§ The second apparent impossibility seemed to present far greater difficulty. Fortunately it was not one of immediate _practical_ importance, although as a question of philosophical inquiry it possessed the highest interest. I had frequently discussed with Mrs. Somerville and my highly gifted friend the late Professor M‘Cullagh of Dublin, the question whether it was possible that we should be able to treat algebraic formulæ by means of machinery. The result of many inquiries led to the conclusion, that if not really impossible, it was almost hopeless. The first difficulty was that of representing an indefinite number in a machine of finite size. It was readily admitted that if a machine afforded means of operating on _all_ numbers under twenty places of figures, then that any number, or _an indefinite_ number, of less than twenty places or figures might be represented by it. But such number will not be really indefinite. It would be possible to make a machine capable of operating upon numbers of forty, sixty, or one hundred places of figures: still, however, a limit must at last be reached, and the numbers represented would not be really _indefinite_. After lengthened consideration of this subject, the solution of the difficulty was discovered; and it presented the appearance of reasoning in a circle. Algebraical operations in their most general form cannot be carried on by machinery without the capability of expressing _indefinite_ constants. On the other hand, the only way of arriving at the expression of an indefinite constant, was through the intervention of Algebra itself. This is not a fit place to enter into the detail of the means employed, further than to observe, that it was found possible to evade the difficulty, by connecting _indefinite_ number with the _infinite in time_ instead of with the _infinite in space_. The solution of this difficulty being found, and the discovery of another principle having been made, namely—that _the nature of a function might be indicated by its position_—algebra, in all its most abstract forms, was placed completely within the reach of mechanism. § The third difficulty that presented itself was one which I had long before anticipated. It was proposed to me nearly at the same time by three of the most eminent cultivators of analysis then existing, M. Jacobi, M. Bessel, and Professor M‘Cullagh, who were examining the drawings of the Analytical Engine. The question they proposed was this:—How would the Analytical Engine be able to treat calculations in which the use of tables of logarithms, sines, &c. or any other tabular numbers should be required?