Locke remarks, "He that will not stir till he infallibly knows the business he goes about will succeed, will have but little else to do but to sit still and perish."1 There is not a moment of our lives when we do not lie under a slight danger of death, or some most terrible fate. There is not a single action of eating, drinking, sitting down, or standing up, which has not proved fatal to some person. Several philosophers have tried to assign the limit of the probabilities which we regard as zero; Buffon named, because it is the probability, practically disregarded, that a man of 56 years of age will die the next day. Pascal remarked that a man would be esteemed a fool for hesitating to accept death when three dice gave sixes twenty times running, if his reward in case of a different result was to be a crown; but as the chance of death in question is only I ÷ 660, or unity divided by a number of 47 places of figures, we may be said to incur greater risks every day for less motives. There is far greater risk of death, for instance, in a game of cricket or a visit to the rink,

Nothing is more requisite than to distinguish carefully between the truth of a theory and the truthful application of the theory to actual circumstances. As a general rule, events in nature and art will present a complexity of relations exceeding our powers of treatment. The intricate action of the mind often intervenes and renders complete analysis hopeless. If, for instance, the probability that a marksman shall hit the target in a single shot be I in 10, we might seem to have no difficulty in calculating the probability of any sucession of hits; thus the probability of three successive hits would be one in a thousand. But, in reality, the confidence and experience derived from the first successful shot would render a second success more probable. The events are not really independent, and there would generally be a far greater preponderance of runs of apparent luck, than a simple calculation of probabilities could account for. In some persons, however, a remarkable series of successes will produce a degree of excitement rendering continued success almost impossible. Attempts to apply the theory of probability to the

1 Essay concerning Human Understanding, bk. iv. ch. 14. § 1.

results of judicial proceedings have proved of little value, simply because the conditions are far too intricate. As Laplace said, "Tant de passions, d'intérêts divers et de circonstances compliquent les questions relatives à ces objets, qu'elles sont presque toujours insolubles." Men acting on a jury, or giving evidence before a court, are subject to so many complex influences that no mathematical formulas can be framed to express the real conditions. Jurymen or even judges on the bench cannot be regarded as acting independently, with a definite probability in favour of each delivering a correct judgment. Each man of the jury is more or less influenced by the opinion of the others, and there are subtle effects of character and manner and strength of mind which defy analysis. Even in physical science we can in comparatively few cases apply the theory in a definite manner, because the data required are too complicated and difficult to obtain. But such failures in no way diminish the truth and beauty of the theory itself; in reality there is no branch of science in which our symbols can cope with the complexity of Nature. Donkin said,

As

"I do not see on what ground it can be doubted that every definite state of belief concerning a proposed hypothesis, is in itself capable of being represented by a nuinerical expression, however difficult or impracticable it may be to ascertain its actual value. It would be very difficult to estimate in numbers the vis viva of all the particles of a human body at any instant; but no one doubts that it is capable of numerical expression.'

"1

The difficulty, in short, is merely relative to our knowledge and skill, and is not absolute or inherent in the subject. We must distinguish between what is theoretically conceivable and what is practicable with our present mental resources. Provided that our aspirations are pointed in a right direction, we must not allow them to be damped by the consideration that they pass beyond what can now be turned to immediate use. In spite of its immense difficulties of application, and the aspersions which have been mistakenly cast upon it, the theory of probabilities, I repeat, is the noblest, as it will in course

1 Philosophical Magazine, 4th Series, vol. i. p. 354

of time prove, perhaps the most fruitful branch of mathematical science. It is the very guide of life, and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of probabilities. In the next chapter we proceed to consider how the whole cogency of inductive reasoning rests upon probabilities. The truth or untruth of a natural law, when carefully investigated, resolves itself into a high or low degree of probability, and this is the case whether or not we are capable of producing precise numerical data.

CHAPTER XI.

PHILOSOPHY OF INDUCTIVE INFERENCE.

WE have inquired into the nature of perfect induction, whereby we pass backwards from certain observed combinations of events, to the logical conditions governing such combinations. We have also investigated the grounds of that theory of probability, which must be our guide when we leave certainty behind, and dilute knowledge with ignorance. There is now before us the difficult task of endeavouring to decide how, by the aid of that theory, we can ascend from the facts to the laws of nature; and may then with more or less success anticipate the future course of events. All our knowledge of natural objects must be ultimately derived from observation, and the difficult question arises-How can we ever know anything which we have not directly observed through one of our senses, the apertures of the mind? The utility of reasoning is to assure ourselves that, at a determinate time and place, or under specified conditions, a certain phenomenon will be observed. When we can use our senses and perceive that the phenomenon does occur, reasoning is superfluous. If the senses cannot be used, because the event is in the future, or out of reach, how can reasoning take their place? Apparently, at least, we must infer the unknown from the known, and the mind must itself create an addition to the sum of knowledge. But I hold that it is quite impossible to make any real additions to the contents of our knowledge, except through new impressions upon the senses, or upon some seat of feeling. I shall

attempt to show that inference, whether inductive or deductive, is never more than an unfolding of the contents of our experience, and that it always proceeds upon the assumption that the future and the unperceived will be governed by the same conditions as the past and the perceived, an assumption which will often prove to be mistaken.

In inductive as in deductive reasoning the conclusion never passes beyond the premises. Reasoning adds no more to the implicit contents of our knowledge, than the arrangement of the specimens in a museum adds to the number of those specimens. Arrangement adds to our knowledge in a certain sense: it allows us to perceive the similarities and peculiarities of the specimens, and on the assumption that the museum is an adequate representation of nature, it enables us to judge of the prevailing forms of natural objects. Bacon's first aphorism holds perfectly true, that man knows nothing but what he has observed, provided that we include his whole sources of experience, and the whole implicit contents of his knowledge. Inference but unfolds the hidden meaning of our observations, and the theory of probability shows how far we go beyond our data in assuming that new specimens will resemble the old ones, or that the future may be regarded as proceeding uniformly with the past.

Various Classes of Inductive Truths.

It will be desirable, in the first place, to distinguish between the several kinds of truths which we endeavour to establish by induction. Although there is a certain common and universal element in all our processes of reasoning, yet diversity arises in their application. Similarity of condition between the events from which we argue, and those to which we argue, must always be the ground of inference; but this similarity may have regard either to time or place, or the simple logical combination of events, or to any conceivable junction of circumstances involving quality, time, and place. Having met with many pieces of substance possessing ductility and a bright yellow colour, and having discovered, by perfect induction, that they all possess a high specific