association, or any rude process of inference, infer the future directly from the past, it is the work of logic to analyse the conditions on which the correctness of this inference depends. Even Mr. Mill would admit that such analysis involves the consideration of general truths", and in this, as in several other important points, we might controvert Mr. Mill's own views by his own statements.

On the Grounds of Inductive Inference.

I hold that, in all cases of inductive inference, we must invent hypotheses, until we fall upon some hypothesis which yields deductive results in accordance with experience. Such accordance renders the chosen hypothesis more or less probable, and we may then deduce, with some degree of likelihood, the nature of our future experience, on the assumption that no arbitrary change takes place in the conditions of nature. We can only argue from the past to the future, on the general principle set forth in the commencement of this work, that what is true of a thing will be true of the like. So far then as one object or event differs from another, all inference is impossible; particulars as particulars can no more make an inference than grains of sand can make a rope. We must always rise to something which is general or same in the cases, and assuming that sameness to be extended to new cases we learn their nature. Hearing a clock tick five thousand times without exception or variation, we adopt the very probable hypothesis that there is some invariably acting machine which produces those uniform sounds, and which will, in the absence of change, go on producing them. Meeting twenty times with a bright yellow ductile substance, and finding it to be always very heavy and incorrodible, I infer that there was some natural condition,

n System of Logic,' bk. II. chap. iii. § 5. pp. 225, &c.

which tended, in the creation of things, to associate these properties together, and I expect to find them associated in the next instance. But there always is the possibility that some unknown change may take place between past and future cases. The clock may run down, or be subject to any one of a hundred accidents altering its condition. There is no reason in the nature of things, so far as known to us, why yellow colour, ductility, high specific gravity, and incorrodibility, should always be associated together; and in other like cases, if not in this, men's expectations have been deceived. Our inferences, therefore, always retain more or less of a hypothetical character, and are so far open to doubt. Only in proportion as our induction approximates to the character of perfect induction, does it approximate to certainty. The amount of uncertainty corresponds to the probability that other objects than those examined, may exist and falsify our inferences; the amount of probability corresponds to the amount of information yielded by our examination; and the theory of probability will be needed to prevent our over-estimating or under-estimating the knowledge we possess.

Illustrations of the Inductive Process.

To illustrate the passage from the known to the apparently unknown, let us suppose that the phenomena under investigation consist of numbers, and that the following six numbers being exhibited to us, we are required to infer the character of the next in the series:

5, 15, 35, 45, 65, 95.

The question first of all arises, How may we describe this series of numbers? What is uniformly true of them? The reader cannot fail to perceive at the first glance that they all end in five, and the problem is, from the proper

principles of number, no one can show that numbers ending in 7 should be primes.

From the history of the theory of numbers some good examples of false induction can be adduced. Taking the following series of prime numbers

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c., it will be found that they all agree in being values of the general expression x2+x+41, putting for x in succession the values, 0, 1, 2, 3, 4, &c. We thus seem always to obtain a prime number, and the induction is apparently very strong, to the effect that this expression always will give primes. Yet a few more trials will disprove this false conclusion. Put x = 40, and we obtain 40 × 40 + 40 + 41, or 41 x 41. Now such a failure could never have happened, had we shown any deductive reason why x2 + x + 41 should give primes.

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. Mr. Babbage admirably 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 sure knowledge of even one other instance. General mathematical theorems have indeed been dis

covered 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°, 11', 112, 113, 114; that is, in the first I; 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.° 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

It is pretty evident, from this most interesting statement, that Newton having simply observed the succession of the numbers, tried various formulæ until he found one which agreed with them all. He was so little satisfied with this process, however, that he verified particular results of his new theorem by comparison with the results of common multiplication, and the rule for the extraction of the square root. Newton, in fact, gave no demonstration of his theorem; and a number of the first mathematicians of the last century, James Bernouilli, Maclaurin, Landen, Euler, Lagrange, &c., occupied themselves with discovering a conclusive method of deductive proof.

o These are the figurate numbers considered in pages 206-216.

p 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.