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point, as steady, as the soberest of business men in the most old-fashioned of cities could reasonably desire.

It will naturally be asked, What are the conditions upon which this certainty and security of anticipation depend? We would lead up to the reply to this question by asking another; on what conditions does certainty depend in any other department of thought and practice? The answer must be, simply and entirely, on regularity in the phenomena; on law, as it is commonly expressed. Where regularity can be traced among phenomena, there can inferences be drawn : where no regularity can be detected, there we have nothing before us but guesswork. Regularity on the part of the phenomena, and potential certainty about them on our part, are absolutely coextensive and correspondent. We need not pause to define what we here mean by regularity, since every one has a sufficient rough conception of it for our present purpose. We may, however, just point out, that within the sphere of ordinary reasoning it presents itself in practice in two different ways. Sometimes we find two instances or groups of phenomena which resemble one another so closely in their details, that we may reason directly from one to the other. Here we are said to employ Induction. For instance, a man performs a chemical experiment. He mixes a certain proportion of oxygen and hydrogen in a vessel, and then sets fire to it. The result is an explosion, and the production of a certain quantity of water. Here we know that we can do the same thing over again; that is, that we can reproduce precisely the same group of phenomena whenever we please. Hence we feel perfect confidence that the same result as before will follow. But there are a great many cases in which we never have an opportunity of finding any thing approaching to the same precise group of phenomena occurring even twice over, and therefore we are not able to argue from the similarity of one to the other. Take for instance an eclipse of the sun. The entire history of our race on earth would not furnish range enough for the occurrence of even two eclipses which could be considered “alike” according to the standard of precision demanded by modern astronomy. No doubt we may express our conception of uniformity here in a hypothetical form, and say that if two cases occurred in which the sun and moon and earth

case.

occupied exactly the same relative positions, we should experi. ence an eclipse of the same approach to totality prevailing over the same area of the earth's surface: And there is no harm in illustrating, as is often done in works on inductive logic, our conception of the uniformity of nature by such an imaginary

But such cases do not occur, and therefore we do not argue directly from similar examples. As almost every one knows, the astronomer starts from certain broad generalizations, first established by Newton, and deduces his conclusion by a train of reasoning from these. That is, he argues deductively.

The above examples are not of course intended as a philosophical account of the distinction between induction and deduction, but they will serve to indicate to the reader, in a general way, the distinction between the two classes of cases in which these modes of reasoning are employed. Now when we come to questions of chance, both of these modes, in their commonly understood application, fail entirely. In the first place, we can never find two cases alike. In such a simple instance as that of the tossing of a coin: no doubt, if we held it twice successively in rigidly the same position, and projected it with precisely the same velocity of rotation and translation, it would always yield the same face. But this cannot be done; if it could, coins would not be tossed any more to decide questions of choice. Nor have we, again, any generalizations which are known to be trustworthy in every case, and to which, there. fore, appeal can be confidently made. No doubt there are such, lying deep down under the phenomena, but their applications and developments are so infinitely complex that they can never be appealed to. Here again we may say that if they could be determined and employed, all occasion for them would at once cease, at least as regards games of chance ; for, when both sides could anticipate the result with certainty, one side would at once see its disadvantage, and decline the game or pursuit.

What, then, exactly is the uniformity or regularity which pervades the region of chance? for, as we have already seen, enormous fortunes are steadily earned by safe prevision in this region, and trustworthy prevision must necessarily be built upon the foundation of objective regularity. This will need a

little discussion, for the clear and adequate appreciation of this kind of uniformity may be regarded as the great logical achievement of modern times. It is like the discovery of a new continent, since it is not merely the extension of old methods over adjacent territories, but rather the discovery of a new method. It is the acquisition, for purposes of inference, of a region where no possibility of inference was formerly known to exist. This new conception is, in a word, that of average regularity-of regularity in the long-run combined with perfect irregularity in the details. Take the throw of a die. No human being knows, or is ever likely to know, what number will turn up on any specified occasion. From knowledge of such details we are as absolutely debarred as we are from knowledge of the nature of the country at any specified point on the hinder side of the moon's surface. But here comes in a difference: whereas in the latter case the uncertainty of detail develops into no kind of aggregate certainty (for we are just as much in the dark about the nature of the whole surface as about that of any part of it) in the latter case, when we begin to extend our observation, a very striking and important kind of regularity begins to emerge. Take a score of throws of the die, and we are not quite as uncertain as we were about one. Take a few hundreds, and the uncertainty begins to give way to tolerably strong conviction. Take a few thousands, and the chaos which might have been possibly anticipated, is seen to be replaced by a very marked order. Out of six thousand throws, we know that about one thousand will yield ace, and so with the other numbers.

This characteristic is not by any means confined to games of chance. Our only reason for thinking more of it in that connection lies in the fact that no other kind of regularity is to be found there. But the slightest observation will detect it, in some direction or other, in almost every class of phenomena. It is found in pursuits, where skill and chance go together; the skill directing the general aim, and the chance showing itself in a multitude of more or less minute and incalculable disturbances. Instances of this are furnished by rifle-shooting, sextant observations, and so on; in fact, by any kind of practice or observation in which instruments have to be employed. The single result is, within certain limits, more or less narrow, absolutely indeterminable; but when we multiply these single results we find them grouping themselves with a systematic and progressive regularity. So again with affairs which turn almost entirely upon the will and choice of man. The number of crimes such as suicides, of marriages, etc., show the same description of regularity. No one can say exactly when and where any one such event will occur, but any one who appeals to statistics can say when and where averages will occur. And to add one more class of instances, that furnished by returns of mortality. Here we have a class of cases where human will has but little causative influence, at least directly. But it is only necessary to mention this class for every one to bring into immediate mental conjunction the proverbial uncertainty of the lives of individuals and families, and the well-known certainty of the average duration of life in large towns or States.

Average regularity, then, or regularity which gradually asserts itself after many repetitions, is the thing to be expected. But in saying this we must insist upon a very needful caution. The general conditions, or broad determining forces of the phenomena in question, must remain unchanged. Every one, for instance, can see what would happen if we were to make use of a die which, after long use and consequent wear, gradually began to change its shape and become irregular. This might happen, for instance, if it were cut out of a piece of chalk which was harder on one side than the other. It is clear that we should then have to put some limit upon the length of experience wherein we were to seek our average. This would to a certain extent place us between two opposed difficulties. We must, on the one hand, from the nature and characteristics of the phenomena in question, insist upon a great many throws to give the ace its fair chance of one in six. But if, on the other hand, we go too far in this direction and admit an enormous number of throws, the tendency to obtain one in six might gradually be found to undergo a change. What is called a bias might be seen to be setting in, so that the ace would get more or less than its fair average, according to the way in which the shape of the die had undergone alteration. Just as the ultimate average could not show itself unless we went on long enough, so it would be lost again if we were to go on too long.

In their physical application to such a thing as a die, the above remarks are obvious enough; but when we consider their corresponding application to moral and social agencies, we shall find that they are often overlooked. Statisticians are sometimes charged, and occasionally with justice, with regarding the great social forces under which men act as permanently unchangeable. Some of the determining influences are doubtless subject to little or no variation; for instance, the most fundamental characteristics of the human body and mind must be regarded as practically fixed. But others are distinctly changeable, and, what is most to the point, changeable by human agency. Mr. Buckle, for instance, in his well-known work on the History of Civilization, roused a good deal of very natural obloquy by his remarks on the regularity of the number of suicides in London. The popular and unphilosophical view used doubtless at one time to regard any trustworthy anticipations upon such matters as these as out of the question, for it well knew the mysterious and complicated links of misery or despair and of human will contending against these, through which each individual suicide was actually brought about. The statistician, however, found that this was not so; he was able to establish beyond all doubt that about the same number of such crimes was actually committed year after year. But when, by a reaction against the popular view, Buckle went on to speak as if this annual number was fixed by a sort of sate, against which all moral and religious efforts would be found in vain, he sell into a serious error. We may be baffled in our struggle with any given individual, but by a judicious alteration of their general surroundings we may succeed in making our efforts tell on the average. No really permanent average can ever be looked for where human actions are concerned. To speak metaphorically, the dice in this case change their shape but very slowly. But they do change, and therefore no predictions will hold good which refer to times or places very remote from the present.

The above caution is the more necessary owing to the fact that the pure theory of Chance contemplates an indefinite suc

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