Sketch of the Analytical Engine Invented by Charles Babbage

Sketch of the Analytical Engine invented by Charles Babbage Esq. By L. F. MENABREA, Officer of the Military Engineers. [From the Bibliothèque Universelle de Génève, No. 82. October 1812.] [BEFORE submitting to our readers the translation of M. Menabrea’s memoir ‘On the Mathematical Principles of the ANALYTICAL ENGINE’ invented by Mr. Babbage, we shall present to them a list of the printed papers connected with the subject, and also of those relating to the Difference Engine by which it was preceded. For information on Mr. Babbage’s “Difference Engine,” which is but slightly alluded to by M. Menabrea, we refer the reader to the following sources:— 1. Letter to Sir Humphry Davy, Bart., P.R.S., on the Application of Machinery to Calculate and Print Mathematical Tables. By Charles Babbage, Esq., F.R.S. London, July 1822. Re-printed, with a Report of the Council of the Royal Society, by order of the House of Commons, May 1823. 2. On the Application of Machinery to the Calculation of Astronomical and Mathematical Tables. By Charles Babbage, Esq.—Memoirs of the Astronomical Society, vol. I. part 2. London, 1822. 3. Address to the Astronomical Society by Henry Thomas Colebrooke, Esq., F.R.S., President, on presenting the first Gold Medal of the Society to Charles Babbage, Esq., for the invention of the Calculating Engine.—Memoirs of the Astronomical Society. London, 1822. 4. On the Determination of the General Term of a New Class of Infinite Series. By Charles Babbage, Esq.—Transactions of the Cambridge Philosophical Society. 5. On Mr. Babbage’s New Machine for Calculating and Printing Mathematical Tables.—Letter from Francis Baily, Esq., F.R.S., to M. Schumacher. No. 46, Astronomische Nachrichten. Reprinted in the Philosophical Magazine, May 1824. 6. On a Method of expressing by Signs the Action of Machinery. By [Pg 2] Charles Babbage, Esq.—Philosophical Transactions. London, 1826. 7. On Errors common to many Tables of Logarithms. By Charles Babbage, Esq.—Memoirs of the Astronomical Society, London, 1827. 8. Report of the Committee appointed by the Council of the Royal Society to consider the subject referred to in a communication received by them from the Treasury respecting Mr. Babbage’s Calculating Engine, and to report thereon. London, 1829. 9. Economy of Manufactures, chap. XX. 8vo. London, 1832. 10. Article on Babbage’s Calculating Engine.—Edinburgh Review, July 1834. No. 120. vol. LIX. The present state of the Difference Engine, which has always been the property of Government, is as follows:—The drawings are nearly finished, and the mechanical notation of the whole, recording every motion of which it is susceptible, is completed. A part of that Engine, comprising sixteen figures, arranged in three orders of differences, has been put together, and has frequently been used during the last eight years. It performs its work with absolute precision. This portion of the Difference Engine, together with all the drawings, are at present deposited in the Museum of King’s College, London. Of the ANALYTICAL ENGINE, which forms the principal object of the present memoir, we are not aware that any notice has hitherto appeared, except a Letter from the Inventor to M. Quetelet, Secretary to the Royal Academy of Sciences at Brussels, by whom it was communicated to that body. We subjoin a translation of this Letter, which was itself a translation of the original, and was not intended for publication by its author. Royal Academy of Sciences at Brussels. General Meeting of the 7th and 8th of May, 1835. “A Letter from Mr. Babbage announces that he has for six months been engaged in making the drawings of a new calculating machine of far greater power than the first. “‘I am myself astonished,’ says Mr. Babbage, ‘at the power I have been enabled to give to this machine; a year ago I should not have believed this result possible. This machine is intended to contain a hundred variables (or numbers susceptible of changing); each of these numbers [Pg 3] may consist of twenty-five figures, being any numbers whatever, being less than a hundred; if be any given function which can be formed by addition, subtraction, multiplication, division, extraction of roots, or elevation to powers, the machine will calculate its numerical value; it will afterwards substitute this value in the place of , or of any other variable, and will calculate this second function with respect to . It will reduce to tables almost all equations of finite differences. Let us suppose that we have observed a thousand values of , , , , and that we wish to calculate them by the formula , the machine must be set to calculate the formula; the first series of the values of , , , must be adjusted to it; it will then calculate them, print them, and reduce them to zero; lastly, it will ring a bell to give notice that a new set of constants must be inserted. When there exists a relation between any number of successive coefficients of a series, provided it can be expressed as has already been said, the machine will calculate them and make their terms known in succession; and it may afterwards be disposed so as to find the value of the series for all the values of the variable.’ “Mr. Babbage announces, in conclusion, that the greatest difficulties of the invention have already been surmounted, and that the plans will be finished in a few months.” In the Ninth Bridgewater Treatise, Mr. Babbage has employed several arguments deduced from the Analytical Engine, which afford some idea of its powers. See Ninth Bridgewater Treatise, 8vo, second edition. London, 1834. Some of the numerous drawings of the Analytical Engine have been engraved on wooden blocks, and from these (by a mode contrived by Mr. Babbage) various stereotype plates have been taken. They comprise— 1. Plan of the figure wheels for one method of adding numbers. 2. Elevation of the wheels and axis of ditto. 3. Elevation of framing only of ditto. 4. Section of adding wheels and framing together. 5. Section of the adding wheels, sign wheels and framing complete. 6. Impression from the original wood block. [Pg 4] 7. Impressions from a stereotype cast of No. 6, with the letters and signs inserted. Nos. 2, 3, and 5 were stereotypes taken from this. 8. Plan of adding wheels and of long and short pinions, by means of which stepping is accomplished. N.B. This process performs the operation of multiplying or dividing a number by any power of ten. 9. Elevation of long pinions in the position for addition. 10. Elevation of long pinions in the position for stepping. 11. Plan of mechanism for carrying the tens (by anticipation), connected with long pinions. 12. Section of the chain of wires for anticipating carriage. 13. Sections of the elevation of parts of the preceding carriage. All these were executed about five years ago. At a later period (August 1840) Mr. Babbage caused one of his general plans (No. 25) of the whole Analytical Engine to be lithographed at Paris. Although these illustrations have not been published, on account of the time which would be required to describe them, and the rapid succession of improvements made subsequently, yet copies have been freely given to many of Mr. Babbage’s friends, and were in August 1838 presented at Newcastle to the British Association for the Advancement of Science, and in August 1840 to the Institute of France through M. Arago, as well as to the Royal Academy of Turin through M. Plana.—EDITOR.] THOSE labours which belong to the various branches of the mathematical sciences, although on first consideration they seem to be the exclusive province of intellect, may, nevertheless, be divided into two distinct sections; one of which may be called the mechanical, because it is subjected to precise and invariable laws, that are capable of being expressed by means of the operations of matter; while the other, demanding the intervention of reasoning, belongs more specially to the domain of the understanding. This admitted, we may propose to execute, by means of machinery, the mechanical branch of these labours, reserving for pure intellect that which depends on the reasoning faculties. Thus the rigid exactness of those laws which regulate numerical calculations must frequently have suggested the employment of material instruments, either for executing the whole of such [Pg 5] calculations or for abridging them; and thence have arisen several inventions having this object in view, but which have in general but partially attained it. For instance, the much-admired machine of Pascal is now simply an object of curiosity, which, whilst it displays the powerful intellect of its inventor, is yet of little utility in itself. Its powers extended no further than the execution of the four[1] first operations of arithmetic, and indeed were in reality confined to that of the two first, since multiplication and division were the result of a series of additions and subtractions. The chief drawback hitherto on most of such machines is, that they require the continual intervention of a human agent to regulate their movements, and thence arises a source of errors; so that, if their use has not become general for large numerical calculations, it is because they have not in fact resolved the double problem which the question presents, that of correctness in the results, united with economy of time. Struck with similar reflections, Mr. Babbage has devoted some years to the realization of a gigantic idea. He proposed to himself nothing less than the construction of a machine capable of executing not merely arithmetical calculations, but even all those of analysis, if their laws are known. The imagination is at first astounded at the idea of such an undertaking; but the more calm reflection we bestow on it, the less impossible does success appear, and it is felt that it may depend on the discovery of some principle so general, that if applied to machinery, the latter may be capable of mechanically translating the operations which may be indicated to it by algebraical notation. The illustrious inventor having been kind enough to communicate to me some [Pg 6] of his views on this subject during a visit he made at Turin, I have, with his approbation, thrown together the impressions they have left on my mind. But the reader must not expect to find a description of Mr. Babbage’s engine; the comprehension of this would entail studies of much length; and I shall endeavour merely to give an insight into the end proposed, and to develope the principles on which its attainment depends. I must first premise that this engine is entirely different from that of which there is a notice in the ‘Treatise on the Economy of Machinery’ by the same author. But as the latter gave rise[2] to the idea of the engine in question, I consider it will be a useful preliminary briefly to recall what were Mr. Babbage’s first essays, and also the circumstances in which they originated. It is well known that the French government, wishing to promote the extension of the decimal system, had ordered the construction of logarithmical and trigonometrical tables of enormous extent. M. de Prony, who had been entrusted with the direction of this undertaking, divided it into three sections, to each of which were appointed a special class of persons. In the first section the formulæ were so combined as to render them subservient to the purposes of numerical calculation; in the second, these same formulæ were calculated for values of the variable, selected at certain successive distances; and under the third section, comprising about eighty individuals, who were most of them only acquainted with the two first rules of arithmetic, the values which were intermediate to those calculated by the second section were interpolated by means of simple additions and subtractions. An undertaking similar to that just mentioned having been entered upon in England, Mr. Babbage conceived that the operations performed under the third section might be executed by a machine; and this idea he realized by means of mechanism, which has been in part put together, [Pg 7] and to which the name Difference Engine is applicable, on account of the principle upon which its construction is founded. To give some notion of this, it will suffice to consider the series of whole square numbers, 1, 4, 9, 16, 25, 36, 49, 64, &c. By subtracting each of these from the succeeding one, we obtain a