Sketch of the Analytical Engine Invented by Charles Babbage

for Data. Working Variables. + + + + + + + + + + + + + + + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 10 0 11 0 0 Number of operations. Nature of operations. Variables for Results. + + 0 0 0 0 0 0 0 0 1 2 3 4 5 1 7 8 9 10 11 NOTE E.—Page 19. [Pg 46] This example has evidently been chosen on account of its brevity and simplicity, with a view merely to explain the manner in which the engine would proceed in the case of an analytical calculation containing variables, rather than to illustrate the extent of its powers to solve cases of a difficult and complex nature. The equations of page 14 are in fact a more complicated problem than the present one. We have not subjoined any diagram of its development for this new example, as we did for the former one, because this is unnecessary after the full application already made of those diagrams to the illustration of M. Menabrea’s excellent tables. It may be remarked that a slight discrepancy exists between the formulæ given in the Memoir as the data for calculation, and the results of the calculation as developed in the last division of the table which accompanies it. To agree perfectly with this latter, the data should have been given as The following is a more complicated example of the manner in which the engine would compute a trigonometrical function containing variables. To multiply Let the resulting products be represented under the general form This trigonometrical series is not only in itself very appropriate for illustrating the processes of the engine, but is likewise of much practical interest from its frequent use in astronomical computations. Before proceeding further with it, we shall point out that there are three very distinct classes of ways in which it may be desired to deduce numerical values from any analytical formula. First. We may wish to find the collective, numerical value of the whole formula, without any reference to the quantities of which that formula is a function, or to the particular mode of their combination and distribution, of which the formula is the result and representative. Values of this kind are of a strictly arithmetical nature in the most limited sense of the term, and retain no trace whatever of the processes through which they have been deduced. In fact, any one such numerical value may have been attained from an infinite variety of data, or of problems. The values for and in the two equations (see Note D.), come under this class of numerical results. Secondly. We may propose to compute the collective numerical value of each term of a formula, or of a series, and to keep these results [Pg 47] separate. The engine must in such a case appropriate as many columns to results as there are terms to compute. Thirdly. It may be desired to compute the numerical value of various subdivisions of each term, and to keep all these results separate. It may be required, for instance, to compute each coefficient separately from its variable, in which particular case the engine must appropriate two result-columns to every term that contains both a variable and coefficient. There are many ways in which it may be desired in special cases to distribute and keep separate the numerical values of different parts of an algebraical formula; and the power of effecting such distributions to any extent is essential to the algebraical character of the Analytical Engine. Many persons who are not conversant with mathematical studies, imagine that because the business of the engine is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical. This is an error. 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 algebraical notation, were provisions made accordingly. It might develope 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, partly because the necessary arrangements for effecting it would increase the complexity and extent of the mechanism to a degree that would not be commensurate with the advantages, where the main object of the invention is to translate into numerical language general formulæ of analysis already known to us, or whose laws of formation are known to us. But it would be a mistake to suppose that because its results are given in the notation of a more restricted science, its processes are therefore restricted to those of that science. The object of the engine is in fact to give the utmost practical efficiency to the resources of numerical interpretations of the higher science of analysis, while it uses the processes and combinations of this latter. To return to the trigonometrical series. We shall only consider the four first terms of the factor (), since this will be sufficient to show the method. We propose to obtain separately the numerical value of each coefficient , , &c. of (1.). The direct multiplication of the two factors gives a result which would stand thus on the engine:— [Pg 48] () () () The variable belonging to each coefficient is written below it, as we have done in the diagram, by way of memorandum. The only further reduction which is at first apparently possible in the preceding result, would be the addition of to (in which case should be effaced from ). The whole operations from the beginning would then be— First Series of Operations. Second Series of Operations. Third Series, which contains only one (final) operation. We do not enter into the same detail of every step of the processes as in the examples of Notes D. and G., thinking it unnecessary and tedious to do so. The reader will remember the meaning and use of the upper and lower indices, &c., as before explained. To proceed: we know that Consequently, a slight examination of the second line of (2.) will show that by making the proper substitutions, (2.) will become These coefficients should respectively appear on We shall perceive, if we inspect the particular arrangement of the results in (2.) on the Result-columns as represented in the diagram, that, in order to effect this transformation, each successive coefficient upon , , &c. (beginning with ), must through means of proper cards be divided by two[25]; and that one of the halves thus[Pg 49] obtained must be added to the coefficient on the Variable which precedes it by ten columns, and the other half to the coefficient on the Variable which precedes it by twelve columns; , , &c. themselves becoming zeros during the process. This series of operations may be thus expressed:— Fourth Series.[26] The calculation of the coefficients , , &c. of (1.), would now be completed, and they would stand ranged in order on , , &c. It will be remarked, that from the moment the fourth series of operations is ordered, the Variables , , &c. cease to be Result-Variables, and become mere Working-Variables. The substitution made by the engine of the processes in the second side of (3.) for those in the first side, is an excellent illustration of the manner in which we may arbitrarily order it to substitute any function, number, or process, at pleasure, for any other function, number or process, on the occurrence of a specified contingency. We will now suppose that we desire to go a step further, and to obtain the numerical value of each complete term of the product (1.), that is of each coefficient and variable united, which for the ()th term would be . We must for this purpose place the variables themselves on another set of columns, , , &c., and then order their successive multiplication by , , &c., each for each. There would thus be a final series of operations as follows:— Fifth and Final Series of Operations. (N.B. that being intended to receive the coefficient on which has no variable, will only have inscribed on it, preparatory to commencing the fifth series of operations.) From the moment that the fifth and final series of operations is ordered, the Variables , , &c. then in their turn cease to be Result-Variables and become mere [Pg 50] Working-Variables; , , &c. being now the recipients of the ultimate results. We should observe, that if the variables , , , &c. are furnished, they would be placed directly upon , , &c., like any other data. If not, a separate computation might be entered upon in a separate part of the engine, in order to calculate them, and place them on , &c. We have now explained how the engine might compute (1.) in the most direct manner, supposing we knew nothing about the general term of the resulting series. But the engine would in reality set to work very differently, whenever (as in this case) we do know the law for the general term. The two first terms of (1.) are and the general term for all after these is which is the coefficient of the ( term. The engine would calculate the two first terms by means of a separate set of suitable Operation-cards, and would then need another set for the third term; which last set of Operation-cards would calculate all the succeeding terms ad infinitum; merely requiring certain new Variable-cards for each term to direct the operations to act on the proper columns. The following would be the successive sets of operations for computing the coefficients of terms— Or we might represent them as follows, according to the numerical order of the operations:— The brackets, it should be understood, point out the relation in which the operations may be grouped, while the comma marks succession. The symbol + might be used for this latter purpose, but this would be liable to produce confusion, as + is also necessarily used to represent one class of the actual operations which are the subject of that succession. In accordance with this meaning attached to the comma, care must be taken when any one group of operations recurs more than once, as is represented above by (11 ... 15), not to insert a comma after the number or letter prefixed to that group. ,(11 ... 15) would stand for an operation followed by the group of operations (11 ... 15); instead of denoting the number of groups which are to follow each other. Wherever a general term exists, there will be a recurring group of operations, as in the above example. Both for brevity and for distinctness, a recurring group is called a cycle. A cycle of operations, then, must be understood to signify any set of operations which is repeated more than once. It is equally a cycle, whether it be repeated twice only, or an indefinite number of times; for it is the fact of a repetition occurring at all that constitutes it such. In many cases of analysis there is a recurring group of one or more [Pg 51] cycles; that is, a cycle of a cycle, or a cycle of cycles. For instance: suppose we wish to divide a series by a series, it being required that the result shall be developed, like the dividend and the divisor, in successive powers of . A little consideration of (1.), and of the steps through which algebraical division is effected, will show that (if the denominator be supposed to consist of terms) the first partial quotient will be completed by the following operations:— that the second partial quotient will be completed by an exactly similar set of operations, which acts on the remainder obtained by the first set, instead of on the original dividend. The whole of the processes therefore that have been gone through, by the time the second partial quotient has been obtained, will be,— which is a cycle that includes a cycle, or a cycle of the second order. The operations for the complete division, supposing we propose to obtain terms of the series constituting the quotient, will be,— It is of course to be remembered that the process of algebraical division in reality continues ad infinitum, except in the few exceptional cases which admit of an exact quotient being obtained. The number in the formula (4.), is always