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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 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.

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 very simple example, in which we are to multiply (a + bx1) by (A + B cos1 x).

We shall begin by writing x0 , x1, cos0 x, cos1 x, above the columns V0, V1, V2, V3; 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, x0·cos0 x, x0·cos1 x, x1·cos0 x, x1·cos1 x, these will be inscribed above the columns V4, V5, V6, V7.

The coefficients of x0, x1, cos0 x, cos1 x being given, they will, by means of the mill, be passed to the columns V0, V1, V2 and V3.

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 V4, V5, V6, V7.

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:—

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Read footnote in diagram.

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 abn; 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 n; that is to say, whatever be the number of multiplications required for elevating b to the nth power (we are supposing for the moment that n is a whole number).

Now, since the exponent n indicates that b is to be multiplied 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 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 a, b and n will be written on as many distinct columns of the store; we shall designate them V0, V1, V2; the result abn will place itself on the column V3.

When the number n has 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 of b by b. 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 number b written on the column V1 to multiply itself with the product b2 written on the column V3, which will give b3.

Another unit is then effaced from the registering-apparatus, and the same processes are continually repeated until it only marks zero.

Thus the number bn will be found inscribed on V3, when the machine, pursuing its course of operations, will order the product of bn by 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 n.

If n were negative, the cards, instead of ordering the multiplication of a by bn, 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 n were fractional, of the form p/q, an additional column would be used for the inscription of q, and the machine would bring into action two sets of processes, one for raising b to the power p, the other for extracting the qth root of the number so obtained.

Again, it may be required, for example, to multiply an expression of the form axm+bxn by another Axp+Bxq, 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 x, the general result of the multiplication would be Aaxm+p+Abxn+p+Baxm+q+Bbxn+q.

Up to this point the process presents no difficulties; but suppose that we have m=p and n=q, and that we wish to reduce the two middle terms to a single one (Ab+Ba)xm+q.

For this purpose, the cards may order m+q and 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

Ab and Ba, that it may add them together and give them in this state as a coefficient for the single term xn+p=xm+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\cdot\frac{2^2\cdot 4^2\cdot 6^2\cdot 8^2\cdot 10^2\ldots (2n)^2}{1^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2\ldots (2n-1)^2\cdot (2n+1)^2}

which we know becomes equal to the ratio of the circumference to the diameter, when n is infinite.

We may require the machine not only to perform the calculation of this fractional expression, but further to give indication as soon as the value becomes identical with that of the ratio of the circumference to the diameter when 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 cos of n=1/0 has been foreseen, a card may immediately order the substitution of the value of π (π 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 π 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 first four 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.

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.

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 presupposes 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 and mechanical notation.

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.

Portrait of Ada Augusta, Countess of Lovelace