Sketch of the Analytical Engine Invented by Charles Babbage

receive them are rightly named Working-Variables, for their office is in its nature purely subsidiary to other purposes. They develope an intermediate and transient class of results, which unite the original data with the final results. The Result-Variables sometimes partake of the nature of Working-Variables. It frequently happens that a Variable destined to receive a final result is the recipient of one or more intermediate values successively, in the course of the processes. Similarly, the Variables for data often become Working-Variables, or Result-Variables, or even both in succession. It so happens, however, that in the case of the present equations the three sets of offices remain throughout perfectly separate and independent. It will be observed, that in the squares below the Working-Variables nothing is inscribed. Any one of these Variables is in many cases destined to pass through various values successively during the performance of a calculation (although in these particular equations no instance of this occurs). Consequently no one fixed symbol, or combination of symbols, should be considered as properly belonging to a merely Working-Variable; and as a general rule their squares are left blank. Of course in this, as in all other cases where we mention a general rule, it is understood that many particular exceptions may be expedient. In order that all the indications contained in the diagram may be completely understood, we shall now explain two or three points, not hitherto touched on. When the value on any Variable is called into use, one of two consequences may be made to result. Either the value may return to the Variable after it has been used, in which case it is ready for a second use if needed; or the Variable may be made zero. (We are of course not considering a third case, of not unfrequent occurrence, in which the same Variable is destined to receive the result of the very operation which it has just supplied with a number.) Now the ordinary rule is, that the value returns to the Variable; unless it has been foreseen that no use for that value can recur, in which case zero is substituted. At [Pg 42] the end of a calculation, therefore, every column ought as a general rule to be zero, excepting those for results. Thus it will be seen by the diagram, that when , the value on , is used for the second time by Operation 5, becomes 0, since is not again needed; that similarly, when (), on , is used for the third time by Operation 11, becomes zero, since () is not again needed. In order to provide for the one or the other of the courses above indicated, there are two varieties of the Supplying Variable-cards. One of these varieties has provisions which cause the number given off from any Variable to return to that Variable after doing its duty in the mill. The other variety has provisions which cause zero to be substituted on the Variable, for the number given off. These two varieties are distinguished, when needful, by the respective appellations of the Retaining Supply-cards and the Zero Supply-cards. We see that the primary office (see Note B.) of both these varieties of cards is the same; they only differ in their secondary office. Every Variable thus has belonging to it one class of Receiving Variable-cards and two classes of Supplying Variable-cards. It is plain however that only the one or the other of these two latter classes can be used by any one Variable for one operation; never both simultaneously; their respective functions being mutually incompatible. It should be understood that the Variable-cards are not placed in immediate contiguity with the columns. Each card is connected by means of wires with the column it is intended to act upon. Our diagram ought in reality to be placed side by side with M. Menabrea’s corresponding table, so as to be compared with it, line for line belonging to each operation. But it was unfortunately inconvenient to print them in this desirable form. The diagram is, in the main, merely another manner of indicating the various relations denoted in M. Menabrea’s table. Each mode has some advantages and some disadvantages. Combined, they form a complete and accurate method of registering every step and sequence in all calculations performed by the engine. No notice has yet been taken of the upper indices which are added to the left of each in the diagram; an addition which we have also taken the liberty of making to the ’s in M, Menabrea’s tables of pages 16, 19, since it does not alter anything therein represented by him, but merely adds something to the previous indications of those tables. The lower indices are obviously indices of locality only, and are wholly independent of the operations performed or of the results obtained, their value continuing unchanged during the performance of calculations. The upper indices, however, are of a different nature. Their office is to indicate any alteration in the value which a Variable represents; and they are of course liable to changes during the processes of a calculation. Whenever a Variable has only zeros upon it, it is called ; the moment a value appears on it (whether that value be placed there arbitrarily, or appears in the natural course of a calculation), it becomes . If this value gives place to another value, the Variable becomes , and so forth. Whenever a value again gives place to zero, the Variable again becomes , even if it have been the moment before. If a value then again be substituted, the Variable becomes (as it would have done if it had not passed through the intermediate ); &c. &c. Just before any calculation is commenced, and after the data have been given, and everything adjusted and prepared for setting the mechanism in action, the upper indices of [Pg 43] the Variables for data are all unity, and those for the Working and Result-variables are all zero. In this state the diagram represents them[24]. There are several advantages in having a set of indices of this nature; but these advantages are perhaps hardly of a kind to be immediately perceived, unless by a mind somewhat accustomed to trace the successive steps by means of which the engine accomplishes its purposes. We have only space to mention in a general way, that the whole notation of the tables is made more consistent by these indices, for they are able to mark a difference in certain cases, where there would otherwise be an apparent identity confusing in its tendency. In such a case as there is more clearness and more consistency with the usual laws of algebraical notation, in being able to write . It is also obvious that the indices furnish a powerful means of tracing back the derivation of any result; and of registering various circumstances concerning that series of successive substitutions, of which every result is in fact merely the final consequence; circumstances that may in certain cases involve relations which it is important to observe, either for purely analytical reasons, or for practically adapting the workings of the engine to their occurrence. The series of substitutions which lead to the equations of the diagram are as follow:— There are three successive substitutions for each of these equations. The formulæ (2.), (3.), and (4.) are implicitly contained in (1.), which latter we may consider as being in fact the condensed expression of any of the former. It will be observed that every succeeding substitution must contain twice as many ’s as its predecessor. So that if a problem require substitutions, the successive series of numbers for the ’s in the whole of them will be 2, 4, 8, 16 ... . The substitutions in the preceding equations happen to be of little value towards illustrating the power and uses of the upper indices; for owing to the nature of these particular equations the indices are all unity throughout. We wish we had space to enter more fully into the relations which these indices would in many cases enable us to trace. M. Menabrea incloses the three centre columns of his table under the general title Variable-cards. The ’s however in reality all represent the actual Variable-columns of the engine, and not the cards that belong to them. Still the title is a very just one, since it is through the special action of certain Variable-cards (when combined with the more generalised agency of the Operation-cards) that every one of the particular relations he has indicated under that title is brought about. Suppose we wish to ascertain how often any one quantity, or combination of quantities, is brought into use during a calculation. [Pg 44] We easily ascertain this, from the inspection of any vertical column or columns of the diagram in which that quantity may appear. Thus, in the present case, we see that all the data, and all the intermediate results likewise, are used twice, excepting ), which is used three times. The order in which it is possible to perform the operations for the present example, enables us to effect all the eleven operations of which it consists, with only three Operation-cards; because the problem is of such a nature that it admits of each class of operations being performed in a group together; all the multiplications one after another, all the subtractions one after another, &c. The operations are . Since the very definition of an operation implies that there must be two numbers to act upon, there are of course two Supplying Variable-cards necessarily brought into action for every operation, in order to furnish the two proper numbers. (See Note B.) Also, since every operation must produce a result, which must be placed somewhere, each operation entails the action of a Receiving Variable-card, to indicate the proper locality for the result. Therefore, at least three times as many Variable-cards as there are operations (not Operation-cards, for these, as we have just seen, are by no means always as numerous as the operations) are brought into use in every calculation. Indeed, under certain contingencies, a still larger proportion is requisite; such, for example, would probably be the case when the same result has to appear on more than one Variable simultaneously (which is not unfrequently a provision necessary for subsequent purposes in a calculation), and in some other cases which we shall not here specify. We see therefore that a great disproportion exists between the amount of Variable and of Operation-cards requisite for the working of even the simplest calculation. All calculations do not admit, like this one, of the operations of the same nature being performed in groups together. Probably very few do so without exceptions occurring in one or other stage of the progress; and some would not admit it at all. The order in which the operations shall be performed in every particular case, is a very interesting and curious question, on which our space does not permit us fully to enter. In almost every computation a great variety of arrangements for the succession of the processes is possible, and various considerations must influence the selection amongst them for the purposes of a Calculating Engine. One essential object is to choose that arrangement which shall tend to reduce to a minimum the time necessary for completing the calculation. It must be evident how multifarious and how mutually complicated are the considerations which the workings of such an engine involve. There are frequently several distinct sets of effects going on simultaneously; all in a manner independent of each other, and yet to a greater or less degree exercising a mutual influence. To adjust each to every other, and indeed even to perceive and trace them out with perfect correctness and success, entails difficulties whose nature partakes to a certain extent of those involved in every question where conditions are very numerous and inter-complicated; such as for instance the estimation of the mutual relations amongst statistical phænomena, and of those involved in many other classes of facts. A. A. L. DIAGRAM BELONGING TO NOTE D. [Pg 45] Number of operations. Nature of operations. Variables