The calculation of the coefficients C_0, C_1, &c. of (1.), would now be completed, and they would stand ranged in order on V_20, V_21, &c. It will be remarked, that from the moment the fourth series of operations is ordered, the Variables V_31, V_32, &c. cease to be Result-Variables, and become mere Working-Variables.
The substitution made by the engine of the processes in the second side of (3.) for those in the first side, is an excellent illustration of the manner in which we may arbitrarily order it to substitute any function, number, or process, at pleasure, for any other function, number or process, on the occurrence of a specified contingency.
We will now suppose that we desire to go a step further, and to obtain the numerical value of each complete term of the product (1.), that is of each coefficient and variable united, which for the (n+1)th term would be C_n . cos ntheta.
We must for this purpose place the variables themselves on another set of columns, V_41, V_42, &c., and then order their successive multiplication by V_21, V_22, &c., each for each. There would thus be a final series of operations as follows:—
Fifth and Final Series of Operations.
(N.B. that V_40 being intended to receive the coefficient on V_20 which has no variable, will only have cos theta (=1) inscribed on it, preparatory to commencing the fifth series of operations.)
From the moment that the fifth and final series of operations is ordered, the Variables V_20, V_21, &c. then in their turn cease to be Result-Variables and become mere Working-Variables; V_40, V_41, &c. being now the recipients of the ultimate results.
We should observe, that if the variables cos theta, cos 2theta, cos 3theta, &c. are furnished, they would be placed directly upon V_41, V_42, &c., like any other data. If not, a separate computation might be entered upon in a separate part of the engine, in order to calculate them, and place them on V_41, &c.
We have now explained how the engine might compute (1.) in the most direct manner, supposing we knew nothing about the general term of the resulting series. But the engine would in reality set to work very differently, whenever (as in this case) we do know the law for the general term.
The two first terms of (1.) are array of equations and the general term for all after these is array of equations which is the coefficient of the ((n+1)^th term. The engine would calculate the two first terms by means of a separate set of suitable Operation-cards, and would then need another set for the third term; which last set of Operation-cards would calculate all the succeeding terms ad infinitum; merely requiring certain new Variable-cards for each term to direct the operations to act on the proper columns. The following would be the successive sets of operations for computing the coefficients of n+2 terms— (x, /, +) (x, x, x, /, +, +) n(x, +, x, /, +) Or we might represent them as follows, according to the numerical order of the operations:— (1, 2 ... 4), (5, 6, ... 15), n(11, 12 ... 15) The brackets, it should be understood, point out the relation in which the operations may be grouped, while the comma marks succession. The symbol + might be used for this latter purpose, but this would be liable to produce confusion, as + is also necessarily used to represent one class of the actual operations which are the subject of that succession. In accordance with this meaning attached to the comma, care must be taken when any one group of operations recurs more than once, as is represented above by n (11 ... 15), not to insert a comma after the number or letter prefixed to that group. n,(11 ... 15) would stand for an operation n followed by the group of operations (11 ... 15); instead of denoting the number of groups which are to follow each other.
Wherever a general term exists, there will be a recurring group of operations, as in the above example. Both for brevity and for distinctness, a recurring group is called a cycle. A cycle of operations, then, must be understood to signify any set of operations which is repeated more than once. It is equally a cycle, whether it be repeated twice only, or an indefinite number of times; for it is the fact of a repetition occurring at all that constitutes it such. In many cases of analysis there is a recurring group of one or more cycles; that is, a cycle of a cycle, or a cycle of cycles. For instance: suppose we wish to divide a series by a series, (a+bx+cx^2+...)/(a′ +b′x+c′x^2+...) it being required that the result shall be developed, like the dividend and the divisor, in successive powers of x. A little consideration of (1.), and of the steps through which algebraical division is effected, will show that (if the denominator be supposed to consist of p terms) the first partial quotient will be completed by the following operations:— /p(x,-) or 1/p(2,3) that the second partial quotient will be completed by an exactly similar set of operations, which acts on the remainder obtained by the first set, instead of on the original dividend. The whole of the processes therefore that have been gone through, by the time the second partial quotient has been obtained, will be,— /p(x,-) or 2,1 p(2,3) which is a cycle that includes a cycle, or a cycle of the second order. The operations for the complete division, supposing we propose to obtain n terms of the series constituting the quotient, will be,— n(/, p(x,-) or n(1), p(2,3) It is of course to be remembered that the process of algebraical division in reality continues ad infinitum, except in the few exceptional cases which admit of an exact quotient being obtained. The number n in the formula (4.), is always that of the number of terms we propose to ourselves to obtain; and the nth partial quotient is the coefficient of the (n-1)th power of x.
There are some cases which entail cycles of cycles of cycles, to an indefinite extent. Such cases are usually very complicated, and they are of extreme interest when considered with reference to the engine. The algebraical development in a series, of the nth function of any given function, is of this nature. Let it be proposed to obtain the nth function of phi(a,b,c ... x) x being the variable We should premise that we suppose the reader to understand what is meant by an nth function. We suppose him likewise to comprehend distinctly the difference between developing an nth function algebraically, and merely calculating an nth function arithmetically. If he does not, the following will be by no means very intelligible; but we have not space to give any preliminary explanations. To proceed: the law, according to which the successive functions of (5.) are to be developed, must of course first be fixed on. This law may be of very various kinds. We may propose to obtain our results in successive powers of x, in which case the general form would be C+C_1x+C_2x^2+&c or in successive powers of n itself, the index of the function we are ultimately to obtain, in which case the general form would be C+C_1n+C_2n^2+&c and x would only enter in the coefficients. Again, other functions of x or of n instead of powers, might be selected. It might be in addition proposed, that the coefficients themselves should be arranged according to given functions of a certain quantity. Another mode would be to make equations arbitrarily amongst the coefficients only, in which case the several functions, according to either of which it might be possible to develop the nth function of (5.), would have to be determined from the combined consideration of these equations and of (5.) itself.
The algebraical nature of the engine (so strongly insisted on in a previous part of this Note) would enable it to follow out any of these various modes indifferently; just as we recently showed that it can distribute and separate the numerical results of any one prescribed series of processes, in a perfectly arbitrary manner. Were it otherwise, the engine could merely compute the arithmetical nth function, a result which, like any other purely arithmetical results, would be simply a collective number, bearing no traces of the data or the processes which had led to it.
Secondly, the law of development for the nth function being selected, the next step would obviously be to develope (5.) itself, according to this law. This result would be the first function, and would be obtained by a determinate series of processes. These in most cases would include amongst them one or more cycles of operations.
The third step (which would consist of the various processes necessary for effecting the actual substitution of the series constituting the first function, for the variable itself) might proceed in either of two ways. It might make the substitution either wherever x occurs in the original (5.), or it might similarly make it wherever x occurs in the first function itself which is the equivalent of (5.). In some cases the former mode might be best, and in others the latter.
Whichever is adopted, it must be understood that the result is to appear arranged in a series following the law originally prescribed for the development of the nth function. This result constitutes the second function; with which we are to proceed exactly as we did with the first function, in order to obtain the third function; and so on, n-1 times, to obtain the nth function. We easily perceive that since every successive function is arranged in a series following the same law, there would (after the first function is obtained) be a cycle, of a cycle, of a cycle, &c. of operations[27], one, two, three, up to n-1 times, in order to get the nth function. We say, after the first function is obtained, because (for reasons on which we cannot here enter) the first function might in many cases be developed through a set of processes peculiar to itself, and not recurring for the remaining functions.
We have given but a very slight sketch, of the principal general steps which would be requisite for obtaining an nth function of such a formula as (5.). The question is so exceedingly complicated, that perhaps few persons can be expected to follow, to their own satisfaction, so brief and general a statement as we are here restricted to on this subject. Still it is a very important case as regards the engine, and suggests ideas peculiar to itself, which we should regret to pass wholly without allusion. Nothing could be more interesting than to follow out, in every detail, the solution by the engine of such a case as the above; but the time, space and labour this would necessitate, could only suit a very extensive work.
To return to the subject of cycles of operations: some of the notation of the integral calculus lends itself very aptly to express them: (2.) might be thus written:— (/), sum(+1)^p(x,-) or (1),sum(+1)^p(2,3) where p stands for the variable; (+1^p for the function of the variable, that is, for phi p; and the limits are from 1 to p, or from 0 to p-1, each increment being equal to unity. Similarly, (4.) would be,— sum(+1)^n~{(/), sum(+1)^p(x,-)} the limits of n being from 1 to n, or from 0 to n-1, or sum(+1)^n{(1), sum(+1)^p(2,3)}
Perhaps it may be thought that this notation is merely a circuitous way of expressing what was more simply and as effectually expressed before; and, in the above example, there may be some truth in this. But there is another description of cycles which can only effectually be expressed, in a condensed form, by the preceding notation. We shall call them varying cycles. They are of frequent occurrence, and include successive cycles of operations of the following nature:— p(1,2,...m), overline over p-1 (1,2...m,) overline over p-2(1,2...m) overline over p-n(1,2...m) where each cycle contains the same group of operations, but in which the number of repetitions of the group varies according to a fixed rate, with every cycle. (9.) can be well expressed as follows:— sump(1,2...m), the limits of p being p-n to p
Independent of the intrinsic advantages which we thus perceive to result in certain cases from this use of the notation of the integral calculus, there are likewise considerations which make it interesting, from the connections and relations involved in this new application. It has been observed in some of the former Notes, that the processes used in analysis form a logical system of much higher generality than the applications to number merely. Thus, when we read over any algebraical formula, considering it exclusively with reference to the processes of the engine, and putting aside for the moment its abstract signification as to the relations of quantity, the symbols +, x, &c., in reality represent (as their immediate and proximate effect, when the formula is applied to the engine) that a certain prism which is a part of the mechanism (see Note C.), turns a new face, and thus presents a new card to act on the bundles of levers of the engine; the new card being perforated with holes, which are arranged according to the peculiarities of the operation of addition, or of multiplication, &c. Again, the numbers in the preceding formula (8.), each of them really represents one of these very pieces of card that are hung over the prism.
Now in the use made in the formulæ (7.). (8.) and (10.), of the notation of the integral calculus, we have glimpses of a similar new application of the language of the higher mathematics. sum, in reality, here indicates that when a certain number of cards have acted in succession, the prism over which they revolve must rotate backwards, so as to bring those cards into their former position; and the limits 1 to n, 1 to p &c., regulate how often this backward rotation is to be repeated.
A.A.L.
There is in existence a beautiful woven portrait of Jacquard, in the fabrication of which 24,000 cards were required.
The power of repeating the cards, alluded to by M. Menabrea in page 15, and more fully explained in Note C., reduces to an immense extent the number of cards required. It is obvious that this mechanical improvement is especially applicable wherever cycles occur in the mathematical operations, and that, in preparing data for calculations by the engine, it is desirable to arrange the order and combination of the processes with a view to obtain them as much as possible symmetrically and in cycles, in order that the mechanical advantages of the backing system may be applied to the utmost. It is here interesting to observe the manner in which the value of an analytical resource is met and enhanced by an ingenious mechanical contrivance. We see in it an instance of one of those mutual adjustments between the purely mathematical and the mechanical departments, mentioned in Note A. as being a main and essential condition of success in the invention of a calculating engine. The nature of the resources afforded by such adjustments would be of two principal kinds. In some cases, a difficulty (perhaps in itself insurmountable) in the one department, would be overcome by facilities in the other; and sometimes (as in the present case) a strong point in the one, would be rendered still stronger and more available, by combination with a corresponding strong point in the other.
As a mere example of the degree to which the combined systems of cycles and of backing can diminish the number of cards requisite, we shall choose a case which places it in strong evidence, and which has likewise the advantage of being a perfectly different kind of problem from those that are mentioned in any of the other Notes. Suppose it be required to eliminate nine variables from ten simple equations of the form— array of equations We should explain, before proceeding, that it is not our object to consider this problem with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution.
The first step would be the elimination of the first unknown quantity x_0 between the two first equations. This would be obtained by the form— array of equations for which the operations 10 (x,x,-) would be needed. The second step would be the elimination of x_0, between the second and third equations, for which the operations would be precisely the same. We should then have had altogether the following operations:— 10(x,x,-), 10(x,x,-)=20(x,x,-) Continuing in the same manner, the total number of operations for the complete elimination of x_0 between all the successive pairs of equations, would be— 9.10(x,x,-)=90(x,x,-) We should then be left with nine simple equations of nine variables from which to eliminate the next variable x_1; for which the total of the processes would be— 8.9(x,X,-)=72(x,x,-) We should then be left with eight simple equations of eight variables from which to eliminate x_2, for which the processes would be— 7.8(x,x-)=56(x,x,-) and so on. The total operations for the elimination of all the variables would thus be— 9.10+8.9+7.8+6.7+5.6+4.5+3.4+2.3+1.2=330 So that three Operation-cards would perform the office of 330 such cards.
If we take n simple equations containing n-1 variables, n being a number unlimited in magnitude, the case becomes still more obvious, as the same three cards might then take the place of thousands or millions of cards.
We shall now draw further attention to the fact, already noticed, of its being by no means necessary that a formula proposed for solution should ever have been actually worked out, as a condition for enabling the engine to solve it. Provided we know the series of operations to be gone through, that is sufficient. In the foregoing instance this will be obvious enough on a slight consideration. And it is a circumstance which deserves particular notice, since herein may reside a latent value of such an engine almost incalculable in its possible ultimate results. We already know that there are functions whose numerical value it is of importance for the purposes both of abstract and of practical science to ascertain, but whose determination requires processes so lengthy and so complicated, that, although it is possible to arrive at them through great expenditure of time, labour and money, it is yet on these accounts practically almost unattainable; and we can conceive there being some results which it may be absolutely impossible in practice to attain with any accuracy, and whose precise determination it may prove highly important for some of the future wants of science in its manifold, complicated and rapidly-developing fields of inquiry, to arrive at.
Without, however, stepping into the region of conjecture, we will mention a particular problem which occurs to us at this moment as being an apt illustration of the use to which such an engine may be turned for determining that which human brains find it difficult or impossible to work out unerringly. In the solution of the famous problem of the Three Bodies, there are, out of about 295 coefficients of lunar perturbations given by M. Clausen (Astroe. Nachrichten, No. 406) as the result of the calculations by Burg, of two by Damoiseau, and of one by Burckhardt, fourteen coefficients that differ in the nature of their algebraic sign; and out of the remainder there are only 101 (or about one-third) that agree precisely both in signs and in amount. These discordances, which are generally small in individual magnitude, may arise either from an erroneous determination of the abstract coefficients in the development of the problem, or from discrepancies in the data deduced from observation, or from both causes combined. The former is the most ordinary source of error in astronomical computations, and this the engine would entirely obviate.
We might even invent laws for series or formulæ in an arbitrary manner, and set the engine to work upon them, and thus deduce numerical results which we might not otherwise have thought of obtaining. But this would hardly perhaps in any instance be productive of any great practical utility, or calculated to rank higher than as a kind of philosophical amusement.
A. A. L.
It is desirable to guard against the possibility of exaggerated ideas that might arise as to the powers of the Analytical Engine. In considering any new subject, there is frequently a tendency, first, to overrate what we find to be already interesting or remarkable; and, secondly, by a sort of natural reaction, to undervalue the true state of the case, when we do discover that our notions have surpassed those that were really tenable.
The Analytical Engine has no pretensions whatever to originate any thing. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with. This it is calculated to effect primarily and chiefly of course, through its executive faculties; but it is likely to exert an indirect and reciprocal influence on science itself in another manner. For, in so distributing and combining the truths and the formulæ of analysis, that they may become most easily and rapidly amenable to the mechanical combinations of the engine, the relations and the nature of many subjects in that science are necessarily thrown into new lights, and more profoundly investigated. This is a decidedly indirect, and a somewhat speculative, consequence of such an invention. It is however pretty evident, on general principles, that in devising for mathematical truths a new form in which to record and throw themselves out for actual use, views are likely to be induced, which should again react on the more theoretical phase of the subject. There are in all extensions of human power, or additions to human knowledge, various collateral influences, besides the main and primary object attained.
To return to the executive faculties of this engine: the question must arise in every mind, are they really even able to follow analysis in its whole extent? No reply, entirely satisfactory to all minds, can be given to this query, excepting the actual existence of the engine, and actual experience of its practical results. We will however sum up for each reader’s consideration the chief elements with which the engine works:—
1. It performs the four operations of simple arithmetic upon any numbers whatever.
2. By means of certain artifices and arrangements (upon which we cannot enter within the restricted space which such a publication as the present may admit of), there is no limit either to the magnitude of the numbers used, or to the number of quantities (either variables or constants) that may be employed.
3. It can combine these numbers and these quantities either algebraically or arithmetically, in relations unlimited as to variety, extent, or complexity.
4. It uses algebraic signs according to their proper laws, and developes the logical consequences of these laws.
5. It can arbitrarily substitute any formula for any other; effacing the first from the columns on which it is represented, and making the second appear in its stead.
6. It can provide for singular values. Its power of doing this is referred to in M. Menabrea’s memoir, page 20, where he mentions the passage of values through zero and infinity. The practicability of causing it arbitrarily to change its processes at any moment, on the occurrence of any specified contingency (of which its substitution of 1/2 cos.Overline(n+1)theta+1/2 cos.overline(n-1)theta for ((cos.ntheta.cos.theta)) explained in Note E., is in some degree an illustration), at once secures this point.
The subject of integration and of differentiation demands some notice. The engine can effect these processes in either of two ways:—
First. We may order it, by means of the Operation and of the Variable-cards, to go through the various steps by which the required limit can be worked out for whatever function is under consideration.
Secondly. It may (if we know the form of the limit for the function in question) effect the integration or differentiation by direct[28] substitution. We remarked in Note B., that any set of columns on which numbers are inscribed, represents merely a general function of the several quantities, until the special function have been impressed by means of the Operation and Variable-cards. Consequently, if instead of requiring the value of the function, we require that of its integral, or of its differential coefficient, we have merely to order whatever particular combination of the ingredient quantities may constitute that integral or that coefficient. In ax^{n}, for instance, instead of the quantities
| V_0 | V_1 | V_2 | V_3 |
| underbrace envolving a, n, x and a x^{n} enclosed in box | a x^{n} | ||
being ordered to appear on V_3 in the combination a x^{n}, they would be ordered to appear in that of anx^{n-1}
They would then stand thus:—
| V_0 | V_1 | V_2 | V_3 |
| underbrace envolving a, n, x and anx^{n-1} enclosed in box | {an x^{n-1} enclosed in a box | ||
Similarly, we might have a/nx^(n+1), the integral of ax_{n}.
An interesting example for following out the processes of the engine would be such a form as int {x^{n}dx}/sqrt{a^{2}-x^{2}} or any other cases of integration by successive reductions, where an integral which contains an operation repeated n times can be made to depend upon another which contains the same n-1 or n-1 times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined.
The methods in Arbogat’s Calcul des Dérivations are peculiarly fitted for the notation and the processes of the engine. Likewise the whole of the Combinatorial Analysis, which consists first in a purely numerical calculation of indices, and secondly in the distribution and combination of the quantities according to laws prescribed by these indices.
We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers. The simplest manner of computing those numbers would be from the direct expansion of x/e^x-1=1/1-x/2+x^2/2.3+x^3/2.34+&c which is in fact a particular case of the development of {a+b x+c x^{2}+\& c.}}/{a′+b′x+c'x^{2}+\&c.}} mentioned in Note E. Or again, we might compute them from the well-known form B_{2 n-1}=2.{1.2.3... 2n}/{(2pi)^{2n}}.{1+{1}/{2^{2 n}}+{1/}{3^{2n}}+...} or from the form array of equations or from many others. As however our object is not simplicity or facility of computation, but the illustration of the powers of the engine, we prefer selecting the formula below, marked (8.). This is derived in the following manner:—
If in the equation {x}/{epsilon^{x}-1}=1-{x}{2}+{B}_1{x^{2}}/{2}+{B}_3{x^{4}}/{2.3.4}+{B}_5{x^{6}/{2.3.4.5.6+...} (in which B_1, B_3 ..., &c. are the Numbers of Bernoulli), we expand the denominator of the first side in powers of x, and then divide both numerator and denominator by x, we shall derive 1=(1-x/2+B_1x^2/2+B_3x^4/2.34+...)(1+x/2+x^2/2.3+x^3/2.34+...
If this latter multiplication be actually performed, we shall have a series of the general form 1+D_1x+D_2x^2+D_3x^3+... in which we see, first, that all the coefficients of the powers of x are severally equal to zero; and secondly, that the general form for D_{2n} the coefficient of the 2(n+1)th term (that is of x^{2n} even any power of x), is the following:— array of equations Multiplying every term by (2.3...2n) we have array of equations which it may be convenient to write under the general form:— 0=A_0A_1B_1+A_3B_3A_5+...+B^{2n-1} A_1, A_3, &c. being those functions of n which respectively belong to B_1, B_3, &c.
We might have derived a form nearly similar to (8.), from {D}_{2n-1} the coefficient of any odd power of x in (6.); but the general form is a little different for the coefficients of the odd powers, and not quite so convenient.
On examining (7.) and (8.), we perceive that, when these formulæ are isolated from (6.) whence they are derived, and considered in themselves separately and independently, n may be any whole number whatever; although when (7.) occurs as one of the D′s’s in (6.), it is obvious that n is then not arbitrary, but is always a certain function of the distance of that D from the beginning. If that distance be = d, then array of equations It is with the independent formula (8.) that we have to do. Therefore it must be remembered that the conditions for the value of n are now modified, and that n is a perfectly arbitrary whole number. This circumstance, combined with the fact (which we may easily perceive) that whatever n is, every term of (8.) after the (n+1)th is = 0, and that the (n+1 )th term itself is always {B}_{2n-1}.1/{B}_{2n-1} enables us to find the value (either numerical or algebraical) of any nth Number of Bernoulli {B}_{2n-1}, in terms of all the preceding ones, if we but know the values of B_1, B_3 ... {B}_{2n-3}. We append to this Note a Diagram and Table, containing the details of the computation for B_7, (B_1, B_3, B_5 being supposed given).
On attentively considering (8.), we shall likewise perceive that we may derive from it the numerical value of every Number of Bernoulli in succession, from the very beginning, ad infinitum, by the following series of computations:—
1st Series.—Let n=1, and calculate (8.) for this value of n. The result is B_1.
2nd Series.—Let n=2. Calculate (8.) for this value of n substituting the value of B_1, just obtained. The result is B_3.
3rd Series.—Let n=3. Calculate (8.) for this value of n, substituting the values of B_1, B_3 before obtained. The result is B_5. And so on, to any extent.
The diagram[30] represents the columns of the engine when just prepared for computing {B}_{2n-1}, (in the case of n=4); while the table beneath them presents a complete simultaneous view of all the successive changes which these columns then severally pass through in order to perform the computation. (The reader is referred to Note D, for explanations respecting the nature and notation of such tables.)
Six numerical data are in this case necessary for making the requisite combinations. These data are 1, 2, n (= 4), B_1, B_3, B_5. Were n = 5, the additional datum B_7, would be needed. Were n = 6, the datum B_9, would be needed; and so on. Thus the actual number of data needed will always be n+2, for n=n; and out of these n+2 data, (overline{n + 2} — 3) of them are successive Numbers of Bernoulli. The reason why the Bernoulli Numbers used as data, are nevertheless placed on Result-columns in the diagram, is because they may properly be supposed to have been previously computed in succession by the engine itself; under which circumstances each B will appear as a result, previous to being used as a datum for computing the succeeding B. Here then is an instance (of the kind alluded to in Note D.) of the same Variables filling more than one office in turn. It is true that if we consider our computation of B, as a perfectly isolated calculation, we may conclude B_1, B_3, B_5, to have been arbitrarily placed on the columns; and it would then perhaps be more consistent to put them on V_4, V_5, V_6 as data and not results. But we are not taking this view. On the contrary, we suppose the engine to be in the course of computing the Numbers to an indefinite extent, from the very beginning; and that we merely single out, by way of example, one amongst the successive but distinct series of computations it is thus performing. Where the B’s are fractional, it must be understood that they are computed and appear in the notation of decimal fractions. Indeed this is a circumstance that should be noticed with reference to all calculations. In any of the examples already given in the translation and in the Notes, some of the data, or of the temporary or permanent results, might be fractional, quite as probably as whole numbers. But the arrangements are so made, that the nature of the processes would be the same as for whole numbers.
In the above table and diagram we are not considering the signs of any of the B’s, merely their numerical magnitude. The engine would bring out the sign for each of them correctly of course, but we cannot enter on every additional detail of this kind, as we might wish to do. The circles for the signs are therefore intentionally left blank in the diagram.
Operation-cards 1, 2, 3, 4, 5, 6 prepare are -{1}/{2}.{2n - 1}/{2n + 1} Thus, Card 1 multiplies two into n, and the three Receiving Variable-cards belonging respectively to V_4, V_5, V_6, allow the result 2n to be placed on each of these latter columns (this being a case in which a triple receipt of the result is needed for subsequent purposes); we see that the upper indices of the two Variables used, during Operation 1, remain unaltered.
We shall not go through the details of every operation singly, since the table and diagram sufficiently indicate them; we shall merely notice some few peculiar cases.
By Operation 6, a positive quantity is turned into a negative quantity, by simply subtracting the quantity from a column which has only zero upon it. (The sign at the top of V_8 would become—during this process.)
Operation 7 will be unintelligible, unless it be remembered that if we were calculating for n=1 instead of n=4, Operation 6 would have completed the computation of B_1 itself; in which case the engine, instead of continuing its processes, would have to put B_1 on V_21; and then either to stop altogether, or to begin Operations 1, 2 ... 7 all over again for value of n (= 2), in order to enter on the computation of B_3; (having however taken care, previous to this recommencement, to make the number on V_3, equal to two, by the addition of unity to the former n=1 on that column). Now Operation 7 must either bring out a result equal to zero (if n=1); or a result greater than zero, as in the present case; and the engine follows the one or the other of the two courses just explained, contingently on the one or the other result of Operation 7. In order fully to perceive the necessity of this experimental operation, it is important to keep in mind what was pointed out, that we are not treating a perfectly isolated and independent computation, but one out of a series of antecedent and prospective computations.
Cards 8, 9, 10 produce 1/2.{2n - 1}/{2n + 1}+B_1{2n}/{2}. In Operation 9 we see an example of an upper index which again becomes a value after having passed front preceding values to zero. V_11 has successively been ^0V_11, ^1V_11, ^2V_11, ^0V_11, ^3V_11; and, from the nature of the office which V_11, performs in the calculation, its index will continue to go through further changes of the same description, which, if examined, will be found to be regular and periodic.
Card 12 has to perform the same office as Card 7 did in the preceding section; since, if n had been = 2, the 11th operation would have completed the computation of B_3.
Cards 13 to 20 make A_3. Since {A}_{2n-1} always consists of 2n-1 factors, A_3 has three factors; and it will be seen that Cards 13, 14, 15, 16 make the second of these factors, and then multiply it with the first; and that 17, 18, 19, 20 make the third factor, and then multiply this with the product of the two former factors.
Card 23 has the office of Cards 11 and 7 to perform, since if n were = 3, the 21st and 22nd operations would complete the computation of B_5. As our case is B_7, the computation will continue one more stage; and we must now direct attention to the fact, that in order to compute A_7 it is merely necessary precisely to repeat the group of Operations 13 to 20; and then, in order to complete the computation of B_7, to repeat Operations 21, 22.
It will be perceived that every unit added to n in {B}_{2n-1}, entails an additional repetition of operations (13 ... 23) for the computation of {B}_{2n-1}. Not only are all the operations precisely the same however for every such repetition, but they require to be respectively supplied with numbers from the very same pairs of columns; with only the one exception of Operation 21, which will of course need B_5 (from V_23) instead of B_3 (from V_22). This identity in the columns which supply the requisite numbers, must not be confounded with identity in the values these columns have upon them and give out to the mill. Most of those values undergo alterations during a performance of the operations (13 ... 23), and consequently the columns present a new set of values for the next performance of (13 ... 23) to work on.
At the termination of the repetition of operations (13 ... 23) in computing B_7, the alterations in the values on the Variables are, that