Those who may desire to study the principles of the Jacquard-loom in the most effectual manner, viz. that of practical observation, have only to step into the Adelaide Gallery or the Polytechnic Institution. In each of these valuable repositories of scientificillustration, a weaver is constantly working at a Jacquard-loom, and is ready to give any information that may be desired as to the construction and modes of acting of his apparatus. The volume on the manufacture of silk, in Lardner’s Cyclopædia, contains a chapter on the Jacquard-loom, which may also be consulted with advantage.
The mode of application of the cards, as hitherto used in the art of weaving, was not found, however, to be sufficiently powerful for all the simplifications which it was desirable to attain in such varied and complicated processes as those required in order to fulfil the purposes of an Analytical Engine. A method was devised of what was technically designatedbackingthe cards in certain groups according to certain laws. The object of this extension is to secure the possibility of bringing any particular card or set of cards into useany number of times successivelyin the solution of one problem. Whether this power shall be taken advantage of or not, in each particular instance, will depend on the nature of the operations which the problem under consideration may require. The process is alluded to by M. Menabrea inpage 15, and it is a very important simplification. It has been proposed to use it for the reciprocal benefit of that art, which, while it has itself no apparent connexion with the domains of abstract science, has yet proved so valuable to the latter, in suggesting the principles which, in their new and singular field of application, seem likely to placealgebraicalcombinations not less completely within the province of mechanism, than are all those varied intricacies of whichintersecting threadsare susceptible. By the introduction of the system ofbackinginto the Jacquard-loom itself, patterns which should possess symmetry, and follow regular laws of any extent, might be woven by means of comparatively few cards.
Those who understand the mechanism of this loom will perceive that the above improvement is easily effected in practice, by causing the prism over which the train of pattern-cards is suspended, to revolvebackwardsinstead offorwards, at pleasure, under the requisite circumstances; until, by so doing, any particular card, or set of cards, that has done duty once, and passed on in the ordinary regular succession, is brought back to the position it occupied just before it was used the preceding time. The prism then resumes itsforwardrotation, and thus brings the card or set of cards in question into play a second time. The prism then resumes itsforwardrotation, and thus brings the card or set of cards in question into play a second time. This process may obviously be repeated any number of times.
A. A. L.
We have represented the solution of these two equations, with every detail, in a diagram[23]similar to those used inNote B.; but additional explanations are requisite, partly in order to make this more complicated case perfectly clear, and partly for the comprehension of certain indications and notations not used in the preceding diagrams. Those who may wish to understandNote G.completely, are recommended to pay particular attention to the contents of the present Note, or they will not otherwise comprehend the similar notation and indications when applied to a much more complicated case.
In all calculations, the columns of Variables used may be divided into three classes:—
1st. Those on which the data are inscribed:
2ndly. Those intended to receive the final results:
3dly. Those intended to receive such intermediate and temporary combinations of the primitive data as are not to be permanently retained, but are merely needed forworking with, in order to attain the ultimate results. Combinations of this kind might properly be calledsecondary data. They are in fact so manysuccessive stagestowards the final result. The columns which receive them are rightly namedWorking-Variables, for their office is in its nature purelysubsidiaryto other purposes. They develope an intermediate and transient class of results, which unite the original data with the final results.
The Result-Variables sometimes partake of the nature of Working-Variables. It frequently happens that a Variable destined to receive a final result is the recipient of one or more intermediate values successively, in the course of the processes. Similarly, the Variables for data often become Working-Variables, or Result-Variables, or even both in succession. It so happens, however, that in the case of the present equations the three sets of offices remain throughout perfectly separate and independent.
It will be observed, that in the squares below theWorking-Variables nothing is inscribed. Any one of these Variables is in many cases destined to pass through various values successively during the performance of a calculation (although in these particular equations no instance of this occurs). Consequently noone fixedsymbol, or combination of symbols, should be considered as properly belonging to a merelyWorking-Variable; and as a general rule their squares are left blank. Of course in this, as in all other cases where we mention ageneralrule, it is understood that many particular exceptions may be expedient.
In order that all the indications contained in the diagram may be completely understood, we shall now explain two or three points, not hitherto touched on. When the value on any Variable is called into use, one of two consequences may be made to result. Either the value mayreturnto the Variable after it has been used, in which case it is ready for a second use if needed; or the Variable may be made zero. (We are of course not considering a third case, of not unfrequent occurrence, in which the same Variable is destined to receive theresultof the very operation which it has just supplied with a number.) Now the ordinary rule is, that the valuereturnsto the Variable; unless it has been foreseen that no use for that value can recur, in which case zero is substituted. Attheendof a calculation, therefore, every column ought as a general rule to be zero, excepting those for results. Thus it will be seen by the diagram, that whenm,the value onV_0,is used for the second time by Operation 5,V_0becomes 0, sincemis not again needed; that similarly, when (mm′-m′n), onV_12,is used for the third time by Operation 11,V_12becomes zero, since (mm′-m′n)is not again needed. In order to provide for the one or the other of the courses above indicated, there aretwovarieties of theSupplyingVariable-cards. One of these varieties has provisions which cause the number given off from any Variable toreturnto that Variable after doing its duty in the mill. 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 theRetainingSupply-cards and theZeroSupply-cards. We see that theprimaryoffice (seeNote B.) of both these varieties of cards is the same; they only differ in theirsecondaryoffice.
Every Variable thus has belonging to it one class ofReceivingVariable-cards andtwoclasses ofSupplyingVariable-cards. It is plain however that only theoneor theotherof these two latter classes can be used by any one Variable foroneoperation; neverbothsimultaneously; their respective functions being mutually incompatible.
It should be understood that the Variable-cards are not placed inimmediate contiguitywith 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 theupperindices which are added to the left of eachVin the diagram; an addition which we have also taken the liberty of making to theV’s in M, Menabrea’s tables ofpages 16,19, since it does notalteranything therein represented by him, but merely adds something to the previous indications of those tables. Thelowerindices are obviously indices oflocalityonly, and are wholly independent of the operations performed or of the results obtained, their value continuing unchanged during the performance of calculations. Theupperindices, however, are of a different nature. Their office is to indicate anyalterationin 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^1V,and so forth. Whenever avalueagain gives place tozero, the Variable again becomes^0V,even if it have been^nVthe moment before. If avaluethen again be substituted, the Variable becomes^{n+1}V(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 ofthe Variables for data are all unity, and those for the Working and Result-variables are all zero. In this state the diagram represents them[24].
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 adifferencein certain cases, where there would otherwise be an apparentidentityconfusing in its tendency. In such a case asV_n = V_p + V_nthere is more clearness and more consistency with the usual laws of algebraical notation, in being able to write^{m+1}V_{n} = ^qV_p + ^{m}V_{n}.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 thatseries of successive substitutions, of which everyresultis 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:—series of successive substitutions
There arethreesuccessive substitutions for each of these equations. The formulæ (2.), (3.), and (4.) areimplicitlycontained in (1.), which latter we may consider as being in fact thecondensedexpression of any of the former. It will be observed that every succeeding substitution must containtwiceas manyV’s as its predecessor. So that if a problem requirensubstitutions, the successive series of numbers for theV’s in the whole of them will be 2, 4, 8, 16 ...2^{n}.
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 titleVariable-cards. TheV’s however in reality all represent the actualVariable-columnsof 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 (whencombinedwith the more generalised 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 anyonequantity, 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, exceptingmn′-m′n), which is used three times.
Theorderin 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 onlythree 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 are6x, 3-, 2/.
Since the very definition of an operation implies that there must betwonumbers to act upon, there are of coursetwo SupplyingVariable-cards necessarily brought into action for every operation, in order to furnish the two proper numbers. (SeeNote B.) Also, since every operation must produce a result, which must be placedsomewhere, each operation entails the action of aReceivingVariable-card, to indicate the proper locality for the result. Therefore, at least three times as many Variable-cards as there areoperations(notOperation-cards, for these, as we have just seen, are by no means always as numerous as theoperations) 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 ofVariableand ofOperation-cards requisite for the working of even the simplest calculation.
Allcalculations 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. Theorderin 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 greatvarietyof 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 thetimenecessary for completing the calculation.
It must be evident how multifarious and how mutually complicated are the considerations which the workings of such an engine involve. There are frequently several distinctsets of effectsgoing 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 whereconditionsare very numerous and inter-complicated; such as for instance the estimation of the mutual relations amongststatisticalphænomena, and of those involved in many other classes of facts.
A. A. L.
DIAGRAM BELONGING TO NOTE D.
This example has evidently been chosen on account of its brevity and simplicity, with a view merely to explain themannerin which the engine would proceed in the case of ananalytical calculation containing variables, rather than to illustrate the extent of its powers to solve cases of a difficult and complex nature. The equations ofpage 14are 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 formulædiscrepancies between two equationsgiven in the Memoir as thedatafor calculation, and theresultsof 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 asarray of equations
The following is a more complicated example of the manner in which the engine would compute a trigonometrical function containing variables. To multiplyarray of equationsLet the resulting products be represented under the general formarray of equations
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 thewhole 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 aninfinite varietyof data, or of problems. The values forxandyin the two equations (seeNote D.), come under this class of numerical results.
Secondly. We may propose to compute the collective numerical value ofeach termof a formula, or of a series, and to keep these resultsseparate. The engine must in such a case appropriate as many columns toresultsas there are terms to compute.
Thirdly. It may be desired to compute the numerical value of varioussubdivisions 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 appropriatetworesult-columns toevery 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 thealgebraicalcharacter 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 innumerical notation, thenature of its processesmust consequently bearithmeticalandnumerical, rather thanalgebraicalandanalytical. This is an error. The engine can arrange and combine its numerical quantities exactly as if they werelettersor any othergeneralsymbols; and in fact it might bring out its results in algebraicalnotation, were provisions made accordingly. It might develope three sets of results simultaneously, viz.symbolicresults (as already alluded to inNotes A.andB.);numericalresults (its chief and primary object); andalgebraicalresults inliteralnotation, 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 itsresultsare given in thenotationof a more restricted science, itsprocessesare therefore restricted to those of that science. The object of the engine is in fact to give theutmost practical efficiencyto the resources ofnumerical interpretationsof 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 four first terms of the factor (A+A_1 cos theta &c.), since this will be sufficient to show the method. We propose to obtain separately the numerical value ofeach coefficientC_0,C_1,&c. of (1.). The direct multiplication of the two factors givesarray of equationsa result which would stand thus on the engine:—
The variable belonging to each coefficient is written below it, as we have done in the diagram, by way of memorandum. The only further reduction which is at first apparently possible in the preceding result, would be the addition ofV_21toV_31(in which caseB_1Ashould be effaced fromV_31). The whole operations from the beginning would then be—
We do not enter into the same detail ofeverystep of the processes as in the examples ofNotes D.andG., thinking it unnecessary and tedious to do so. The reader will remember the meaning and use of the upper and lower indices, &c., as before explained.
To proceed: we know thatarray of equationsConsequently, a slight examination of the second line of (2.) will show that by making the proper substitutions, (2.) will become
These coefficients should respectively appear onV_20 V_21 V_22 V_23 V_24We shall perceive, if we inspect the particular arrangement of the results in (2.) on the Result-columns as represented in the diagram, that, in order to effect this transformation, each successive coefficient uponV_32,V_33,&c. (beginning withV_32), must through means of proper cards be divided bytwo[25]; and that one of the halves thusobtained must be added to the coefficient on the Variable which precedes it by ten columns, and the other half to the coefficient on the Variable which precedes it by twelve columns;V_32,V_33,&c. themselves becoming zeros during the process.
This series of operations may be thus expressed:—
Fourth Series.[26]