.
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 itsedgeat equal intervals; and if he then conceives that the counters do not actually lie one upon another so as to be in contact, which passes perpendicularly through their centres, and around which each disc canrevolve horizontallyso 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. Thelowestof 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 n 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 arrangementwhich 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:—
TheV’s are for the purpose of convenient reference to any column, either in writing or speaking, and are consequently numbered. The reason why the letterVis chosen for this purpose in preference to any other letter, is because these columns are designated (as the reader will find in proceeding with the Memoir) theVariables, and sometimes theVariable columns, or thecolumns of Variables. The origin of this appellation is, that the values on the columns are destined to change, that is tovary, 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 thevariablesin an analytical formula, and not of theconstants. The columns are called Variables on a ground wholly unconnected with theanalyticaldistinction 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 acolumn of the engine, and variable with a small letter when we mean thevariable of a formula. Similarly,Variable-cardssignify 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[20]for the other, according as the number represented on the column below is positive or negative. In a similar manner any other purelysymbolicalresults of algebraical processes might be made to appear in these circles. InNote Athe practicability of developingsymbolicalwith no less ease thannumericalresults has been touched on.
The zeros beneath thesymboliccircles 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[21]as to express any positive or negative number whatever within the limits of the machine; which limits depend on theperpendicularextent 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 anygeneralsymbol 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 quantitiesa,n,x,and let us further suppose thata = 5,n = 7,x = 98.We should have—
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 theresults, and must inscribe the function in the square below this column. In the above instance we might have any one of the following functions:—a x^{n}, x^{a n},a x n x x, {a}/{n} x, a+n+x,\& c. \& c.Let us select the first. It would stand as follows, previous to calculation;—
The data being given, we must now put into the engine the cards proper for directing the operations in the case of the particular function chosen. These operations would in this instance be,—
Firstly, six multiplications in order to getx^n (= 98^{7}))for the above particular data).
Secondly, one multiplication in order then to geta x x^n (= 98^{7})).
In all, seven multiplications to complete the whole process. We may thus represent them:—(+,+,+,+,+,+,+) or 7x
The multiplications would, however, at successive stages in the solution of the problem, operate on pairs of numbers, derived fromdifferentcolumns. In other words, thesame operationwould be performed on differentsubjects of operation. Andhere again is an illustration of the remarks made in the preceding Note on the independent manner in which the engine directs itsoperations. In determining the value ofa x^{n},theoperationsarehomogeneous, but are distributed amongst differentsubjects of operation, at successive stages of the computation. It is by means of certain punched cards, belonging to the Variables themselves, that the action of the operations is sodistributedas to suit each particular function. TheOperation-cardsmerely determine the succession of operations in a general manner. They in fact throw all that portion of the mechanism included in themill, into a series of differentstates, which we may call theadding state, or themultiplying state, &c. respectively. In each of these states the mechanism is ready to act in the way peculiar to that state, on any pair of numbers which may be permitted to come within its sphere of action. Onlyoneof these operating states of the mill can exist at a time; and the nature of the mechanism is also such that onlyone pair of numberscan be received and acted on at a time. Now, in order to secure that the mill shall receive a constant supply of the proper pairs of numbers in succession, and that it shall also rightly locate the result of an operation performed upon any pair, each Variable has cards of its own belonging to it. It has, first, a class of cards whose business it is toallowthe number on the Variable to pass into the mill, there to be operated upon. These cards may be called theSupplying-cards.Theyfurnish the mill with its proper food. Each Variable has, secondly, another class of cards, whose office it is to allow the Variable to receive a numberfromthe mill. These cards may be called theReceiving-cards.Theyregulate the location of results, whether temporary or ultimate results. The Variable-cards in general (including both the preceding classes) might, it appears to us, be even more appropriately designated the Distributive-cards, since it is through their means that the action of the operations, and the results of this action, are rightlydistributed.
There aretwo varietiesof theSupplyingVariable-cards, respectively adapted for fulfilling two distinct subsidiary purposes: but as these modifications do not bear upon the present subject, we shall notice them in another place.
In the above case ofa x^{n},the Operation-cards merely order seven multiplications, that is, they order the mill to be in themultiplying stateseven successive times (without any reference to the particular columns whose numbers are to be acted upon). The proper Distributive Variable-cards step in at each successive multiplication, and cause the distributions requisite for the particular case.array showing the distributive variables cards stepThe engine might be made to calculate all these in succession. Having completeda x^{n},the functionx^{an}might be written under the brackets instead ofa x^{n},and a new calculation commenced (the appropriate Operation and Variable-cards for the new function of course coming into play). The results would then appear onV_5.So on for any number of different functions of the quantitiesa,x,n.Eachresultmight either permanentlyremain on its column during the succeeding calculations, so that when all the functions had been computed, their values would simultaneously exist onV_4,V_5,V_6,&c.; or each result, might (after being jointed off, or used in any specified manner) be effaced to make way for its successor. The square underV_4ought, for the latter arrangement, to have the functionsa x^{n},x^{an},anx,&c. successively inscribed in it.
Let us now suppose that we havetwoexpressions whose values have been computed by the engine independently of each other (each having its own group of columns for data and results). Let them bea,b.p.y.They would then stand as follows on the columns:—
We may now desire to combine together these two results, in any manner we please; in which case it would only be necessary to have an additional card or cards, which should order the requisite operations to be performed with the numbers on the two result-columns,V_4andV_8,and theresult of these further operationsto appear on a new column,V_9.Say that we wish to dividea x^{n}byb.p.y.The numerical value of this division would then appear on the columnV_9,beneath which we have inscribed{a x^{n}}/{b~p~y}.The whole series of operations from the beginning would be as follow (n being =7being = 7):—7x, 2x, / or 9x, /
This example is introduced merely to show that we may, if we please, retain separately and permanently anyintermediateresults (likea x^{n}, b.p.y), which occur in the course of processes having an ulterior and more complicated result as their chief and final objectlike {a x^{n}}/{bpy}.
Any group of columns may be considered as representing ageneralfunction, until aspecialone has been implicitly impressed upon them through the introduction into the engine of the Operation and Variable-cards made out for aparticularfunction. Thus, in the preceding example,V_1,V_2,V_3,V_5,V_6,V_7represent thegeneralfunctionphi(a, n, b, p, x, y)until the function{a x^{n}}/{b.p.y}has been determined on, andimplicitlyexpressed by the placing of the right cards in the engine. The actual working of the mechanism, as regulated by these cards, thenexplicitlydevelopes the value of the function. The inscription of a function under the brackets, and in the square under the result-column, in no way influences the processes or the results, and is merely a memorandum for the observer, to remind him of what is going on. It is the Operation and the Variable-cards only, which in reality determine the function. Indeed it should be distinctly kept in mind that the inscriptions withinanyof the squares, arequite independent of the mechanism or workings of the engine, and are nothing but arbitrary memorandums placed there at pleasure to assist the spectator.
The further we analyse the manner in which such an engine performs its processes and attains its results, the more we perceive how distinctly it places in a true and just light the mutual relations and connexion of the various steps of mathematical analysis, how clearly it separates those things which are in reality distinct and independent, and unites those which are mutually dependent.
A. A. L.