Passages from the Life of a Philosopher

whether whole numbers or decimals, are arrived at. 7 g. The root thus found must now be used to reduce the original equation to one dimension lower. {133} 8 h. This new equation of one dimension lower must now be treated by sections 3, 4, 5, 6, and 7, until the new root is found. 9 i. The repetition of sections 7 and 8 must go on until all the roots have been found. Now it will be observed that Professor Mosotti was quite ready to admit at once that each of these different processes could be performed by the Analytical Machine through the medium of properly-arranged sets of Jacquard cards. His real difficulty consisted in teaching the engine to know when to change from one set of cards to another, and back again repeatedly, at intervals not known to the person who gave the orders. The dimensions of the algebraic equation being known, the number of arithmetical processes necessary for Sturm’s theorem is consequently known. A set of operation cards can therefore be prepared. These must be accompanied by a corresponding set of variable cards, which will represent the columns in the store, on which the several coefficients of the given equation, and the various combinations required amongst them, are to be placed. The next stage is to find a number greater than the greatest root of the given equation. There are various courses for arriving at such a number. Any one of these being selected, another set of operation and variable cards can be prepared to execute this operation. Now, as this second process invariably follows the first, the second set of cards may be attached to the first set, and the engine will pass on from the first to the second process, and again from the second to the third process. {134} But here a difficulty arises: successive powers of ten are to be substituted for x in the equation, until a certain event happens. A set of cards may be provided to make the substitution of the highest power of ten, and similarly for the others; but on the occurrence of a certain event, namely, the change of a sign from + to −, this stage of the calculation is to terminate. Now at a very early period of the inquiry I had found it necessary to teach the engine to know when any numbers it might be computing passed through zero or infinity. The passage through zero can be easily ascertained, thus: Let the continually-decreasing number which is being computed be placed upon a column of wheels in connection with a carrying apparatus. After each process this number will be diminished, until at last a number is subtracted from it which is greater than the number expressed on those wheels. Thus let it be 00000,00000,00000,00423 Subtract 00000,00000,00000,00511 99999,99999,99999,99912 Now in every case of a carriage becoming due, a certain lever is transferred from one position to another in the cage next above it. Consequently in the highest cage of all (say the fiftieth in the Analytical Engine), an arm will be moved or not moved accordingly as the carriages do or do not run up beyond the highest wheel. This arm can, of course, make any change which has previously been decided upon. In the instance we have been considering it would order the cards to be turned on to the next set. If we wish to find when any number, which is increasing, {135} exceeds in the number of its digits the number of wheels on the columns of the machine, the same carrying arm can be employed. Hence any directions may be given which the circumstances require. It will be remarked that this does not actually prove, even in the Analytical Engine of fifty figures, that the number computed has passed through infinity; but only that it has become greater than any number of fifty places of figures. There are, however, methods by which any machine made for a given number of figures may be made to compute the same formulæ with double or any multiple of its original number. But the nature of this work prevents me from explaining that method. It may here be remarked that in the process, the cards employed to make the substitutions of the powers of ten are operation cards. They are, therefore, quite independent of the numerical values substituted. Hence the same set of operation cards which order the substitutions 1 × 10n will, if backed, order the substitution of 2 × 10n, &c. We may, therefore, avail ourselves of mechanism for backing these cards, and call it into action whenever the circumstances themselves require it. The explanation of M. Mosotti’s difficulty is this:—Mechanical means have been provided for backing or advancing the operation cards to any extent. There exist means of expressing the conditions under which these various processes are required to be called into play. It is not even necessary that two courses only should be possible. Any number of courses may be possible at the same time; and the choice of each may depend upon any number of conditions. 〈GENERAL MENABREA’S DESCRIPTION.〉 It was during these meetings that my highly valued friend, M. Menabrea, collected the materials for that lucid and {136} admirable description which he sub­se­quent­ly published in the Bibli. Univ. de Genève, t. xli. Oct. 1842. The elementary principles on which the Analytical Engine rests were thus in the first instance brought before the public by General Menabrea. 〈THE COUNTESS OF LOVELACE’S NOTES.〉 Some time after the appearance of his memoir on the subject in the “Bibliothèque Universelle de Genève,” the late Countess of Lovelace25 informed me that she had translated the memoir of Menabrea. I asked why she had not herself written an original paper on a subject with which she was so intimately acquainted? To this Lady Lovelace replied that the thought had not occurred to her. I then suggested that she should add some notes to Menabrea’s memoir; an idea which was immediately adopted. 25 Ada Augusta, Countess of Lovelace, only child of the Poet Byron. We discussed together the various illustrations that might be introduced: I suggested several, but the selection was entirely her own. So also was the algebraic working out of the different problems, except, indeed, that relating to the numbers of Bernouilli, which I had offered to do to save Lady Lovelace the trouble. This she sent back to me for an amendment, having detected a grave mistake which I had made in the process. The notes of the Countess of Lovelace extend to about three times the length of the original memoir. Their author has entered fully into almost all the very difficult and abstract questions connected with the subject. These two memoirs taken together furnish, to those who are capable of understanding the reasoning, a complete demonstration—That the whole of the developments and operations of analysis are now capable of being executed by machinery. 〈VARIOUS APPLICATIONS.〉 There are various methods by which these developments {137} are arrived at:—1. By the aid of the Differential and Integral Calculus. 2. By the Combinatorial Analysis of Hindenburg. 3. By the Calculus of Derivations of Arbogast. Each of these systems professes to expand any function according to any laws. Theoretically each method may be admitted to be perfect; but practically the time and attention required are, in the greater number of cases, more than the human mind is able to bestow. Consequently, upon several highly interesting questions relative to the Lunar theory, some of the ablest and most indefatigable of existing analysts are at variance. The Analytical Engine is capable of executing the laws prescribed by each of these methods. At one period I examined the Combinatorial Analysis, and also took some pains to ascertain from several of my German friends, who had had far more experience of it than myself, whether it could be used with greater facility than the Differential system. They seemed to think that it was more readily applicable to all the usual wants of analysis. I have myself worked with the system of Arbogast, and if I were to decide from my own limited use of the three methods, I should, for the purposes of the Analytical Engine, prefer the Calcul des Derivations. As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise—By what course of calculation can these results be arrived at by the machine in the shortest time? In the drawings I have prepared I proposed to have a thousand variables, upon each of which any number not having more than fifty figures can be placed. This machine would multiply 50 figures by other 50, and print the product {138} of 100 figures. Or it would divide any number having 100 figures by any other of 50 figures, and print the quotient of 50 figures. Allowing but a moderate velocity for the machine, the time occupied by either of these operations would be about one minute. The whole of the numerical constants throughout the works of Laplace, Plana, Le Verrier, Hansen, and other eminent men whose indefatigable labours have brought astronomy to its present advanced state, might easily be recomputed. They are but the numerical coefficients of the various terms of functions developed according to certain series. In all cases in which these numerical constants can be calculated by more than one method, it might be desirable to compute them by several processes until frequent practice shall have confirmed our belief in the infallibility of mechanism. 〈ERRORS OF TABLES.〉 The great importance of having accurate Tables is admitted by all who understand their uses; but the multitude of errors really occurring is comparatively little known. Dr. Lardner, in the “Edinburgh Review,” has made some very instructive remarks on this subject. I shall mention two within my own experience: these are selected because they occurred in works where neither care nor expense were spared on the part of the Government to insure perfect accuracy. It is, however, but just to the eminent men who presided over the preparation of these works for the press to observe, that the real fault lay not in them but in the nature of things. In 1828 I lent the Government an original MS. of the table of Logarithmic Sines, Cosines, &c., computed to every second of the quadrant, in order that they might have it compared with Taylor’s Logarithms, 4to., 1792, of which they possessed a considerable number of copies. Nineteen {139} errors were thus detected, and a list of these errata was published in the Nautical Almanac for 1832: these may be called Nineteen errata of the first order . . 1832. An error being detected in one of these errata, in the following Nautical Almanac we find an Erratum of the errata in N. Alm. 1832 . . 1833. But in this very erratum of the second order a new mistake was introduced larger than any of the original mistakes. In the year next following there ought to have been found Erratum in the erratum of the errata in N. Alm. 1832 . . 1834. In the “Tables de la Lune,” by M. P. A. Hansen, 4to, 1857, published at the expense of the English Government, under the direction of the Astronomer Royal, is to be found a list of errata amounting to 155. In the 21st of these original errata there have been found three mistakes. These are duly noted in a newly-printed list of errata discovered during computations made with them in the “Nautical Almanac;” so that we now have the errata of an erratum of the original work. This list of errata from the office of the “Nautical Almanac” is larger than the original list. The total number of errors at present (1862) discovered in Hansen’s “Tables of the Moon” amounts to above three hundred and fifty. In making these remarks I have no intention of imputing the slightest blame to the Astronomer Royal, who, like other men, cannot avoid submitting to inevitable fate. The only circumstance which is really extraordinary is that, when it was dem­on­strated that all tables are capable of being computed by machinery, and even when