Scientific Memoirs/3/Sketch of the Analytical Engine invented by Charles Babbage, Esq.

Article XXIX.

Sketch of the Analytical Engine invented by Charles Babbage Esq. By L. F. Menabrea, of Turin, Officer of the Military Engineers.

[From the Bibliothèque Universelle de Génève, No. 82. October 1842.]

[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. Reprinted, 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 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 may consist of twenty-five figures, $\scriptstyle {v_{1},v_{2},\ldots \ldots v_{n}}$ being any numbers whatever, $\scriptstyle {n}$ being less than a hundred; if $\scriptstyle {f(v_{1},v_{2},v_{3},..v_{n})}$ 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 $\scriptstyle {v}$, or of any other variable, and will calculate this second function with respect to $\scriptstyle {v}$. It will reduce to tables almost all equations of finite differences. Let us suppose that we have observed a thousand values of $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$, $\scriptstyle {d}$, and that we wish to calculate them by the formula $\scriptstyle {p={\sqrt {\frac {a+b}{cd}}}}$, the machine must be set to calculate the formula; the first series of the values of $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$, $\scriptstyle {d}$ 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.

7. Impressions from a stereotype cast of No. 6, with the letters and signs inserted. Nos. 2, 3, 4 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.]

 

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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 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 œconomy 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 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 Œconomy 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, 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 new series, which we will name the Series of First Differences, consisting of the numbers 3, 5, 7, 9, 11, 13, 15, &c. On subtracting from each of these the preceding one, we obtain the Second Differences, which are all constant and equal to 2. We may represent this succession of operations, and their results, in the following table:—

	A. Column of Square Numbers.	B. First Differ­ences.	C. Second Differ­ences.
	1
		3
	4		2b
		5
a	9		2d
		7
c	16		2
		9
	25		2
		11
	36

From the mode in which the two last 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 difference 2; similarly, if from the square number 9 we would pass to the following one 16, we must add to the former the difference 7, which difference is in other words the preceding difference 5, plus the constant difference 2; or again, which comes to the same thing, to obtain 16 we have only to add together the three numbers 2, 5, 9, placed obliquely in the direction a b. Similarly, we obtain the number 25 by summing up the three numbers placed in the oblique direction d c: commencing by the addition 2 + 7, we have the first difference 9 consecutively to 7; adding 16 to the 9 we have the square 25. 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. 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 B. 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 9. 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 is the square number consecutive to 9. If we now recommence these operations, 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 mechanism.

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 $\scriptstyle {m}$, this same variable be increased by equal degrees; the corresponding values of the polynomial then calculated, and the first, second, third, &c. differences of these be taken (as for the preceding series of squares); the $\scriptstyle {m}$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 $\scriptstyle {(m+1)}$ 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 a series, the mechanism must include an infinite number of dials, which would in fact render the thing impossible. 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 of a certain number of terms of the series, neglecting the rest. 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 the value of the function by different methods, and then respectively to use the results thus obtained, as data whence to deduce, by means of the machine, the other intermediate values. 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 by it.

Such is the nature of the first machine which Mr. Babbage conceived. 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^[3] particular theorem of analysis; and that, in short, its operations cannot be extended so as to embrace the solution of an infinity of other questions included within the domain of mathematical analysis. It was while contemplating the vast field which yet remained to be traversed, that Mr. 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.

Having now explained the state of the question, it is time for me to develope 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: firstly, the numerical 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 $\scriptstyle {(a+bx)~(m+nx)}$, the result will be represented by $\scriptstyle {am+(an+bm)x+bnx^{2}}$, in which expression we must first calculate $\scriptstyle {am}$, $\scriptstyle {an}$, $\scriptstyle {bm}$, $\scriptstyle {bn}$; then take the sum of $\scriptstyle {am+bm}$; and lastly, respectively distribute the coefficients thus obtained, amongst the powers of the variable. 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 œconomy of time; the machine must therefore have the additional requisite of executing by itself all the successive operations required for the solution of a problem proposed to it, when once the primitive numerical data for this same problem have been introduced. 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 to the proposed result, it must exclude all methods of trial and guess-work, and can only admit the direct processes of calculation^[4].

It is necessarily thus; for the machine is not a thinking being, but simply an automaton which acts according to the laws imposed upon it. 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 four first operations of arithmetic, that is addition, subtraction, multiplication and division, can be performed in a direct manner through the intervention of the machine. 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. 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 so on. 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^[5] consisting of discs, which columns we will designate as $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$, $\scriptstyle {\mathbf {V} _{3}}$, $\scriptstyle {\mathbf {V} _{4}}$, &c., we may require, for instance, to divide the number written on the column $\scriptstyle {\mathbf {V} _{1}}$ by that on the column $\scriptstyle {\mathbf {V} _{4}}$, and to obtain the result on the column $\scriptstyle {\mathbf {V} _{7}}$. 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 result is to be represented. 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. 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 of the machine, may be distinguished into two principal classes:

First, that relative to the Operations.

Secondly, that relative to the Variables.

By this latter we mean that which indicates the columns to be operated on. 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^[6] on every disc of the two vertical lines on which the numbers were represented; and it transfers the numbers to the mill. 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 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. 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^[7], used for the manufacture of brocaded stuffs, in the following manner:—

Two species of threads are usually distinguished in woven stuffs; one is the warp or longitudinal thread, the other the woof or transverse thread, which is conveyed by the instrument called the shuttle, and which crosses the longitudinal thread or warp. 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. 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 places. 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 face for every stroke of the shuttle, which face shall then be impelled parallelly to itself against the bundle of lever-arms, the operation of raising the threads will be regularly performed. Thus we see that brocaded tissues may be manufactured with a precision and rapidity formerly difficult to obtain.

Arrangements analogous to those just described have been introduced into the Analytical Engine. It contains two principal species of cards: first, Operation cards, by means of which the parts of the machine are so disposed as to execute any determinate series of operations, such as additions, subtractions, multiplications, and divisions; secondly, cards of the Variables, which indicate to the machine the columns on which the results are to be represented. The cards, when put in motion, successively arrange the various portions of the machine according to the nature of the processes that are to be effected, and the machine at the same time executes these processes by means of the various pieces of mechanism of which it is constituted.

In order more perfectly to conceive the thing, let us select as an example the resolution of two equations of the first degree with two unknown quantities. Let the following be the two equations, in which $\scriptstyle {x}$ and $\scriptstyle {y}$ are the unknown quantities:—

$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {mx+ny=d}\\&\scriptstyle {m'x+n'y=d'.}\end{aligned}}\right.}$

We deduce $\scriptstyle {x={\frac {dn'-d'n}{n'm-nm'}}}$, and for $\scriptstyle {y}$ an analogous expression. Let us continue to represent by $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$, &c. the different columns which contain the numbers, and let us suppose that the first eight columns have been chosen for expressing on them the numbers represented by $\scriptstyle {m}$, $\scriptstyle {n}$, $\scriptstyle {d}$, $\scriptstyle {m'}$, $\scriptstyle {n'}$, $\scriptstyle {d'}$, $\scriptstyle {n}$ and $\scriptstyle {n'}$, which implies that $\scriptstyle {\mathbf {V} _{0}=m}$, $\scriptstyle {\mathbf {V} _{1}=n}$, $\scriptstyle {\mathbf {V} _{2}=d}$, $\scriptstyle {\mathbf {V} _{3}=m'}$, $\scriptstyle {\mathbf {V} _{4}=n'}$, $\scriptstyle {\mathbf {V} _{5}=d'}$, $\scriptstyle {\mathbf {V} _{6}=n}$, $\scriptstyle {\mathbf {V} _{7}=n'}$.

The series of operations commanded by the cards, and the results obtained, may be represented in the following table:—

Number of the operations.	Operation-cards.	Cards of the variables.		Progress of the operations.
	Symbols indicating the nature of the operations.	Columns on which operations are to be performed.	Columns which receive results of operations.
1	$\scriptstyle {\times }$	$\scriptstyle {\mathbf {V} _{2~}\times \mathbf {V} _{4~}=}$	$\scriptstyle {\mathbf {V} _{8~}\ldots \ldots \ldots }$	$\scriptstyle {=dn'}$
2	$\scriptstyle {\times }$	$\scriptstyle {\mathbf {V} _{5~}\times \mathbf {V} _{1~}=}$	$\scriptstyle {\mathbf {V} _{9~}\ldots \ldots \ldots }$	$\scriptstyle {=d'n}$
3	$\scriptstyle {\times }$	$\scriptstyle {\mathbf {V} _{4~}\times \mathbf {V} _{0~}=}$	$\scriptstyle {\mathbf {V} _{10}\ldots \ldots \ldots }$	$\scriptstyle {=n'm}$
4	$\scriptstyle {\times }$	$\scriptstyle {\mathbf {V} _{1~}\times \mathbf {V} _{3~}=}$	$\scriptstyle {\mathbf {V} _{11}\ldots \ldots \ldots }$	$\scriptstyle {=nm'}$
5	$\scriptstyle {-}$	$\scriptstyle {\mathbf {V} _{8~}-\mathbf {V} _{9~}=}$	$\scriptstyle {\mathbf {V} _{12}\ldots \ldots \ldots }$	$\scriptstyle {=dn'-d'n}$
6	$\scriptstyle {-}$	$\scriptstyle {\mathbf {V} _{10}-\mathbf {V} _{11}=}$	$\scriptstyle {\mathbf {V} _{13}\ldots \ldots \ldots }$	$\scriptstyle {=n'm-nm'}$
7	$\scriptstyle {\div }$	$\scriptstyle {{\frac {\mathbf {V} _{12}}{\mathbf {V} _{13}}}\quad =}$	$\scriptstyle {\mathbf {V} _{14}\ldots \ldots \ldots }$	$\scriptstyle {=x={\frac {dn'-d'n}{n'm-nm'}}}$

Since the cards do nothing but indicate in what manner and on what columns the machine shall act, it is clear that we must still, in every particular case, introduce the numerical data for the calculation. Thus, in the example we have selected, we must previously inscribe the numerical values of $\scriptstyle {m}$, $\scriptstyle {n}$, $\scriptstyle {d}$, $\scriptstyle {m'}$, $\scriptstyle {n'}$, $\scriptstyle {d'}$, in the order and on the columns indicated, after which the machine when put in action will give the value of the unknown quantity $\scriptstyle {x}$ for this particular case. To obtain the value of $\scriptstyle {y}$, another series of operations analogous to the preceding must be performed. But we see that they will be only four in number, since the denominator of the expression for $\scriptstyle {y}$, excepting the sign, is the same as that for $\scriptstyle {x}$, and equal to $\scriptstyle {n'm-nm'}$. In the preceding table it will be remarked that the column for operations indicates four successive multiplications, two subtractions, and one division. Therefore, if desired, we need only use three operation cards; to manage which, it is sufficient to introduce into the machine an apparatus which shall, after the first multiplication, for instance, retain the card which relates to this operation, and not allow it to advance so as to be replaced by another one, until after this same operation shall have been four times repeated. In the preceding example we have seen, that to find the value of $\scriptstyle {x}$ we must begin by writing the coefficients $\scriptstyle {m}$, $\scriptstyle {n}$, $\scriptstyle {d}$, $\scriptstyle {m'}$, $\scriptstyle {n'}$, $\scriptstyle {d'}$, upon eight columns, thus repeating $\scriptstyle {n}$ and $\scriptstyle {n'}$ twice. According to the same method, if it were required to calculate $\scriptstyle {y}$ likewise, these coefficients must be written on twelve different columns. But it is possible to simplify this process, and thus to diminish the chances of errors, which chances are greater, the larger the number of the quantities that have to be inscribed previous to setting the machine in action. To understand this simplification, we must remember that every number written on a column must, in order to be arithmetically combined with another number, be effaced from the column on which it is, and transferred to the mill. Thus, in the example we have discussed, we will take the two coefficients $\scriptstyle {m}$ and $\scriptstyle {n'}$, which are each of them to enter into two different products, that is $\scriptstyle {m}$ into $\scriptstyle {mn'}$ and $\scriptstyle {md'}$, $\scriptstyle {n'}$ into $\scriptstyle {mn'}$ and $\scriptstyle {n'd}$. These coefficients will be inscribed on the columns $\scriptstyle {\mathbf {V} _{0}}$ and $\scriptstyle {\mathbf {V} _{4}}$. If we commence the series of operations by the product of $\scriptstyle {m}$ into $\scriptstyle {n'}$, these numbers will be effaced from the columns $\scriptstyle {\mathbf {V} _{0}}$ and $\scriptstyle {\mathbf {V} _{4}}$, that they may be transferred to the mill, which will multiply them into each other, and will then command the machine to represent the result, say on the column $\scriptstyle {\mathbf {V} _{6}}$. But as these numbers are each to be used again in another operation, they must again be inscribed somewhere; therefore, while the mill is working out their product, the machine will inscribe them anew on any two columns that may be indicated to it through the cards; and, as in the actual case, there is no reason why they should not resume their former places, we will suppose them again inscribed on $\scriptstyle {\mathbf {V} _{0}}$ and $\scriptstyle {\mathbf {V} _{4}}$, whence in short they would not finally disappear, to be reproduced no more, until they should have gone through all the combinations in which they might have to be used.

We see, then, that the whole assemblage of operations requisite for resolving the two^[8] above equations of the first degree, may be definitively represented in the following table:—

Columns on which are in­scribed the primitive data.	Number of the operations.	Cards of the operations.		Variable cards.			Statement of results.
		Number of the Operation cards.	Nature of each operation.	Columns acted on by each operation.	Columns that receive the result of each operation.	Indication of change of value on any column.
$\scriptstyle {^{1}\mathbf {V} _{0~}=m}$	1	1	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{0~}\times ~^{1}\mathbf {V} _{4~}=}$	$\scriptstyle {^{1}\mathbf {V} _{6~}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{0~}=~^{1}\mathbf {V} _{0~}}\\&\scriptstyle {^{1}\mathbf {V} _{4~}=~^{1}\mathbf {V} _{4~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{6~}=mn'}$
$\scriptstyle {^{1}\mathbf {V} _{1~}=n}$	2	"	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{3~}\times ~^{1}\mathbf {V} _{1~}=}$	$\scriptstyle {^{1}\mathbf {V} _{7~}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{3~}=~^{1}\mathbf {V} _{3~}}\\&\scriptstyle {^{1}\mathbf {V} _{1~}=~^{1}\mathbf {V} _{1~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{7~}=m'n}$
$\scriptstyle {^{1}\mathbf {V} _{2~}=d}$	3	"	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{2~}\times ~^{1}\mathbf {V} _{4~}=}$	$\scriptstyle {^{1}\mathbf {V} _{8~}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{2~}=~^{1}\mathbf {V} _{2~}}\\&\scriptstyle {^{1}\mathbf {V} _{4~}=~^{0}\mathbf {V} _{4~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{8~}=dn'}$
$\scriptstyle {^{1}\mathbf {V} _{3~}=m'}$	4	"	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{5~}\times ~^{1}\mathbf {V} _{1~}=}$	$\scriptstyle {^{1}\mathbf {V} _{9~}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{5~}=~^{1}V_{5~}}\\&\scriptstyle {^{1}\mathbf {V} _{1~}=~^{1}\mathbf {V} _{1~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{9~}=d'n}$
$\scriptstyle {^{1}\mathbf {V} _{4~}=n'}$	5	"	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{0~}\times ~^{1}\mathbf {V} _{5~}=}$	$\scriptstyle {^{1}\mathbf {V} _{10}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{0~}=~^{0}\mathbf {V} _{0~}}\\&\scriptstyle {^{1}\mathbf {V} _{5~}=~^{0}\mathbf {V} _{5~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{10}=d'm}$
$\scriptstyle {^{1}\mathbf {V} _{5~}=d'}$	6	"	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{2~}\times ~^{1}\mathbf {V} _{3~}=}$	$\scriptstyle {^{1}\mathbf {V} _{11}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{2~}=~^{0}\mathbf {V} _{2~}}\\&\scriptstyle {^{1}\mathbf {V} _{3~}=~^{0}\mathbf {V} _{3~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{11}=dm'}$
	7	2	$\scriptstyle {-}$	$\scriptstyle {^{1}\mathbf {V} _{6~}-~^{1}\mathbf {V} _{7~}=}$	$\scriptstyle {^{1}\mathbf {V} _{12}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{6~}=~^{0}\mathbf {V} _{6~}}\\&\scriptstyle {^{1}\mathbf {V} _{7~}=~^{0}\mathbf {V} _{7~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{12}=mn'-m'n}$
	8	"	$\scriptstyle {-}$	$\scriptstyle {^{1}\mathbf {V} _{8~}-~^{1}\mathbf {V} _{9~}=}$	$\scriptstyle {^{1}\mathbf {V} _{13}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{8~}=~^{0}\mathbf {V} _{8~}}\\&\scriptstyle {^{1}\mathbf {V} _{9~}=~^{0}\mathbf {V} _{9~}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{13}=dn'-d'n}$
	9	"	$\scriptstyle {-}$	$\scriptstyle {^{1}\mathbf {V} _{10}-^{1}\mathbf {V} _{11}=}$	$\scriptstyle {^{1}\mathbf {V} _{14}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{10}=~^{0}\mathbf {V} _{10}}\\&\scriptstyle {^{1}\mathbf {V} _{11}=~^{0}\mathbf {V} _{11}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{14}=d'm-dm'}$
	10	3	$\scriptstyle {\div }$	$\scriptstyle {^{1}\mathbf {V} _{13}\div ~^{1}\mathbf {V} _{12}=}$	$\scriptstyle {^{1}\mathbf {V} _{15}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{13}=~^{0}\mathbf {V} _{13}}\\&\scriptstyle {^{1}\mathbf {V} _{12}=~^{1}\mathbf {V} _{12}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{15}={\frac {dn'-d'n}{mn'-m'n}}=x}$
	11	"	$\scriptstyle {\div }$	$\scriptstyle {^{1}\mathbf {V} _{14}-~^{1}\mathbf {V} _{12}=}$	$\scriptstyle {^{1}\mathbf {V} _{16}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{14}=~^{0}\mathbf {V} _{14}}\\&\scriptstyle {^{1}\mathbf {V} _{12}=~^{0}\mathbf {V} _{12}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{16}={\frac {d'm-dm'}{mn'-m'n}}=y}$
1	2	3	4	5	6	7	8

In order to diminish to the utmost the chances of error in inscribing the numerical data of the problem, they are successively placed on one of the columns of the mill; then, by means of cards arranged for this purpose, these same numbers are caused to arrange themselves on the requisite columns, without the operator having to give his attention to it; so that his undivided mind may be applied to the simple inscription of these same numbers.

According to what has now been explained, we see that the collection of columns of Variables may be regarded as a store of numbers, accumulated there by the mill, and which, obeying the orders transmitted to the machine by means of the cards, pass alternately from the mill to the store, and from the store to the mill, that they may undergo the transformations demanded by the nature of the calculation to be performed.

Hitherto no mention has been made of the signs in the results, and the machine would be far from perfect were it incapable of expressing and combining amongst each other positive and negative quantities. To accomplish this end, there is, above every column, both of the mill and of the store, a disc, similar to the discs of which the columns themselves consist. According as the digit on this disc is even or uneven, the number inscribed on the corresponding column below it will be considered as positive or negative. This granted, we may, in the following manner, conceive how the signs can be algebraically combined in the machine. When a number is to be transferred from the store to the mill, and vice versâ, it will always be transferred with its sign, which will be effected by means of the cards, as has been explained in what precedes. Let any two numbers then, on which we are to operate arithmetically, be placed in the mill with their respective signs. Suppose that we are first to add them together; the operation-cards will command the addition: if the two numbers be of the same sign, one of the two will be entirely effaced from where it was inscribed, and will go to add itself on the column which contains the other number; the machine will, during this operation, be able, by means of a certain apparatus, to prevent any movement in the disc of signs which belongs to the column on which the addition is made, and thus the result will remain with the sign which the two given numbers originally had. When two numbers have two different signs, the addition commanded by the card will be changed into a subtraction through the intervention of mechanisms which are brought into play by this very difference of sign. Since the subtraction can only be effected on the larger of the two numbers, it must be arranged that the disc of signs of the larger number shall not move while the smaller of the two numbers is being effaced from its column and subtracted from the other, whence the result will have the sign of this latter, just as in fact it ought to be. The combinations to which algebraical subtraction give 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 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 to multiply $\scriptstyle {(a+bx^{1})}$ by $\scriptstyle {(A+B\cos ^{1}x)}$. We shall begin by writing $\scriptstyle {x^{0}}$, $\scriptstyle {x^{1}}$, $\scriptstyle {\cos ^{0}x}$, $\scriptstyle {\cos ^{1}x}$, above the columns $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$, $\scriptstyle {\mathbf {V} _{3}}$; 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, $\scriptstyle {x^{0}.\cos ^{0}x}$, $\scriptstyle {x^{0}.\cos ^{1}x}$, $\scriptstyle {x^{1}.\cos ^{0}x}$, $\scriptstyle {x^{1}.\cos ^{1}x}$; these will be inscribed above the columns $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, $\scriptstyle {\mathbf {V} _{7}}$. The coefficients of $\scriptstyle {x^{0}}$, $\scriptstyle {x^{1}}$, $\scriptstyle {\cos ^{0}x}$, $\scriptstyle {\cos ^{1}x}$ being given, they will, by means of the mill, be passed to the columns $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$ and $\scriptstyle {\mathbf {V} _{3}}$. 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 $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, $\scriptstyle {\mathbf {V} _{7}}$. 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 variable.	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 operation.
$\scriptstyle {x^{0}\ldots \ldots \ldots ^{1}\mathbf {V} _{0}}$	$\scriptstyle {a}$	"	"	"	"	"	"	"⁠"
$\scriptstyle {x^{1}\ldots \ldots \ldots ^{1}\mathbf {V} _{1}}$	$\scriptstyle {b}$	"	"	"	"	"	"	"⁠"
$\scriptstyle {\mathrm {Cos} ^{0}x\ldots ^{1}\mathbf {V} _{2}}$	A	"	"	"	"	"	"	"⁠"
$\scriptstyle {\mathrm {Cos} ^{1}x\ldots ^{1}\mathbf {V} _{3}}$	B	"	"	"	"	"	"	"⁠"
$\scriptstyle {x^{0}\cos ^{0}x..^{0}\mathbf {V} _{4}}$	…	$\scriptstyle {a}$A	1	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{0}\times ^{1}\mathbf {V} _{2}=}$	$\scriptstyle {^{1}\mathbf {V} _{4}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{0}=^{1}\mathbf {V} _{0}}\\&\scriptstyle {^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{2}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{4}=a}$A coefficients of $\scriptstyle {x}$
$\scriptstyle {x^{0}\cos ^{1}x..^{0}\mathbf {V} _{5}}$	…	$\scriptstyle {a}$B	2	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{0}\times ^{1}\mathbf {V} _{3}=}$	$\scriptstyle {^{1}\mathbf {V} _{5}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{0}=^{0}\mathbf {V} _{0}}\\&\scriptstyle {^{1}\mathbf {V} _{3}=^{1}\mathbf {V} _{3}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{5}=a}$B … … $\scriptstyle {x}$
$\scriptstyle {x^{1}\cos ^{0}x..^{0}\mathbf {V} _{6}}$	…	$\scriptstyle {b}$A	3	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{1}\times ^{1}\mathbf {V} _{2}=}$	$\scriptstyle {^{1}\mathbf {V} _{6}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\\&\scriptstyle {^{1}\mathbf {V} _{2}=^{0}\mathbf {V} _{2}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{6}=b}$A … … $\scriptstyle {x}$
$\scriptstyle {x^{1}\cos ^{1}x..^{0}\mathbf {V} _{7}}$	…	$\scriptstyle {b}$B	4	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{1}\times ^{1}\mathbf {V} _{3}=}$	$\scriptstyle {^{1}\mathbf {V} _{7}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{1}=^{0}\mathbf {V} _{1}}\\&\scriptstyle {^{1}\mathbf {V} _{3}=^{0}\mathbf {V} _{3}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{7}=b}$B … … $\scriptstyle {x}$

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 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 $\scriptstyle {ab^{n}}$; 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 of $\scriptstyle {n}$; that is to say, whatever be the number of multiplications required for elevating $\scriptstyle {b}$ to the $\scriptstyle {n}$th power (we are supposing for the moment that $\scriptstyle {n}$ is a whole number). Now, since the exponent $\scriptstyle {n}$ indicates that $\scriptstyle {b}$ is to be multiplied $\scriptstyle {n}$ times by itself, and all these operations are of the same nature, it will be sufficient to employ one single operation-card, viz. that which orders the multiplication.

But when $\scriptstyle {n}$ 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 $\scriptstyle {a}$, $\scriptstyle {b}$ and $\scriptstyle {n}$ will be written on as many distinct columns of the store; we shall designate them $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$; the result $\scriptstyle {ab^{n}}$ will place itself on the column $\scriptstyle {\mathbf {V} _{3}}$. When the number $\scriptstyle {n}$ has been introduced into the machine, a card will order a certain registering-apparatus to mark $\scriptstyle {(n-1)}$, and will at the same time execute the multiplication of $\scriptstyle {b}$ by $\scriptstyle {b}$. When this is completed, it will be found that the registering-apparatus has effaced a unit, and that it only marks $\scriptstyle {(n-2)}$; while the machine will now again order the number $\scriptstyle {b}$ written on the column $\scriptstyle {\mathbf {V} _{1}}$ to multiply itself with the product $\scriptstyle {b^{2}}$ written on the column $\scriptstyle {\mathbf {V} _{3}}$, which will give $\scriptstyle {b^{3}}$. Another unit is then effaced from the registering-apparatus, and the same processes are continually repeated until it only marks zero. Thus the number $\scriptstyle {b^{n}}$ will be found inscribed on $\scriptstyle {\mathbf {V} _{3}}$, when the machine, pursuing its course of operations, will order the product of $\scriptstyle {b^{n}}$ by $\scriptstyle {a}$; 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 $\scriptstyle {n}$. If $\scriptstyle {n}$ were negative, the cards, instead of ordering the multiplication of $\scriptstyle {a}$ by $\scriptstyle {b^{n}}$, 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, if $\scriptstyle {n}$ were fractional, of the form $\scriptstyle {\frac {p}{q}}$, an additional column would be used for the inscription of $\scriptstyle {q}$, and the machine would bring into action two sets of processes, one for raising $\scriptstyle {b}$ to the power $\scriptstyle {p}$, the other for extracting the $\scriptstyle {q}$th root of the number so obtained.

Again, it may be required, for example, to multiply an expression of the form $\scriptstyle {ax^{m}+bx^{n}}$ by another $\scriptstyle {{\text{A}}x^{p}+{\text{B}}x^{q}}$, 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 to $\scriptstyle {x}$, the general result of the multiplication would be $\scriptstyle {{\text{A}}ax^{m+p}+{\text{A}}bx^{n+p}+{\text{B}}ax^{m+q}+{\text{B}}bx^{n+q}}$. Up to this point the process presents no difficulties; but suppose that we have $\scriptstyle {m=p}$ and $\scriptstyle {n=q}$, and that we wish to reduce the two middle terms to a single one $\scriptstyle {({\text{A}}b+{\text{B}}a)x^{m+q}}$. For this purpose, the cards may order $\scriptstyle {m+q}$ and $\scriptstyle {n+p}$ 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 $\scriptstyle {{\text{A}}b}$ and $\scriptstyle {{\text{B}}a}$, that it may add them together and give them in this state as a coefficient for the single term $\scriptstyle {x^{n+p}=x^{m+q.}}$

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:—

$\scriptstyle {2.{\frac {2^{2}.4^{2}.6^{2}.8^{2}.10^{2}.\ldots .(2n)^{2}}{1^{2}.3^{2}.5^{2}.7^{2}.9^{2}.\ldots .(2n-1)^{2}.(2n+1)^{2}}}}$,

which we know becomes equal to the ratio of the circumference to the diameter, when $\scriptstyle {n}$ is infinite. We may require the machine not only to perform the calculaiton 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 $\scriptstyle {n}$ 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 $\scriptstyle {\cos }$ of $\scriptstyle {n=\infty }$ has been foreseen, a card may immediately order the substitution of the value of $\scriptstyle {\pi }$ ($\scriptstyle {\pi }$ 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 $\scriptstyle {\pi }$ 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 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 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, in three 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 than twenty 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; for such or such a process which might be very easy for a calculator, may be long and complicated for the engine, and vice 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 can be 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.

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 675), where it is explained that an engine which can effect these four operations can in fact effect every 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 the fundamental starting-point, and the object proposed for attainment by the machine is the subsequent combination of these in every possible variety; whereas in the other case the execution of some one of these four operations, selected at pleasure, is the ultimatum, 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 other ends. Should this distinction not now appear perfectly clear, it will become so on 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 no necessary sequence and connexion need exist between two such inventions, and that they may be 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. In Note 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 unconsciously assumed, and would scarcely suggest any inquiry with reference to it.—Note by Translator.
3. See Note 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. See Note B.
6. Zero is not always substituted when a number is transferred to the mill. This is explained further on in the memoir, and still more fully in Note D.—Note by Translator.
7. See Note C.
8. See Note 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 for 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. See Note E.
11. For an explanation of the upper left-hand indices attached to the $\scriptstyle {\mathbf {V} }$'s in this and in the preceding Table, we must refer the reader to Note D, amongst those appended to the memoir.—Note by Translator.
12. See Note F.
13. See Note 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.