“The law which seemed at first to govern this series failed at the hundred million and second term. This term is larger than we expected by 10,000. The next term is larger than was anticipated by 30,000, and the excess of each term above what we had expected forms the following table:— 10,000 30,000 60,000 100,000 150,000 . . . . . . being, in fact, the series of _triangular numbers_, {208} each multiplied by 10,000. “If we now continue to observe the numbers presented by the wheel, we shall find, that for a hundred, or even for a thousand terms, they continue to follow the new law relating to the triangular numbers; but after watching them for 2761 terms, we find that this law fails in the case of the 2762d term. “If we continue to observe, we shall discover another law then coming into action, which also is dependent, but in a different manner, on triangular numbers. This will continue through about 1430 terms, when a new law is again introduced which extends over about 950 terms, and this, too, like all its predecessors, fails, and gives place to other laws, which appear at different intervals. “Now it must be observed that _the law that each number presented by the engine is greater by unity than the preceding number_, which law the observer had deduced from an induction of a hundred million instances, _was not the true law that regulated its action_, and that the occurrence of the number 100,010,002 at the 100,000,002nd term was _as necessary a consequence of the original adjustment_, _and might have been as fully foreknown at the commencement_, _as was the regular succession of any one of the intermediate numbers to its immediate antecedent_. The same remark applies to the next apparent deviation from the new law, which was founded on an induction of 2761 terms, and also to the succeeding law, with this limitation only—that, whilst their consecutive introduction at various definite intervals, is a necessary consequence of the mechanical structure of the engine, our knowledge of analysis does not enable us to predict the periods themselves at which the more distant laws will be introduced.”