for certain special suppositions respecting , and . The general integral itself is of the form, and it is worthy of remark, that the engine might (in a manner more or less similar to the preceding) calculate the value of this formula upon most other hypotheses for the functions in the integral, with as much, or (in many cases) with more, ease than it can formula (8.). A. A. L. [Pg 68] 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.