[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.