There can be no doubt that what here happens with forty instances, might happen with forty thousand or forty million instances. An apparent law never once failing up to a certain point may then suddenly break down, so that inductive reasoning, as it has been described by some writers, can give no sure knowledge of what is to come. Babbage pointed out, in his Ninth Bridgewater Treatise, that a machine could be constructed to give a perfectly regular series of numbers through a vast series of steps, and yet to break the law of progression suddenly at any required point. No number of particular cases as particulars enables us to pass by inference to any new case. It is hardly needful to inquire here what can be inferred from an infinite series of facts, because they are never practically within our power; but we may unhesitatingly accept the conclusion, that no finite number of instances can ever prove a general law, or can give us certain knowledge of even one other instance. General mathematical theorems have indeed been discovered by the observation of particular cases, and may again be so discovered. We have Newton’s own statement, to the effect that he was thus led to the all-important Binomial Theorem, the basis of the whole structure of mathematical analysis. Speaking of a certain series of terms, expressing the area of a circle or hyperbola, he says: “I reflected that the denominators were in arithmetical progression; so that only the numerical co-efficients of the numerators remained to be investigated. But these, in the alternate areas, were the figures of the powers of the number eleven, namely 11^{0}, 11^{1}, 11^{2}, 11^{3}, 11^{4}; that is, in the first 1; in the second 1, 1; in the third 1, 2, 1; in the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.[138] I inquired, therefore, in what manner all the remaining figures could be found from the first two; and I found that if the first figure be called *m*, all the rest could be found by the continual multiplication of the terms of the formula ((*m* - 0)/1) × ((*m* - 1)/2) × ((*m* - 2)/3) × ((*m* - 3)/4) × &c.”[139] [138] These are the figurate numbers considered in pages 183, 187, &c. [139] *Commercium Epistolicum.* *Epistola ad Oldenburgum*, Oct. 24, 1676. Horsley’s *Works of Newton*, vol. iv. p. 541. See De Morgan in *Penny Cyclopædia*, art. “Binomial Theorem,” p. 412.