This remark seems to require further comment, since it is in somedegree calculated to strike the mind as being at variance with thesubsequent passage, where it isexplained thatan engine which can effect these four operations can infact effectevery species of calculation. The apparent discrepancy isstronger too in the translation than in the original, owing to itsbeing impossible to render precisely into the English tongue all theniceties of distinction which the French idiom happens to admit of inthe phrases used for the two passages we refer to. The explanationlies in this: that in the one case the execution of these fouroperations is thefundamental starting-point, and the object proposedfor attainment by the machine is thesubsequent combination of thesein every possible variety; whereas in the other case the execution ofsomeone of these four operations, selected at pleasure, is theultimatum, the sole and utmost result that can be proposed forattainment by the machine referred to, and which result it cannot anyfurther combine or work upon. The onebegins where the otherends.Should this distinction not now appear perfectly clear, it will becomeso on perusing the rest of the Memoir, and the Notes that are appendedto it.—
The idea that the one engine is the offspring and has grown out of the other, is an exceedingly natural and plausible supposition, until reflection reminds us that nonecessary sequence and connexion need exist between two such inventions, and that they may be wholly independent. M. Menabrea has shared this idea in common with persons who have not his profound and accurate insight into the nature of either engine.InNote A. (see the Notes at the end of the Memoir) it will be found sufficiently explained, however, that this supposition is unfounded. M. Menabrea's opportunities were by no means such as could be adequate to afford himinformation on a point like this, which would be naturally and almost unconsciouslyassumed, andwould scarcely suggest any inquiry with reference to it.—
This must not be understood in too unqualified a manner. The engine is capable under certain circumstances, of feeling about to discover which of two or more possible contingencies has occurred, and of then shaping its future course accordingly.—
Zero is notalways substituted when a number istransferred to the mill. This is explained further on in the memoir,and still more fully inNote D.—
Not having had leisure to discuss with Mr. Babbage the manner of introducing into his machine the combination of algebraical signs, I do not pretend here to expose the method he uses for this purpose; but I considered that I ought myself to supply the deficiency, conceiving that this paper would have been imperfect if I had omitted to point out one means that might be employed for resolving this essential part of the problem in question.
For an explanation of the upper left-hand indices attached to the V's in this and in the preceding Table, we must refer the reader toNote D, amongst those appended to the memoir.—
This sentence has been slightly altered in the translation in order to express more exactly the present state of the engine.—
The notation here alluded to is a most interesting and important subject, and would have well deserved a separate and detailed Note upon it amongst those appended to the Memoir. It has, however, been impossible, within the space allotted, even to touch upon so wide a field.—
We do not mean to imply that theonly use made of theJacquard cards is that of regulating the algebraicaloperations; but we mean to explain thatthose cardsand portions of mechanism which regulate theseoperations arewholly independent of those which are used for other purposes. M.Menabrea explains that there arethree classes of cards usedin the engine for three distinct sets of objects, viz.Cards ofthe Operations,Cards of the Variables, and certainCards of Numbers.
In fact, such an extension as we allude to would merely constitute a further and more perfected development of any system introduced for making the proper combinations of the signsplus andminus.How ably M. Menabrea has touched on this restricted case is pointed out inNote B.
The machine might have been constructed so as to tabulate for ahigher value ofn than seven. Since, however, every unitadded to the value ofn increases the extent of the mechanismrequisite, there would on this account be a limit beyond which itcould not be practically carried. Seven is sufficiently high for thecalculation of all ordinary tables.
The fact that, in the Analytical Engine, the same extent of mechanism suffices for the solution of
,whethern=7,n=100,000, orn=any number whatever, at once suggests how entirely distinct must be thenature of the principles through whose application matter has been enabled to become the working agent of abstract mental operations in each of these engines respectively, and it affords an equally obvious presumption, that in the case of the Analytical Engine, not only are those principles in themselves of a higher and more comprehensive description, but also such as must vastly extend thepractical value of the engine whose basis they constitute.
A fuller account of themanner in whichthe signs are regulated isgiven in M. Menabrea's Memoir. He himself expressesdoubts (in anote of his own) as to his havingbeen likely to hit on the precise methods really adopted; hisexplanation being merely a conjectural one. That itdoesaccord precisely with the fact is a remarkable circumstance, andaffords a convincing proof how completely M. Menabrea has been imbuedwith the true spirit of the invention. Indeed the whole of theabove Memoir is a striking production, when we consider that M.Menabrea had had but very slight means for obtaining anyadequate ideas respecting the Analytical Engine. It requires howevera considerable acquaintance with the abstruse and complicated natureof such a subject, in order fully to appreciate the penetration of thewriter who could take so just and comprehensive a view of it uponsuch limited opportunity.
This adjustment is done by hand merely.
It is convenient to omit the circles whenever the signs + or − can be actually represented.
We recommend the reader to trace the successive substitutions backwards from (1) to (4), inM. Menabrea'sTable. This he will easily do by means of the upper and lower indices, and it is interesting to observe how each V successively ramifies (so to speak) into two other V's in some other column of the Table, until at length the V's of the original data are arrived at.
This division would be managed by ordering the number 2 to appearon any separate new column which should be conveniently situated forthe purpose, and then directing this column (which is in thestrictest sense aWorking-Variable) to divide itselfsuccessively with V32, V33, &c.
It should be observed, that were the rest of the factor(A + A cos θ + &c.) taken into account, instead offour terms only, C3would have the additional term ½B1A4; andC4 the two additional terms, BA4,½B1A5. This would indeed have been the case hadevensix terms been multiplied.
A cycle that includesn other cycles, successivelycontained one within another, is called a cycle of then+1th order. A cycle may simplyinclude many othercycles, and yet only be of the second order. If a series follows acertain law for a certain number of terms, and then another law foranother number of terms, there will be a cycle of operations for everynew law; but these cycles will not becontained one withinanother,—they merelyfollow each other. Thereforetheir number may be infinite without influencing theorder ofa cycle that includes a repetition of such a series.
The engine cannot of course compute limits for perfectlysimple anduncompounded functions, except in thismanner. It is obvious that it has no power of representing or ofmanipulating with any butfinite increments or decrements,and consequently that wherever the computation of limits (or of anyother functions) depends upon thedirect introduction ofquantities which either increase or decreaseindefinitely, weare absolutely beyond the sphere of its powers. Its nature andarrangements are remarkably adapted for taking into account allfinite increments or decrements (however small or large), andfor developing the true and logical modifications of form or valuedependent upon differences of this nature. The engine may indeed beconsidered as including the whole Calculus of Finite Differences; manyof whose theorems would be especially and beautifully fitted fordevelopment by its processes, and would offer peculiarly interestingconsiderations. We may mention, as an example the calculation of theNumbers of Bernoulli by means of theDifferences of Zero.
It is interesting to observe, that so complicated a case as thiscalculation of the Bernoullian Numbers nevertheless presents aremarkable simplicity in one respect; viz. that during the processesfor the computation ofmillions of these Numbers, no otherarbitrary modification would be requisite in the arrangements,excepting the above simple and uniform provision for causing one ofthe data periodically to receive the finite increment unity.