[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.