Sketch of the Analytical Engine Invented by Charles Babbage

determination requires processes so lengthy and so complicated, that, although it is possible to arrive at them through great expenditure of time, labour and money, it is yet on these accounts practically almost unattainable; and we can conceive there being some results which it may be absolutely impossible in practice to attain with any accuracy, and whose precise determination it may prove highly important for [Pg 56] some of the future wants of science in its manifold, complicated and rapidly-developing fields of inquiry, to arrive at. Without, however, stepping into the region of conjecture, we will mention a particular problem which occurs to us at this moment as being an apt illustration of the use to which such an engine may be turned for determining that which human brains find it difficult or impossible to work out unerringly. In the solution of the famous problem of the Three Bodies, there are, out of about 295 coefficients of lunar perturbations given by M. Clausen (Astroe. Nachrichten, No. 406) as the result of the calculations by Burg, of two by Damoiseau, and of one by Burckhardt, fourteen coefficients that differ in the nature of their algebraic sign; and out of the remainder there are only 101 (or about one-third) that agree precisely both in signs and in amount. These discordances, which are generally small in individual magnitude, may arise either from an erroneous determination of the abstract coefficients in the development of the problem, or from discrepancies in the data deduced from observation, or from both causes combined. The former is the most ordinary source of error in astronomical computations, and this the engine would entirely obviate. We might even invent laws for series or formulæ in an arbitrary manner, and set the engine to work upon them, and thus deduce numerical results which we might not otherwise have thought of obtaining. But this would hardly perhaps in any instance be productive of any great practical utility, or calculated to rank higher than as a kind of philosophical amusement. A. A. L. NOTE G.—Page 24. It is desirable to guard against the possibility of exaggerated ideas that might arise as to the powers of the Analytical Engine. In considering any new subject, there is frequently a tendency, first, to overrate what we find to be already interesting or remarkable; and, secondly, by a sort of natural reaction, to undervalue the true state of the case, when we do discover that our notions have surpassed those that were really tenable. The Analytical Engine has no pretensions whatever to originate any thing. 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 formulæ 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. It is however pretty evident, on general principles, that in devising for mathematical truths a new form in which to record and throw themselves out for actual use, views are likely to be induced, which should again react on the more theoretical phase of the subject. There are in all extensions of human power, or additions to human [Pg 57] knowledge, various collateral influences, besides the main and primary object attained. To return to the executive faculties of this engine: the question must arise in every mind, are they really even able to follow analysis in its whole extent? No reply, entirely satisfactory to all minds, can be given to this query, excepting the actual existence of the engine, and actual experience of its practical results. We will however sum up for each reader’s consideration the chief elements with which the engine works:— 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. 5. It can arbitrarily substitute any formula for any other; effacing the first from the columns on which it is represented, and making the second appear in its stead. 6. It can provide for singular values. Its power of doing this is referred to in M. Menabrea’s memoir, page 20, where he mentions the passage of values through zero and infinity. The practicability of causing it arbitrarily to change its processes at any moment, on the occurrence of any specified contingency (of which its substitution of for () explained in Note E., is in some degree an illustration), at once secures this point. The subject of integration and of differentiation demands some notice. The engine can effect these processes in either of two ways:— First. We may order it, by means of the Operation and of the Variable-cards, to go through the various steps by which the required limit can be worked out for whatever function is under consideration. Secondly. It may (if we know the form of the limit for the function in question) effect the integration or differentiation by direct[28] substitution. We remarked in Note B., that any set of columns [Pg 58] on which numbers are inscribed, represents merely a general function of the several quantities, until the special function have been impressed by means of the Operation and Variable-cards. Consequently, if instead of requiring the value of the function, we require that of its integral, or of its differential coefficient, we have merely to order whatever particular combination of the ingredient quantities may constitute that integral or that coefficient. In , for instance, instead of the quantities being ordered to appear on in the combination , they would be ordered to appear in that of They would then stand thus:— Similarly, we might have , the integral of . An interesting example for following out the processes of the engine would be such a form as or any other cases of integration by successive reductions, where an integral which contains an operation repeated times can be made to depend upon another which contains the same or times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined. The methods in Arbogat’s Calcul des Dérivations are peculiarly fitted for the notation and the processes of the engine. Likewise the whole of the Combinatorial Analysis, which consists first in a purely numerical calculation of indices, and secondly in the distribution and combination of the quantities according to laws prescribed by these indices. We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers. The simplest manner of computing those numbers would be from the direct expansion of [Pg 59] which is in fact a particular case of the development of mentioned in Note E. Or again, we might compute them from the well-known form or from the form or from many others. As however our object is not simplicity or facility of computation, but the illustration of the powers of the engine, we prefer selecting the formula below, marked (8.). This is derived in the following manner:— If in the equation (in which , ..., &c. are the Numbers of Bernoulli), we expand the denominator of the first side in powers of , and then divide both numerator and denominator by , we shall derive If this latter multiplication be actually performed, we shall have a series of the general form in which we see, first, that all the coefficients of the powers of are severally equal to zero; and secondly, that the general form for the coefficient of the 2()th term (that is of even any power of ), is the following:— [Pg 60] Multiplying every term by () we have which it may be convenient to write under the general form:— , , &c. being those functions of which respectively belong to , , &c. We might have derived a form nearly similar to (8.), from the coefficient of any odd power of in (6.); but the general form is a little different for the coefficients of the odd powers, and not quite so convenient. On examining (7.) and (8.), we perceive that, when these formulæ are isolated from (6.) whence they are derived, and considered in themselves separately and independently, may be any whole number whatever; although when (7.) occurs as one of the ’s in (6.), it is obvious that is then not arbitrary, but is always a certain function of the distance of that from the beginning. If that distance be = , then It is with the independent formula (8.) that we have to do. Therefore it must be remembered that the conditions for the value of are now modified, and that is a perfectly arbitrary whole number. This circumstance, combined with the fact (which we may easily perceive) that whatever is, every term of (8.) after the ()th is = 0, and that the ()th term itself is always enables us to find the value (either numerical or algebraical) of any th Number of Bernoulli , in terms of all the preceding ones, if we but know the values of , ... . We append to this Note a Diagram and Table, containing the details of the computation for , (, , being supposed given). On attentively considering (8.), we shall likewise perceive that we may derive from it the numerical value of every Number of Bernoulli in succession, from the very beginning, ad infinitum, by the following series of computations:— 1st Series.—Let , and calculate (8.) for this value of . The result is . 2nd Series.—Let . Calculate (8.) for this value of substituting the value of , just obtained. The result is . 3rd Series.—Let . Calculate (8.) for this value of , substituting the values of , before obtained. The result is . And so on, to any extent. The diagram[30] represents the columns of the engine when just [Pg 61] prepared for computing , (in the case of ); while the table beneath them presents a complete simultaneous view of all the successive changes which these columns then severally pass through in order to perform the computation. (The reader is referred to Note D, for explanations respecting the nature and notation of such tables.) Six numerical data are in this case necessary for making the requisite combinations. These data are 1, 2, (= 4), , , . Were = 5, the additional datum , would be needed. Were = 6, the datum , would be needed; and so on. Thus the actual number of data needed will always be , for ; and out of these data, () of them are successive Numbers of Bernoulli. The reason why the Bernoulli Numbers used as data, are nevertheless placed on Result-columns in the diagram, is because they may properly be supposed to have been previously computed in succession by the engine itself; under which circumstances each will appear as a result, previous to being used as a datum for computing the succeeding . Here then is an instance (of the kind alluded to in Note D.) of the same Variables filling more than one office in turn. It is true that if we consider our computation of , as a perfectly isolated calculation, we may conclude , , , to have been arbitrarily placed on the columns; and it would then perhaps be more consistent to put them on , , as data