Sketch of the Analytical Engine Invented by Charles Babbage

and not results. But we are not taking this view. On the contrary, we suppose the engine to be in the course of computing the Numbers to an indefinite extent, from the very beginning; and that we merely single out, by way of example, one amongst the successive but distinct series of computations it is thus performing. Where the ’s are fractional, it must be understood that they are computed and appear in the notation of decimal fractions. Indeed this is a circumstance that should be noticed with reference to all calculations. In any of the examples already given in the translation and in the Notes, some of the data, or of the temporary or permanent results, might be fractional, quite as probably as whole numbers. But the arrangements are so made, that the nature of the processes would be the same as for whole numbers. In the above table and diagram we are not considering the signs of any of the ’s, merely their numerical magnitude. The engine would bring out the sign for each of them correctly of course, but we cannot enter on every additional detail of this kind, as we might wish to do. The circles for the signs are therefore intentionally left blank in the diagram. Operation-cards 1, 2, 3, 4, 5, 6 prepare are Thus, Card 1 multiplies two into , and the three Receiving Variable-cards belonging respectively to , , , allow the result to be placed on each of these latter columns (this being a case in which a triple receipt of the result is needed for subsequent purposes); we see that the upper indices of the two Variables used, during Operation 1, remain unaltered. We shall not go through the details of every operation singly, since the table and diagram sufficiently indicate them; we shall merely notice some few peculiar cases. By Operation 6, a positive quantity is turned into a negative quantity, by simply subtracting the quantity from a column which has only zero upon it. (The sign at the top of [Pg 62] would become—during this process.) Operation 7 will be unintelligible, unless it be remembered that if we were calculating for instead of , Operation 6 would have completed the computation of itself; in which case the engine, instead of continuing its processes, would have to put on ; and then either to stop altogether, or to begin Operations 1, 2 ... 7 all over again for value of (= 2), in order to enter on the computation of ; (having however taken care, previous to this recommencement, to make the number on , equal to two, by the addition of unity to the former on that column). Now Operation 7 must either bring out a result equal to zero (if ); 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 out of a series of antecedent and prospective computations. Cards 8, 9, 10 produce . In Operation 9 we see an example of an upper index which again becomes a value after having passed front preceding values to zero. has successively been , , , , ; and, from the nature of the office which , performs in the calculation, its index will continue to go through further changes of the same description, which, if examined, will be found to be regular and periodic. Card 12 has to perform the same office as Card 7 did in the preceding section; since, if had been = 2, the 11th operation would have completed the computation of . Cards 13 to 20 make . Since always consists of factors, has three factors; and it will be seen that Cards 13, 14, 15, 16 make the second of these factors, and then multiply it with the first; and that 17, 18, 19, 20 make the third factor, and then multiply this with the product of the two former factors. Card 23 has the office of Cards 11 and 7 to perform, since if were = 3, the 21st and 22nd operations would complete the computation of . As our case is , the computation will continue one more stage; and we must now direct attention to the fact, that in order to compute it is merely necessary precisely to repeat the group of Operations 13 to 20; and then, in order to complete the computation of , to repeat Operations 21, 22. [Pg 63] It will be perceived that every unit added to in , entails an additional repetition of operations (13 ... 23) for the computation of . Not only are all the operations precisely the same however for every such repetition, but they require to be respectively supplied with numbers from the very same pairs of columns; with only the one exception of Operation 21, which will of course need (from ) instead of (from ). This identity in the columns which supply the requisite numbers, must not be confounded with identity in the values these columns have upon them and give out to the mill. Most of those values undergo alterations during a performance of the operations (13 ... 23), and consequently the columns present a new set of values for the next performance of (13 ... 23) to work on. At the termination of the repetition of operations (13 ... 23) in computing , the alterations in the values on the Variables are, that In this state the only remaining processes are first: to transfer the value which is on , to ; and secondly to reduce , , to zero, and to add[30] one to , in order that the engine may be ready to commence computing . Operations 24 and 25 accomplish these purposes. It may be thought anomalous that Operation 25 is represented as leaving the upper index of still = unity. But it must be remembered that these indices always begin anew for a separate calculation, and that Operation 25 places upon , the first value for the new calculation. It should be remarked, that when the group (13 ... 23) is repeated, changes occur in some of the upper indices during the course of the repetition: for example, , would become , and . We thus see that when , nine Operation-cards are used; that when , fourteen Operation-cards are used; and that when , twenty-five Operation-cards are used; but that no more are needed, however great may be; and not only this, but that these same twenty-five cards suffice for the successive computation of all the Numbers from , to , inclusive. With respect to the number of Variable-cards, it will be remembered, from the explanations in previous Notes, that an average of three such cards to each operation (not however to each Operation-card) is the estimate. According to this the computation of will require twenty-seven Variable-cards; forty-two such cards; seventy-five; and for every succeeding after , there would be thirty-three additional Variable-cards (since each repetition of the group (13 ... 23) adds eleven to the number of operations required for computing the previous ). But we must now explain, that whenever there is a cycle of operations, and if these merely require to be supplied with numbers from the same pairs of columns and likewise each operation to place its result on the same column for every repetition of the whole group, the process then admits of a cycle of Variable-cards for effecting its purposes. There is obviously much more symmetry and simplicity in the arrangements, when cases do admit of repeating the Variable as well as the Operation-cards. Our present example is of this nature. The only exception to a perfect identity in all the processes and columns used, for every repetition of Operations (13 ... 23) is, that Operation 21 always requires one of its factors from a new column, and Operation 24 always puts its result on a new column. But as these [Pg 64] variations follow the same law at each repetition, (Operation 21 always requiring its factor from a column one in advance of that which it used the previous time, and Operation 24 always putting its result on the column one in advance of that which received the previous result), they are easily provided for in arranging the recurring group (or cycle) of Variable-cards. We may here remark that the average estimate of three Variable-cards coming into use to each operation, is not to be taken as an absolutely and literally correct amount for all cases and circumstances. Many special circumstances, either in the nature of a problem, or in the arrangements of the engine under certain contingencies, influence and modify this average to a greater or less extent. But it is a very safe and correct general rule to go upon. In the preceding case it will give us seventy-five Variable-cards as the total number which will be necessary for computing any after . This is very nearly the precise amount really used, but we cannot here enter into the minutiæ of the few particular circumstances which occur in this example (as indeed at some one stage or other of probably most computations) to modify slightly this number. It will be obvious that the very same seventy-five Variable-cards may be repeated for the computation of every succeeding Number, just on the same principle as admits of the repetition of the thirty-three Variable-cards of Operations (13 ... 23) in the computation of any one Number. Thus there will be a cycle of a cycle of Variable-cards. If we now apply the notation for cycles, as explained in Note E, we may express the operations for computing the Numbers of Bernoulli in the following manner:— Again, represents the total operations for computing every number in succession, from to inclusive. In this formula we see a varying cycle of the first order, and an ordinary cycle of the second order. The latter cycle in this case includes in it the varying cycle. [Pg 65] On inspecting the ten Working-Variables of the diagram, it will be perceived, that although the value on any one of them (excepting , and ) goes through a series of changes, the office which each performs is in this calculation fixed and invariable. Thus always prepares the numerators of the factors of any ; the denominators. always receives the ()th factor of , and the ()th. always decides which of two courses the succeeding processes are to follow, by feeling for the value of through means of a subtraction; and so on; but we shall not enumerate further. It is desirable in all calculations, so to arrange the processes, that the offices performed by the Variables may be as uniform and fixed as possible. Diagram for the computation by the Engine of the Numbers of Bernoulli. See Note G. (page 67 et seq.) [Pg 66] Number of operation. Nature of operation. Variables acted upon. Variables receiving results. Indication of change in the value of any Variable. Statement of Results. 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 1 2 4 0 0 0 0 0 0 0 0 0 0 1 2 1 3 1 4 0 0 5 2 6 0 7 1 8 9 10 11 0 12 13 14 15 16 0 17 18 19 20 0 21 0 22 0 23 Here follows a repetition of Operations thirteen to twenty-three 24 25 by a Variable-card. 0 0 Number of operation. Result Variables. 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 [Pg 67] Supposing that it was desired not only to tabulate , , &c., but , , &c.; we have only then to appoint another series of Variables, , , &c., for receiving these latter results as they are successively produced upon . Or again, we may, instead of this, or in addition to this second series of results, wish to tabulate the value of each successive total term of the series (8), viz: , , , &c. We have then merely to multiply each with each corresponding , as produced; and to place these successive products on Result-columns appointed for the purpose. The formula (8.) is interesting in another point of view. It is one particular case of the general Integral of the following Equation of Mixed Differences:— for certain special suppositions respecting , and . The general integral itself is of the form, and it is worthy of remark, that the engine might (in a manner more or less similar to the preceding) calculate the value of this formula upon most other hypotheses for the functions in the integral, with as much, or (in many cases) with more, ease than it can formula (8.). A. A. L. [Pg 68] FOOTNOTES: [16] We do not mean to imply that the only use made of the Jacquard cards is that of regulating the algebraical operations. But we mean to explain that those cards and portions of mechanism which regulate these operations, are wholly independent