FOOTNOTES:

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 calculationsrequisite 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 formab^n;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 ofn;that is to say, whatever be the number of multiplications required for elevatingbto thenth power (we are supposing for the moment thatnis a whole number). Now, since the exponentnindicates thatbis to be multipliedntimes by itself, and all these operations are of the same nature, it will be sufficient to employ onesingle operation-card, viz. that which orders the multiplication.

But whennis 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 numbersa,bandnwill be written on as many distinct columns of the store; we shall designate themV_0,V_1,V_2;the resultab^2will place itself on the columnV_3.When the numbernhas been introduced into the machine, a card will order a certain registering-apparatus to mark (n-1), and will at the same time execute the multiplication ofbbyb.When this is completed, it will be found that the registering-apparatus has effaced a unit, and that it only marks (n-2); while the machine will now again order the numberbwritten on the columnV_1to multiply itself with the productb^2written on the columnV_3,which will giveb^3.Another unit is then effaced from the registering-apparatus, and the same processes are continually repeated until it only marks zero. Thus the numberb^nwill be found inscribed onV_3,when the machine, pursuing its course of operations, will order the product ofb^nbya;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 ofn.Ifnwere negative, the cards, instead of ordering the multiplication ofabyb^n,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, ifnwere fractional, of the formp/q,an additional column would be used for the inscription ofq,and the machine would bring into action two sets of processes, one for raisingbto the powerp,the other for extracting theqth root of the number so obtained.

Again, it may be required, for example, to multiply an expression of the formax^m-bx^nby anotherAx^p+Bx^q,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 tox,the general result of the multiplication would beAax^(n+p)+Bax^(n+q).Up to this point the process presents no difficulties; but suppose that we havem=pandn=q,and that we wish to reduce the two middle terms to a single one ({A}b + {B}a)x^{m+q}.For this purpose, the cards may orderm+qandn+pto 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 coefficientsaBandBa,that it may add them together and give them in this state as a coefficient for the single termx^{n+q} = x^{m+q}.

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:—2 x 2^2 x 4^2 x 6^2 x 8^2 x 10^2 ... 2n^2/1^2 x 3^2 x 5^2 x 7^2 x 9^2 ... (2n-1^2which we know becomes equal to the ratio of the circumference to the diameter, whennis 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 whennis 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 thecosofn=inftyhas been foreseen, a card may immediately order the substitution of the value ofpi,(pibeing 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 numberpiin 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 calledcards 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. Itwould 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 the machine might be able to perform the same calculations by other more expeditious combinations, founded on the rapidity with which it executes the four first operations of arithmetic. To give an idea of this rapidity, we need only mention that Mr. Babbage believes he can, by his engine, form the product of two numbers, each containing twenty figures, inthree minutes.

Perhaps the immense number of cards required for the solution of any rather complicated problem may appear to be an obstacle; but this does not seem to be the case. There is no limit to the number of cards that can be used. Certain stuffs require for their fabrication not less thantwenty thousand cards, and we may unquestionably far exceed even this quantity[12].

Resuming what we have explained concerning the Analytical Engine, we may conclude that it is based on two principles: the first, consisting in the fact that every arithmetical calculation ultimately depends on four principal operations—addition, subtraction, multiplication, and division; the second, in the possibility of reducing every analytical calculation to that of the coefficients for the several terms of a series. If this last principle be true, all the operations of analysis come within the domain of the engine. To take another point of view: the use of the cards offers a generality equal to that of algebraical formulæ, since such a formula simply indicates the nature and order of the operations requisite for arriving at a certain definite result, and similarly the cards merely command the engine to perform these same operations; but in order that the mechanisms may be able to act to any purpose, the numerical data of the problem must in every particular case be introduced. Thus the same series of cards will serve for all questions whose sameness of nature is such as to require nothing altered excepting the numerical data. In this light the cards are merely a translation of algebraical formulæ, or, to express it better, another form of analytical notation.

Since the engine has a mode of acting peculiar to itself, it will in every particular case be necessary to arrange the series of calculations conformably to the means which the machine possesses; forsuch or such a process which might be very easy for a calculator, may be long and complicated for the engine, andvice versâ.

Considered under the most general point of view, the essential object of the machine being to calculate, according to the laws dictated to it, the values of numerical coefficients which it is then to distribute appropriately on the columns which represent the variables, it follows that the interpretation of formulæ and of results is beyond its province, unless indeed this very interpretation be itself susceptible of expression by means of the symbols which the machine employs. Thus, although it is not itself the being that reflects, it may yet be considered as the being which executes the conceptions of intelligence[13]. The cards receive the impress of these conceptions, and transmit to the various trains of mechanism composing the engine the orders necessary for their action. When once the engine shall have been constructed, the difficulty will be reduced to the making out of the cards; but as these are merely the translation of algebraical formulæ, it will, by means of some simple notations, be easy to consign the execution of them to a workman. Thus the whole intellectual labour will be limited to the preparation of the formulæ, which must be adapted for calculation by the engine.

Now, admitting that such an engine can be constructed, it may be inquired: what will be its utility? To recapitulate; it will afford the following advantages:—First, rigid accuracy. We know that numerical calculations are generally the stumbling-block to the solution of problems, since errors easily creep into them, and it is by no means always easy to detect these errors. Now the engine, by the very nature of its mode of acting, which requires no human intervention during the course of its operations, presents every species of security under the head of correctness; besides, it carries with it its own check; for at the end of every operation it prints off, not only the results, but likewise the numerical data of the question; so that it is easy to verify whether the question has been correctly proposed. Secondly, economy of time: to convince ourselves of this, we need only recollect that the multiplication of two numbers, consisting each of twenty figures, requires at the very utmost three minutes. Likewise, when a long series of identical computations is to be performed, such as those required for the formation of numerical tables, the machine canbe brought into play so as to give several results at the same time, which will greatly abridge the whole amount of the processes. Thirdly, economy of intelligence: a simple arithmetical computation requires to be performed by a person possessing some capacity; and when we pass to more complicated calculations, and wish to use algebraical formulæ in particular cases, knowledge must be possessed which pre-supposes preliminary mathematical studies of some extent. Now the engine, from its capability of performing by itself all these purely material operations, spares intellectual labour, which may be more profitably employed. Thus the engine may be considered as a real manufactory of figures, which will lend its aid to those many useful sciences and arts that depend on numbers. Again, who can foresee the consequences of such an invention? In truth, how many precious observations remain practically barren for the progress of the sciences, because there are not powers sufficient for computing the results! And what discouragement does the perspective of a long and arid computation cast into the mind of a man of genius, who demands time exclusively for meditation, and who beholds it snatched from him by the material routine of operations! Yet it is by the laborious route of analysis that he must reach truth; but he cannot pursue this unless guided by numbers; for without numbers it is not given us to raise the veil which envelopes the mysteries of nature. Thus the idea of constructing an apparatus capable of aiding human weakness in such researches, is a conception which, being realized, would mark a glorious epoch in the history of the sciences. The plans have been arranged for all the various parts, and for all the wheel-work, which compose this immense apparatus, and their action studied; but these have not yet been fully combined together in the drawings[14]and mechanical notation[15]. The confidence which the genius of Mr. Babbage must inspire, affords legitimate ground for hope that this enterprise will be crowned with success; and while we render homage to the intelligence which directs it, let us breathe aspirations for the accomplishment of such an undertaking.

FOOTNOTES:[1]This remark seems to require further comment, since it is in some degree calculated to strike the mind as being at variance with the subsequent passage (page 10), where it is explained thatan engine which can effect these four operationscan in fact effectevery species of calculation. The apparent discrepancy is stronger too in the translation than in the original, owing to its being impossible to render precisely into the English tongue all the niceties of distinction which the French idiom happens to admit of in the phrases used for the two passages we refer to. The explanation lies in this: that in the one case the execution of these four operations is thefundamental starting-point, and the object proposed for attainment by the machine is thesubsequent combination of thesein every possible variety; whereas in the other case the execution of some one of these four operations, selected at pleasure, is theultimatum, the sole and utmost result that can be proposed for attainment by the machine referred to, and which result it cannot any further combine or work upon. The one begins where the otherends. Should this distinction not now appear perfectly clear, it will become soon perusing the rest of the Memoir, and the Notes that are appended to it.—NOTE BY TRANSLATOR.[2]The idea that the one engine is the offspring and has grown out of the other, is an exceedingly natural and plausible supposition, until reflection reminds us that nonecessarysequence and connexion need exist between two such inventions, and that theymaybe wholly independent, M. Menabrea has shared this idea in common with persons who have not his profound and accurate insight into the nature of either engine. InNote A.(see the Notes at the end of the Memoir) it will be found sufficiently explained, however, that this supposition is unfounded. M. Menabrea’s opportunities were by no means such as could be adequate to afford him information on a point like this, which would be naturally and almost unconsciouslyassumed, and would scarcely suggest any inquiry with reference to it.—NOTE BY TRANSLATOR.[3]SeeNote A.[4]This must not be understood in too unqualified a manner. The engine is capable, under certain circumstances, of feeling about to discover which of two or more possible contingencies has occurred, and of then shaping its future course accordingly.—NOTE BY TRANSLATOR.[5]SeeNote B.[6]Zero is notalwayssubstituted when a number is transferred to the mill. This is explained further on in the memoir, and still more fully inNote D.—NOTE BY TRANSLATOR.[7]SeeNote C.[8]SeeNote D.[9]Not having had leisure to discuss with Mr. Babbage the manner of introducing into his machine the combination of algebraical signs, I do not pretend here to expose the method he uses fur this purpose; but I considered that I ought myself to supply the deficiency, conceiving that this paper would have been imperfect if I had omitted to point out one means that might be employed for resolving this essential part of the problem in question.[10]SeeNote E.[11]For an explanation of the upper left-hand indices attached to theV′s in this and in the preceding table, we must refer the reader toNote D., amongst those appended to the memoir.—NOTE BY TRANSLATOR.[12]SeeNote F.[13]SeeNote G.[14]This sentence has been slightly altered in the translation in order to express more exactly the present state of the engine.—NOTE BY TRANSLATOR.[15]The notation here alluded to is a most interesting and important subject, and would have well deserved a separate and detailed Note upon it, amongst those appended to the Memoir. It has, however, been impossible, within the space allotted, even to touch upon so wide a field.—NOTE BY TRANSLATOR.

[1]This remark seems to require further comment, since it is in some degree calculated to strike the mind as being at variance with the subsequent passage (page 10), where it is explained thatan engine which can effect these four operationscan in fact effectevery species of calculation. The apparent discrepancy is stronger too in the translation than in the original, owing to its being impossible to render precisely into the English tongue all the niceties of distinction which the French idiom happens to admit of in the phrases used for the two passages we refer to. The explanation lies in this: that in the one case the execution of these four operations is thefundamental starting-point, and the object proposed for attainment by the machine is thesubsequent combination of thesein every possible variety; whereas in the other case the execution of some one of these four operations, selected at pleasure, is theultimatum, the sole and utmost result that can be proposed for attainment by the machine referred to, and which result it cannot any further combine or work upon. The one begins where the otherends. Should this distinction not now appear perfectly clear, it will become soon perusing the rest of the Memoir, and the Notes that are appended to it.—NOTE BY TRANSLATOR.

[2]The idea that the one engine is the offspring and has grown out of the other, is an exceedingly natural and plausible supposition, until reflection reminds us that nonecessarysequence and connexion need exist between two such inventions, and that theymaybe wholly independent, M. Menabrea has shared this idea in common with persons who have not his profound and accurate insight into the nature of either engine. InNote A.(see the Notes at the end of the Memoir) it will be found sufficiently explained, however, that this supposition is unfounded. M. Menabrea’s opportunities were by no means such as could be adequate to afford him information on a point like this, which would be naturally and almost unconsciouslyassumed, and would scarcely suggest any inquiry with reference to it.—NOTE BY TRANSLATOR.

[3]SeeNote A.

[4]This must not be understood in too unqualified a manner. The engine is capable, under certain circumstances, of feeling about to discover which of two or more possible contingencies has occurred, and of then shaping its future course accordingly.—NOTE BY TRANSLATOR.

[5]SeeNote B.

[6]Zero is notalwayssubstituted when a number is transferred to the mill. This is explained further on in the memoir, and still more fully inNote D.—NOTE BY TRANSLATOR.

[7]SeeNote C.

[8]SeeNote D.

[9]Not having had leisure to discuss with Mr. Babbage the manner of introducing into his machine the combination of algebraical signs, I do not pretend here to expose the method he uses fur this purpose; but I considered that I ought myself to supply the deficiency, conceiving that this paper would have been imperfect if I had omitted to point out one means that might be employed for resolving this essential part of the problem in question.

[10]SeeNote E.

[11]For an explanation of the upper left-hand indices attached to theV′s in this and in the preceding table, we must refer the reader toNote D., amongst those appended to the memoir.—NOTE BY TRANSLATOR.

[12]SeeNote F.

[13]SeeNote G.

[14]This sentence has been slightly altered in the translation in order to express more exactly the present state of the engine.—NOTE BY TRANSLATOR.

[15]The notation here alluded to is a most interesting and important subject, and would have well deserved a separate and detailed Note upon it, amongst those appended to the Memoir. It has, however, been impossible, within the space allotted, even to touch upon so wide a field.—NOTE BY TRANSLATOR.

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