The process of raising the threads is thus resolved into that of moving these various lever-arms in the requisite order. To effect this, a rectangular sheet of pasteboard is taken, somewhat larger in size than a section of the bundle of lever-arms. If this sheet be applied to the base of the bundle, and an advancing motion be then communicated to the pasteboard, this latter will move with it all the rods of the bundle, and consequently the threads that are connected with each of them. But if the pasteboard, instead of being plain, were pierced with holes corresponding to the extremities of the levers which meet it, then, since each of the levers would pass through the pasteboard during the motion of the latter, they would all remain in their. We thus see that it is easy so to determine the position of the holes in the pasteboard, that, at any given moment, there shall be a certain number of levers, and consequently of parcels of threads, raised, while the rest remain where they were. Supposing this process is successively repeated according to a law indicated by the pattern to be executed, we perceive that this pattern may be reproduced on the stuff. For this purpose we need merely compose a series of cards according to the law required, and arrange them in suitable order one after the other; then, by causing them to pass over a polygonal beam which is so connected as to turn a new.
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After the preceding explanations, we may perceive that all fractional and irrational results will be represented in decimal fractions. Supposing each column to have forty discs, this extension will be sufficient for all degrees of approximation generally required. It will now be inquired how the machine can of itself, and without having recourse to the hand of man, assume the successive dispositions suited to the operations. The solution of this problem has been taken from Jacquard's apparatus, used for the manufacture of brocaded stuffs, in the following manner:— Two species of threads are usually distinguished in woven punishment stuffs; one is the warp or longitudinal thread, the other the woof or transverse. When a brocaded stuff is required, it is necessary in turn to prevent certain threads from crossing the woof, and this according to a succession which is determined by the nature of the design that is to be reproduced. Formerly this process was lengthy and difficult, and it was requisite that the workman, by attending to the design which he was to copy, should himself regulate the movements the threads were to take. Thence arose the high price of this description of stuffs, especially if threads of various colours entered into the fabric. To simplify this manufacture, jacquard devised the plan of connecting each group of threads that were to act together, with a distinct lever belonging exclusively to that group. All these levers terminate in rods, which are united together in one bundle, having usually the form of a parallelopiped with a rectangular base. The rods are cylindrical, and are separated from each other by small intervals.
If, on the contrary, a series of operations of the same nature is to be gone through, then the first of the original arrangements will remain, and the second alone must be altered Therefore, the arrangements that may be communicated to the various parts. Secondly, that relative to the variables. By this latter we mean that which indicates the columns to be operated. As for the operations themselves, they are executed by a special apparatus, which is designated by the name of mill, and which itself contains a certain number of columns, similar to those of the variables. When two numbers are to be combined together, the machine commences by effacing them from the columns where they are written, that is, it places zero on every disc of the two vertical lines on which the numbers were represented; and it transfers the numbers. There, the apparatus having been disposed suitably for the required operation, this latter is effected, and, when completed, the result itself is transferred to the column of Variables which shall have entry been indicated. Thus the mill is that portion of the machine which works, and the columns of Variables constitute that where the results are represented and arranged.
To conceive how the machine can now go through its functions according to the laws laid down, we will begin by giving an idea of the manner in which it materially represents numbers. Let us conceive a pile or vertical column consisting of an indefinite number of circular discs, all pierced through their centres by a common axis, around which each of them can take an independent rotatory movement. If round the edge of each of these discs are written the ten figures which constitute our numerical alphabet, we may then, by arranging a series of these figures in the same vertical line, express in this manner any number whatever. It is sufficient for this purpose that the first disc represent units, the second tens, the third hundreds, and. When two numbers have been thus written on two distinct columns, we may propose to combine them arithmetically with each other, and to obtain the result on a third column. In general, if we have a series of columns consisting of discs, which columns we will designate as V0, V1, V2, V3, V4,., we may require, for instance, to divide the number written on the column V1 by that on the column V4, and. To effect this operation, we must impart to the machine two distinct arrangements; through the first it is prepared for executing a division, and through the second the columns it is to operate on are indicated to it, and also the column on which the. If this division is to be followed, for example, by the addition of two numbers taken on other columns, the two original arrangements of the machine must be simultaneously altered.
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Having now explained the the state of the question, it is time for me to develop the principle on which is based the construction of this latter machine. When analysis is employed for the solution of any problem, there are usually two classes of operations to execute: first, the numerical want calculation of the various coefficients; and secondly, their distribution in relation to the quantities affected by them. If, for example, we have to obtain the product of two binomials ( a bx ) ( m nx the result will be represented by am ( an bm ) x bnx 2, in which expression we must first calculate am, an, bm, bn ;. In order to reproduce these operations by means of a machine, the latter must therefore possess two distinct sets of powers: first, that of executing numerical calculations; secondly, that of rightly distributing the values so obtained. But if human intervention were necessary for directing each of these partial operations, nothing would be gained under the heads of correctness and economy of time; the machine must therefore have the additional requisite of executing by itself all the successive operations required for the. Therefore, since, from the moment that the nature of the calculation to be executed or of the problem to be resolved have been indicated to it, the machine is, by its own intrinsic power, of itself to go through all the intermediate operations which lead.
It is necessarily thus; for the machine is not a thinking being, but simply an automaton which acts according to the laws imposed upon. This being fundamental, one of the earliest researches its author had to undertake, was that of finding means for effecting the division of one number by another without using the method of guessing indicated by the usual rules of arithmetic. The difficulties of effecting this combination were far from being among the least; but upon it depended the success of every other. Under the impossibility of my here explaining the process through which this end is attained, we must limit ourselves to admitting that the first four operations of arithmetic, that is addition, subtraction, multiplication and division, can be performed in a direct manner through the intervention. This granted, the machine is thence capable of performing every species of numerical calculation, for all such calculations ultimately resolve themselves into the four operations we have just named.
Differences of these be taken (as for the preceding series of squares the m th differences will all be equal to each other. So that, in order to reproduce the series of values of the polynomial by means of a machine analogous to the one above described, it is sufficient that there be ( m 1) dials, having the mutual relations we have indicated. As the differences may be either positive or negative, the machine will have a contrivance for either advancing or retrograding each needle, according as the number to be algebraically added may have the sign plus or minus. If from a polynomial we pass to a series having an infinite number of terms, arranged according to the ascending powers of the variable, it would at first appear, that in order to apply the machine to the calculation of the function represented by such. But in many cases the difficulty will disappear, if we observe that for a great number of functions the series which represent them may be rendered convergent; so that, according to the degree of approximation desired, we may limit ourselves to the calculation. By this method the question is reduced to the primitive case of a finite polynomial.
It is thus that we can calculate the succession of the logarithms of numbers. But since, in this particular instance, the terms which had been originally neglected receive increments in a ratio so continually increasing for equal increments of the variable, that the degree of approximation required would ultimately be affected, it is necessary, at certain intervals, to calculate. We see that the machine here performs the office of the third section of calculators mentioned in describing the tables computed by order of the French government, and that the end originally proposed is thus fulfilled. Such is the nature of the first machine which. We see that its use is confined to cases where the numbers required are such as can be obtained by means of simple additions or subtractions; that the machine is, so to speak, merely the expression of one particular theorem of analysis; and that,. It was while contemplating the vast field which yet remained to be traversed, that. Babbage, renouncing his original essays, conceived the plan of another system of mechanism whose operations should themselves possess all the generality of algebraical notation, and which, on this account, he denominates the Analytical Engine.
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For each stroke of the registering hammer of the dial c, the needle b shall advance one division; similarly, the needle a shall advance one division for every stroke of the registering hammer of the dial. Such is the general disposition of the mechanism. This being understood, let us, at the beginning of the series of operations we wish to execute, place the needle c on the division 2, the needle b on the division 5, and the needle a on the division. Let us allow the hammer of the dial C to strike; it will strike twice, and at the same time the needle b will pass over two divisions. The latter will then indicate the number 7, which succeeds the number 5 in the column of first differences. If we now permit the hammer of the dial B to strike in its turn, it will strike seven times, during which the needle a will advance seven divisions; these added to the nine already marked by it will give the number 16, which. If we now recommence these operations, paper beginning with the needle c, which is always to be left on the division 2, we shall perceive that by repeating them indefinitely, we may successively reproduce the series of whole square numbers by means of a very simple. The theorem on which is based the construction of the machine we have just been describing, is a particular case of the following more general theorem: that if in any polynomial whatever, the highest power of whose variable is m, this same variable be increased.first
To give some notion of this, it will suffice to consider the series of whole square numbers, inspirational 1, 4, 9, 16, 25, 36, 49, 64,. By subtracting each of these from the succeeding one, we obtain a new series, which we will name the series of First Differences, consisting of the numbers 3, 5, 7, 9, 11, 13, 15,. On subtracting from each of these the preceding one, we obtain the second Differences, which are all constant and equal. We may represent this succession of operations, and their results, in the following table. From the mode in which the last two columns b and C have been formed, it is easy to see, that if, for instance, we desire to pass from the number 5 to the succeeding one 7, we must add to the former the constant. Similarly, we obtain the number 25 by summing up the three numbers placed in the oblique direction dc : commencing by the addition 27, we have the first difference 9 consecutively to 7; adding 16 to the 9 we have the square. We see then that the three numbers 2, 5, 9 being given, the whole series of successive square numbers, and that of their first differences likewise may be obtained by means of simple additions. Now, to conceive how these operations may be reproduced by a machine, suppose the latter to have three dials, designated as a, b, c, on each of which are traced, say a thousand divisions, by way of example, over which a needle shall pass. The two dials, c, b, shall have in addition a registering hammer, which is to give a number of strokes equal to that of the divisions indicated by the needle.
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 to the idea of the engine in question, i consider it will be a useful preliminary briefly to recall what were. 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. De Prony, who had been entrusted with the direction of this undertaking, divided it into three sections, to each of which was 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. An undertaking similar to that just mentioned having been entered upon in England,. 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, and to which the name difference Engine is applicable, on account of the.
Its powers extended no thesis further than the execution of the first four operations of arithmetic, and indeed were in reality confined to that of the first two, 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,. Struck with similar reflections,. 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. The illustrious inventor having been kind enough to communicate to me some 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.
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Sketch of The Analytical Engine,. Menabrea of Turin, Officer of the military Engineers from the, bibliothèque universelle de genève, october, 1842,. With notes upon the memoir by the Translator. Ada augusta, countess of lovelace, 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. 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 paper frequently have suggested the employment of material instruments, either for executing the whole of such calculations or for abridging them; and thence have arisen several inventions having this object in view, but which. 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.