FOOTNOTES:
[16] 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. (See [pages 13] and [22].)
[17] 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].
[18] The machine might have been constructed so as to tabulate for a higher value of
than seven. Since, however, every unit added to the value of
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
,
, or
= any number whatever, at once suggests how entirely distinct must be the nnature 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.
[19] This subject is further noticed in [Note F].
[20] A fuller account of the manner in which the signs are regulated, is given on in Mons. Menabrea’s Memoir, [pages 17], [18]. He himself expresses doubts (in a note of his own at the bottom of the latter page) as to his having been likely to hit on the precise methods really adopted; his explanation In being merely a conjectural one. That it does accord precisely with the fact is a remarkable circumstance, and affords a convincing proof how completely Mons. 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 Mons. 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.
[21] This adjustment is done by hand merely.
[22] It is convenient to omit the circles whenever the signs + or — can be actually represented.
[23] See the diagram of [page 46].
[24] We recommend the reader to trace the successive substitutions backwards from (1.) to (4.), in Mons. Menabrea’s Table. This he will easily do by means of the upper and lower indices, and it is interesting to observe how each
successively ramifies (so to speak) into two other
’s in some other column of the Table; until at length the
’s of the original data are arrived at.
[25] This division would be managed by ordering the number two 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
,
, &c.
[26] It should be observed, that were the rest of the factor (
) taken into account, instead of four terms only,
would have the additional term
; and
, the two additional terms,
,
. This would indeed have been the case had even six terms been multiplied.
[27] A cycle that includes
other cycles, successively contained one within another, is called a cycle of the
th 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.
[28] 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 Nothing.
[29] See the diagram at the end of these Notes.
[30] 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.
Transcriber’s Notes
“Article XXIX,” extracted from Scientific memoirs, Vol. 3, 1843. Translated, with notes, from the Italian original by Ada King, Countess of Lovelace, daughter of Byron. She is identified on page 35 as “A. A. L.” It was originally published by Luigi Federico Menabrea as ‘Notions sur la machine analytique de M. Charles Babbage,’ pp. 352-376 in: Bibliothèque Universelle de Génève. Nouvelle Série, Tome 41.