This remark seems to require further comment, since it is in some degree calculated to strike the mind as being at variance with the subsequent passage, where it is explained that an engine which can effect these four operations can in fact effect every species of calculation. The apparent discrepancy is stronger too in the translation than in the original, owing to its being impossible to render precisely into the English tongue all the niceties of distinction which the French idiom happens to admit of in the phrases used for the two passages we refer to. The explanation lies in this: that in the one case the execution of these four operations is the fundamental starting-point, and the object proposed for attainment by the machine is the subsequent combination of these in every possible variety; whereas in the other case the execution of some one of these four operations, selected at pleasure, is the ultimatum, the sole and utmost result that can be proposed for attainment by the machine referred to, and which result it cannot any further combine or work upon. The one begins where the other ends. Should this distinction not now appear perfectly clear, it will become so on perusing the rest of the Memoir, and the Notes that are appended to it.—NOTE BY TRANSLATOR.
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 no necessary 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. In Note 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 him information on a point like this, which would be naturally and almost unconsciously assumed, and would scarcely suggest any inquiry with reference to it.—NOTE BY TRANSLATOR.
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.—NOTE BY TRANSLATOR.
Zero is not always substituted when a number is transferred to the mill. This is explained further on in the memoir, and still more fully in Note D.—NOTE BY TRANSLATOR.
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 to Note D, amongst those appended to the memoir.—NOTE BY TRANSLATOR.
This sentence has been slightly altered in the translation in order to express more exactly the present state of the engine.—NOTE BY TRANSLATOR.
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.—NOTE BY TRANSLATOR.
We do not mean to imply that the only use made of the Jacquard cards is that of regulating the algebraical operations; but we mean to explain that those cards and portions of mechanism which regulate these operations are wholly independent of those which are used for other purposes. M. Menabrea explains that there are three classes of cards used in the engine for three distinct sets of objects, viz. Cards of the Operations, Cards of the Variables, and certain Cards 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 signs plus and minus. How ably M. Menabrea has touched on this restricted case is pointed out in Note B.
The machine might have been constructed so as to tabulate for a higher value of n than seven. Since, however, every unit added to the value of n increases the extent of the mechanism requisite, there would on this account be a limit beyond which it could not be practically carried. Seven is sufficiently high for the calculation of all ordinary tables.
The fact that, in the Analytical Engine, the same extent of mechanism suffices for the solution of , whether n=7, n=100,000, or n=any number whatever, at once suggests how entirely distinct must be the nature 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 the practical value of the engine whose basis they constitute.
A fuller account of the manner in which the signs are regulated is given in M. Menabrea's Memoir. He himself expresses doubts (in a note of his own) as to his having been likely to hit on the precise methods really adopted; his explanation being merely a conjectural one. That it does accord precisely with the fact is a remarkable circumstance, and affords a convincing proof how completely M. Menabrea has been imbued with the true spirit of the invention. Indeed the whole of the above Memoir is a striking production, when we consider that M. Menabrea had had but very slight means for obtaining any adequate ideas respecting the Analytical Engine. It requires however a considerable acquaintance with the abstruse and complicated nature of such a subject, in order fully to appreciate the penetration of the writer who could take so just and comprehensive a view of it upon such 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), in M. Menabrea's Table. 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 appear on any separate new column which should be conveniently situated for the purpose, and then directing this column (which is in the strictest sense a Working-Variable) to divide itself successively with V32, V33, &c.
It should be observed, that were the rest of the factor (A + A cos θ + &c.) taken into account, instead of four terms only, C3 would have the additional term ½B1A4; and C4 the two additional terms, BA4, ½B1A5. This would indeed have been the case had even six terms been multiplied.
A cycle that includes n other cycles, successively contained one within another, is called a cycle of the n+1th order. A cycle may simply include many other cycles, and yet only be of the second order. If a series follows a certain law for a certain number of terms, and then another law for another number of terms, there will be a cycle of operations for every new law; but these cycles will not be contained one within another,—they merely follow each other. Therefore their number may be infinite without influencing the order of a cycle that includes a repetition of such a series.
The engine cannot of course compute limits for perfectly simple and uncompounded functions, except in this manner. It is obvious that it has no power of representing or of manipulating with any but finite increments or decrements, and consequently that wherever the computation of limits (or of any other functions) depends upon the direct introduction of quantities which either increase or decrease indefinitely, we are absolutely beyond the sphere of its powers. Its nature and arrangements are remarkably adapted for taking into account all finite increments or decrements (however small or large), and for developing the true and logical modifications of form or value dependent upon differences of this nature. The engine may indeed be considered as including the whole Calculus of Finite Differences; many of whose theorems would be especially and beautifully fitted for development by its processes, and would offer peculiarly interesting considerations. We may mention, as an example the calculation of the Numbers of Bernoulli by means of the Differences of Zero.
It is interesting to observe, that so complicated a case as this calculation of the Bernoullian Numbers nevertheless presents a remarkable simplicity in one respect; viz. that during the processes for the computation of millions of these Numbers, no other arbitrary modification would be requisite in the arrangements, excepting the above simple and uniform provision for causing one of the data periodically to receive the finite increment unity.