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The other variety has provisions which cause zero to be substituted on the Variable, for the number given off.

These two varieties are distinguished, when needful, by the respective appellations of the Retaining Supply-cards and the Zero Supply-cards.

We see that the primary office (see Note B.) of both these varieties of cards is the same; they only differ in their secondary office.

Every Variable thus has belonging to it one class of Receiving Variable-cards and two classes of Supplying Variable-cards.

It is plain however that only the one or the other of these two latter classes can be used by any one Variable for one operation; never both simultaneously, their respective functions being mutually incompatible.

It should be understood that the Variable-cards are not placed in immediate contiguity with the columns.

Each card is connected by means of wires with the column it is intended to act upon.

Our diagram ought in reality to be placed side by side with M. Menabrea's corresponding table, so as to be compared with it, line for line belonging to each operation.

But it was unfortunately inconvenient to print them in this desirable form.

The diagram is, in the main, merely another manner of indicating the various relations denoted in M. Menabrea's table.

Each mode has some advantages and some disadvantages.

Combined, they form a complete and accurate method of registering every step and sequence in all calculations performed by the engine.

No notice has yet been taken of the upper indices which are added to the left of each V in the diagram; an addition which we have also taken the liberty of making to the V's in M. Menabrea's tables 3 and 4, since it does not alter anything therein represented by him, but merely adds something to the previous indications of those tables.

The lower indices are obviously indices of locality only, and are wholly independent of the operations performed or of the results obtained, their value continuing unchanged during the performance of calculations.

The upper indices, however, are of a different nature.

Their office is to indicate any alteration in the value which a Variable represents; and they are of course liable to changes during the processes of a calculation.

Whenever a Variable has only zeros upon it, it is called 0V; the moment a value appears on it (whether that value be placed there arbitrarily, or appears in the natural course of a calculation), it becomes 1V.

If this value gives place to another value, the Variable becomes 2V, and so forth.

Whenever a value again gives place to zero, the Variable again becomes 0V, even if it have been nV the moment before.

If a value then again be substituted, the Variable becomes n+1V (as it would have done if it had not passed through the intermediate 0V); &c. &c. Just before any calculation is commenced, and after the data have been given, and everything adjusted and prepared for setting the mechanism in action, the upper indices of the Variables for data are all unity, and those for the Working and Result-variables are all zero.

In this state the diagram represents them.

There are several advantages in having a set of indices of this nature; but these advantages are perhaps hardly of a kind to be immediately perceived, unless by a mind somewhat accustomed to trace the successive steps by means of which the engine accomplishes its purposes.

We have only space to mention in a general way, that the whole notation of the tables is made more consistent by these indices, for they are able to mark a difference in certain cases, where there would otherwise be an apparent identity confusing in its tendency.

In such a case as Vn=Vp+Vn there is more clearness and more consistency with the usual laws of algebraical notation, in being able to write m+1Vn=qVp+mVn.

It is also obvious that the indices furnish a powerful means of tracing back the derivation of any result; and of registering various circumstances concerning that series of successive substitutions, of which every result is in fact merely the final consequence; circumstances that may in certain cases involve relations which it is important to observe, either for purely analytical reasons, or for practically adapting the workings of the engine to their occurrence.

The series of substitutions which lead to the equations of the diagram are as follow:—

Read footnote.

There are three successive substitutions for each of these equations.

The formulæ (2.), (3.) and (4.) are implicitly contained in (1.), which latter we may consider as being in fact the condensed expression of any of the former.

It will be observed that every succeeding substitution must contain twice as many V's as its predecessor.

So that if a problem require n substitutions, the successive series of numbers for the V's in the whole of them will be 2, 4, 8, 16…2n.

The substitutions in the preceding equations happen to be of little value towards illustrating the power and uses of the upper indices, for, owing to the nature of these particular equations, the indices are all unity throughout.

We wish we had space to enter more fully into the relations which these indices would in many cases enable us to trace.

M. Menabrea incloses the three centre columns of his table under the general title Variable-cards.

The V's however in reality all represent the actual Variable-columns of the engine, and not the cards that belong to them.

Still the title is a very just one, since it is through the special action of certain Variable-cards (when combined with the more generalized agency of the Operation-cards) that every one of the particular relations he has indicated under that title is brought about.

Suppose we wish to ascertain how often any one quantity, or combination of quantities, is brought into use during a calculation.

We easily ascertain this, from the inspection of any vertical column or columns of the diagram in which that quantity may appear.

Thus, in the present case, we see that all the data, and all the intermediate results likewise, are used twice, excepting (mn' − m'n), which is used three times.

The order in which it is possible to perform the operations for the present example, enables us to effect all the eleven operations of which it consists with only three Operation cards; because the problem is of such a nature that it admits of each class of operations being performed in a group together; all the multiplications one after another, all the subtractions one after another, &c.

The operations are {6(×), 3(-), 2(÷)}.

Since the very definition of an operation implies that there must be two numbers to act upon, there are of course two

Supplying Variable-cards necessarily brought into action for every operation, in order to furnish the two proper numbers.

(See Note B.)

Also, since every operation must produce a result, which must be placed somewhere, each operation entails the action of a Receiving Variable-card, to indicate the proper locality for the result.

Therefore, at least three times as many Variable-cards as there are operations (not Operation-cards, for these, as we have just seen, are by no means always as numerous as the operations) are brought into use in every calculation.

Indeed, under certain contingencies, a still larger proportion is requisite; such, for example, would probably be the case when the same result has to appear on more than one Variable simultaneously (which is not unfrequently a provision necessary for subsequent purposes in a calculation), and in some other cases which we shall not here specify.

We see therefore that a great disproportion exists between the amount of Variable and of Operation-cards requisite for the working of even the simplest calculation.

All calculations do not admit, like this one, of the operations of the same nature being performed in groups together.

Probably very few do so without exceptions occurring in one or other stage of the progress; and some would not admit it at all.

The order in which the operations shall be performed in every particular case is a very interesting and curious question, on which our space does not permit us fully to enter.

In almost every computation a great variety of arrangements for the succession of the processes is possible, and various considerations must influence the selection amongst them for the purposes of a Calculating Engine.

One essential object is to choose that arrangement which shall tend to reduce to a minimum the time necessary for completing the calculation.

It must be evident how multifarious and how mutually complicated are the considerations which the working of such an engine involve.

There are frequently several distinct sets of effects going on simultaneously; all in a manner independent of each other, and yet to a greater or less degree exercising a mutual influence.

To adjust each to every other, and indeed even to perceive and trace them out with perfect correctness and success, entails difficulties whose nature partakes to a certain extent of those involved in every question where conditions are very numerous and inter-complicated; such as for instance the estimation of the mutual relations amongst statistical phænomena, and of those involved in many other classes of facts.

A. A. L.

Note E

This example has evidently been chosen on account of its brevity and simplicity, with a view merely to explain the manner in which the engine would proceed in the case of an analytical calculation containing variables, rather than to illustrate the extent of its powers to solve cases of a difficult and complex nature.

The equations in first example in the Memoir are in fact a more complicated problem than the present one.

We have not subjoined any diagram of its development for this new example, as we did for the former one, because this is unnecessary after the full application already made of those diagrams to the illustration of M. Menabrea's excellent tables.

It may be remarked that a slight discrepancy exists between the formulæ

(a + bx1)

(A + B cos1 x)

given in the Memoir as the data for calculation, and the results of the calculation as developed in the last division of the table which accompanies it.

To agree perfectly with this latter, the data should have been given as

(ax0 + bx1)

(A cos0 x + B cos1 x)

The following is a more complicated example of the manner in which the engine would compute a trigonometrical function containing variables.

To multiply

A+A1cos θ + A2cos 2θ + A3cos 3θ + ···

by B + B1cos θ.

Let the resulting products be represented under the general form

C0 + C1cos θ + C2cos 2θ + C3cos 3θ + ··· (1.)

This trigonometrical series is not only in itself very appropriate for illustrating the processes of the engine, but is likewise of much practical interest from its frequent use in astronomical computations.

Before proceeding further with it, we shall point out that there are three very distinct classes of ways in which it may be desired to deduce numerical values from any analytical formula.

First.

We may wish to find the collective numerical value of the whole formula, without any reference to the quantities of which that formula is a function, or to the particular mode of their combination and distribution, of which the formula is the result and representative.

Values of this kind are of a strictly arithmetical nature in the most limited sense of the term, and retain no trace whatever of the processes through which they have been deduced.

In fact, any one such numerical value may have been attained from an infinite variety of data, or of problems.

The values for x and y in the two equations (see Note D.) come under this class of numerical results.

Secondly.

We may propose to compute the collective numerical value of each term of a formula, or of a series, and to keep these results separate.

The engine must in such a case appropriate as many columns to results as there are terms to compute.

Thirdly.

It may be desired to compute the numerical value of various subdivisions of each term, and to keep all these results separate.

It may be required, for instance, to compute each coefficient separately from its variable, in which particular case the engine must appropriate two result-columns to every term that contains both a variable and coefficient.

There are many ways in which it may be desired in special cases to distribute and keep separate the numerical values of different parts of an algebraical formula; and the power of effecting such distributions to any extent is essential to the algebraical character of the Analytical Engine.

Many persons who are not conversant with mathematical studies, imagine that because the business of the engine is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical.

This is an error.

The engine can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols; and in fact it might bring out its results in algebraical notation, were provisions made accordingly.

It might develope three sets of results simultaneously, viz.

symbolic results (as already alluded to in Notes A. and B.), numerical results (its chief and primary object); and algebraical results in literal notation.

This latter however has not been deemed a necessary or desirable addition to its powers, partly because the necessary arrangements for effecting it would increase the complexity and extent of the mechanism to a degree that would not be commensurate with the advantages, where the main object of the invention is to translate into numerical language general formulæ of analysis already known to us, or whose laws of formation are known to us.

But it would be a mistake to suppose that because its results are given in the notation of a more restricted science, its processes are therefore restricted to those of that science.

The object of the engine is in fact to give the utmost practical efficiency to the resources of numerical interpretations of the higher science of analysis, while it uses the processes and combinations of this latter.

To return to the trigonometrical series.

We shall only consider the first four terms of the factor (A + A1 cos θ + &c.), since this will be sufficient to show the method.

We propose to obtain separately the numerical value of each coefficient C0, C1, &c. of (1.).

The direct multiplication of the two factors gives