Sketch of the Analytical Engine Invented by Charles Babbage

rise, are analogous to the preceding. Let us pass on to multiplication. When two numbers to be multiplied are of the same sign, the result is positive; if the signs are different, the product must be negative. In order that the machine may act conformably to this law, we have but to conceive that, on the column containing the product of the two given numbers, the digit which indicates the sign of that product, has been formed by the mutual addition of the two digits that respectively indicated the signs of the two given numbers; it is then obvious that [Pg 18] if the digits of the signs are both even, or both odd, their sum will be an even number, and consequently will express a positive number; but that if, on the contrary, the two digits of the signs are one even and the other odd, their sum will be an odd number, and will consequently express a negative number. In the case of division, instead of adding the digits of the discs, they must be subtracted one from the other, which will produce results analogous to the preceding; that is to say, that if these figures are both even or both uneven, the remainder of this subtraction will be even; and it will be uneven in the contrary case. When I speak of mutually adding or subtracting the numbers expressed by the digits of the signs, I merely mean that one of the sign-discs is made to advance or retrograde a number of divisions equal to that which is expressed by the digit on the other sign-disc. We see, then, from the preceding explanation, that it is possible mechanically to combine the signs of quantities so as to obtain results conformable to those indicated by algebra[9]. The machine is not only capable of executing those numerical calculations which depend on a given algebraical formula, but it is also fitted for analytical calculations in which there are one or several variables to be considered. It must be assumed that the analytical expression to be operated on can be developed according to powers of the variable, or according to determinate functions of this same variable, such as circular functions, for instance; and similarly for the result that is to be attained. If we then suppose that above the columns of the store, we have inscribed the powers or the functions of the variable, arranged according to whatever is the prescribed law of development, the coefficients of these several terms may be respectively placed on the corresponding column below each. In this manner we shall have a representation of an analytical development; and, supposing the position of the several terms composing it to be invariable, the problem will be reduced to that of calculating their coefficients according to the laws demanded by the nature of the question. In order to make this more clear, we shall take the following:[10] very simple example, in which we are [Pg 19] to multiply () by (). We shall begin by writing , , , , above the columns , , , ; then, since from the form of the two functions to be combined, the terms which are to compose the products will be of the following nature, , , , ; these will be inscribed above the columns , , , . The coefficients of , , , being given, they will, by means of the mill, be passed to the columns , , and . Such are the primitive data of the problem. It is now the business of the machine to work out its solution, that is to find the coefficients which are to be inscribed on , , , . To attain this object, the law of formation of these same coefficients being known, the machine will act through the intervention of the cards, in the manner indicated by the following table:— [11] Columns above which are written the functions of the variables. Coefficients. Cards of the operations Cards of the variables. Given. To be formed. Number of the operations. Nature of the operation. Columns on which operations are to be performed. Columns on which are to be inscribed the results of the operations. Indication of change of value on any column submitted to an operation. Results of the operations. " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 1 2 3 4 It will now be perceived that a general application may be made of the principle developed in the preceding example, to every species of process which it may be proposed to effect on series submitted to calculation. It is sufficient that the law of formation of the coefficients be known, and that this law be inscribed on the cards of the machine, which will then of itself execute all the calculations [Pg 20] requisite for arriving at the proposed result. If, for instance, a recurring series were proposed, the law of formation of the coefficients being here uniform, the same operations which must be performed for one of them will be repeated for all the others; there will merely be a change in the locality of the operation, that is it will be performed with different columns. Generally, since every analytical expression is susceptible of being expressed in a series ordered according to certain functions of the variable, we perceive that the machine will include all analytical calculations which can be definitively reduced to the formation of coefficients according to certain laws, and to the distribution of these with respect to the variables. We may deduce the following important consequence from these explanations, viz. that since the cards only indicate the nature of the operations to be performed, and the columns of Variables with which they are to be executed, these cards will themselves possess all the generality of analysis, of which they are in fact merely a translation. We shall now further examine some of the difficulties which the machine must surmount, if its assimilation to analysis is to be complete. There are certain functions which necessarily change in nature when they pass through zero or infinity, or whose values cannot be admitted when they pass these limits. When such cases present themselves, the machine is able, by means of a bell, to give notice that the passage through zero or infinity is taking place, and it then stops until the attendant has again set it in action for whatever process it may next be desired that it shall perform. If this process has been foreseen, then the machine, instead of ringing, will so dispose itself as to present the new cards which have relation to the operation that is to succeed the passage through zero and infinity. These new cards may follow the first, but may only come into play contingently upon one or other of the two circumstances just mentioned taking place. Let us consider a term of the form ; since the cards are but a translation of the analytical formula, their number in this particular case must be the same, whatever be the value of ; that is to say, whatever be the number of multiplications required for elevating to the th power (we are supposing for the moment that is a whole number). Now, since the exponent indicates that is to be multiplied times by itself, and all these operations are of the same nature, it will be sufficient to employ one [Pg 21] single operation-card, viz. that which orders the multiplication. But when is given for the particular case to be calculated, it will be further requisite that the machine limit the number of its multiplications according to the given values. The process may be thus arranged. The three numbers , and will be written on as many distinct columns of the store; we shall designate them , , ; the result will place itself on the column . When the number has been introduced into the machine, a card will order a certain registering-apparatus to mark (), and will at the same time execute the multiplication of by . When this is completed, it will be found that the registering-apparatus has effaced a unit, and that it only marks (); while the machine will now again order the number written on the column to multiply itself with the product written on the column , which will give . Another unit is then effaced from the registering-apparatus, and the same processes are continually repeated until it only marks zero. Thus the number will be found inscribed on , when the machine, pursuing its course of operations, will order the product of by ; and the required calculation will have been completed without there being any necessity that the number of operation-cards used should vary with the value of . If were negative, the cards, instead of ordering the multiplication of by , would order its division; this we can easily conceive, since every number, being inscribed with its respective sign, is consequently capable of reacting on the nature of the operations to be executed. Finally, if were fractional, of the form , an additional column would be used for the inscription of , and the machine would bring into action two sets of processes, one for raising to the power , the other for extracting the th root of the number so obtained. Again, it may be required, for example, to multiply an expression of the form by another , and then to reduce the product to the least number of terms, if any of the indices are equal. The two factors being ordered with respect to , the general result of the multiplication would be . Up to this point the process presents no difficulties; but suppose that we have and , and that we wish to reduce the two middle terms to a single one (. [Pg 22] For this purpose, the cards may order and to be transferred into the mill, and there subtracted one from the other; if the remainder is nothing, as would be the case on the present hypothesis, the mill will order other cards to bring to it the coefficients and , that it may add them together and give them in this state as a coefficient for the single term . This example illustrates how the cards are able to reproduce all the operations which intellect performs in order to attain a determinate result, if these operations are themselves capable of being precisely defined. Let us now examine the following expression:— which we know becomes equal to the ratio of the circumference to the diameter, when is infinite. We may require the machine not only to perform the calculation of this fractional expression, but further to give indication as soon as the value becomes identical with that of the ratio of the circumference to the diameter when is infinite, a case in which the computation would be impossible. Observe that we should thus require of the machine to interpret a result not of itself evident, and that this is not amongst its attributes, since it is no thinking being. Nevertheless, when the of has been foreseen, a card may immediately order the substitution of the value of , ( being the ratio of the circumference to the diameter), without going through the series of calculations indicated. This would merely require that the machine contain a special card, whose office it should be to place the number in a direct and independent manner on the column indicated to it. And here we should introduce the mention of a third species of cards, which may be called cards of numbers. There are certain numbers, such as those expressing the ratio of the circumference to the diameter, the Numbers of Bernoulli, &c., which frequently present themselves in calculations. To avoid the necessity for computing them every time they have to be used, certain cards may be combined specially in order to give these numbers ready made into the mill, whence they afterwards go and place themselves on those columns of the store that are destined for them. Through this means the machine will be susceptible of those simplifications afforded by the use of numerical tables. It [Pg 23] would be equally possible to introduce, by means of these cards, the logarithms of numbers; but perhaps it might not be in this case either the shortest or the most appropriate method; for