Sketch of the Analytical Engine Invented by Charles Babbage

on the perpendicular extent of the mechanism, that is, on the number of discs to a column. [Pg 37] 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 , , , and let us further suppose that , , . We should have— +[22] + + + 0 0 0 0 0 0 9 0 5 7 8 0 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:— Let us select the first. It would stand as follows, previous to calculation;— + + + + 0 0 0 0 0 0 0 0 0 0 9 0 5 7 8 0 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 get ) for the above particular data). Secondly, one multiplication in order then to get ). In all, seven multiplications to complete the whole process. We may thus represent them:— The multiplications would, however, at successive stages in the solution of the problem, operate on pairs of numbers, derived from different columns. In other words, the same operation would be performed on different subjects of operation. And [Pg 38] here again is an illustration of the remarks made in the preceding Note on the independent manner in which the engine directs its operations. In determining the value of , the operations are homogeneous, but are distributed amongst different subjects 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 so distributed as to suit each particular function. The Operation-cards merely determine the succession of operations in a general manner. They in fact throw all that portion of the mechanism included in the mill, into a series of different states, which we may call the adding state, or the multiplying 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. Only one of these operating states of the mill can exist at a time; and the nature of the mechanism is also such that only one pair of numbers can 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 to allow the number on the Variable to pass into the mill, there to be operated upon. These cards may be called the Supplying-cards. They furnish 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 number from the mill. These cards may be called the Receiving-cards. They regulate 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 rightly distributed. There are two varieties of the Supplying Variable-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 of , the Operation-cards merely order seven multiplications, that is, they order the mill to be in the multiplying state seven 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. The engine might be made to calculate all these in succession. Having completed , the function might be written under the brackets instead of , 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 on . So on for any number of different functions of the quantities , , . Each result might either permanently [Pg 39] remain on its column during the succeeding calculations, so that when all the functions had been computed, their values would simultaneously exist on , , , &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 under ought, for the latter arrangement, to have the functions , , , &c. successively inscribed in it. Let us now suppose that we have two expressions 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 be , . They would then stand as follows on the columns:— + + + + + + + + + 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 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, and , and the result of these further operations to appear on a new column, . Say that we wish to divide by . The numerical value of this division would then appear on the column , beneath which we have inscribed . The whole series of operations from the beginning would be as follow ( being = 7):— This example is introduced merely to show that we may, if we please, retain separately and permanently any intermediate results (like ), which occur in the course of processes having an ulterior and more complicated result as their chief and final object . Any group of columns may be considered as representing a general function, until a special one has been implicitly impressed upon them through the introduction into the engine of the Operation and Variable-cards made out for a particular function. Thus, in the preceding example, , , , , , represent the general function until the function has been determined on, and implicitly expressed by the placing of the right cards in the engine. The actual working of the mechanism, as regulated by these cards, then explicitly developes 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 within any of the squares, are [Pg 40] quite 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. NOTE C.—Page 12. Those who may desire to study the principles of the Jacquard-loom in the most effectual manner, viz. that of practical observation, have only to step into the Adelaide Gallery or the Polytechnic Institution. In each of these valuable repositories of scientific illustration, a weaver is constantly working at a Jacquard-loom, and is ready to give any information that may be desired as to the construction and modes of acting of his apparatus. The volume on the manufacture of silk, in Lardner’s Cyclopædia, contains a chapter on the Jacquard-loom, which may also be consulted with advantage. The mode of application of the cards, as hitherto used in the art of weaving, was not found, however, to be sufficiently powerful for all the simplifications which it was desirable to attain in such varied and complicated processes as those required in order to fulfil the purposes of an Analytical Engine. A method was devised of what was technically designated backing the cards in certain groups according to certain laws. The object of this extension is to secure the possibility of bringing any particular card or set of cards into use any number of times successively in the solution of one problem. Whether this power shall be taken advantage of or not, in each particular instance, will depend on the nature of the operations which the problem under consideration may require. The process is alluded to by M. Menabrea in page 15, and it is a very important simplification. It has been proposed to use it for the reciprocal benefit of that art, which, while it has itself no apparent connexion with the domains of abstract science, has yet proved so valuable to the latter, in suggesting the principles which, in their new and singular field of application, seem likely to place algebraical combinations not less completely within the province of mechanism, than are all those varied intricacies of which intersecting threads are susceptible. By the introduction of the system of backing into the Jacquard-loom itself, patterns which should possess symmetry, and follow regular laws of any extent, might be woven by means of comparatively few cards. Those who understand the mechanism of this loom will perceive that the above improvement is easily effected in practice, by causing the prism over which the train of pattern-cards is suspended, to revolve backwards instead of forwards, at pleasure, under the requisite circumstances; until, by so doing, any particular card, or set of cards, that has done duty once, and passed on in the ordinary regular succession, is brought back to the position it occupied just before it was used the preceding time. The prism then resumes its forward rotation, and thus brings the card or set of cards in question into play a second time. The prism then resumes its forward rotation, and thus brings the card or set of cards in question into play a second time. This process may obviously be repeated any number of times. A. A. L. NOTE D.—Page 15. [Pg 41] We have represented the solution of these two equations, with every detail, in a diagram[23] similar to those used in Note B.; but additional explanations are requisite, partly in order to make this more complicated case perfectly clear, and partly for the comprehension of certain indications and notations not used in the preceding diagrams. Those who may wish to understand Note G. completely, are recommended to pay particular attention to the contents of the present Note, or they will not otherwise comprehend the similar notation and indications when applied to a much more complicated case. In all calculations, the columns of Variables used may be divided into three classes:— 1st. Those on which the data are inscribed: 2ndly. Those intended to receive the final results: 3dly. Those intended to receive such intermediate and temporary combinations of the primitive data as are not to be permanently retained, but are merely needed for working with, in order to attain the ultimate results. Combinations of this kind might properly be called secondary data. They are in fact so many successive stages towards the final result. The columns which