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in regular progressive groups, with a view to derive from it numerically the only things which it is really important to know, viz. the most probable value, the probable error of a single determination, and the weight of the result as compared with that similarly derived from a different and independent series. But when the data are otherwise grouped, which is a case by no means of unfrequent occurrence, or when a portion only is regularly arranged in groups, and all above or below certain limits massed together in the gross without regard to grouping, much delicacy subsists in deciding, according to just principles, on the exact amount of all these elements; and it would have added much to the practical utility and value of M. Quetelet's work had he given some examples of this nature, with plain and brief rules or formulæ for their working. This is the more to be regretted, because we are actually left at a loss to decide by what numerical process his mean results, where stated, have been arrived at in some of the examples set down. For instance, in that of the Scotch soldiers, where all the groups are regular and all stated, we find it merely mentioned incidentally that the mean is a little more than 40 inches,' whereas the really most probable mean is 39.830, while that which the course of the figures in the tabulated working of the example would appear to indicate as resulting from an equipartition of the numbers of cases in excess and defect is 39.525. Again, in the example of the conscripts, where the extreme groups are massed undistinguishably, the rule of equipartition, according to its simplest and most obvious application to the tabulated figures, would place the mean at 63.939 inches, whereas we find it indicated, rather than stated, as follows: “If it be observed that the mean
height is about 63.947 inches.' The difference, it is true, is trifling in itself, but becomes of consequence when the object is from the figures set down to discover by what process they have been obtained.
We come now, however, to that highly interesting part of the work before us which treats of the study of causes: in general, and in the peculiarly complex form it assumes, in those moral and social inquiries, the data for which are gathered by statistical enumeration. A few remarks on the part which the theory of Probabilities plays in these inquiries will not be out of place here.
This theory is connected with the general philosophy of causation and with inductive inquiry in two distinct ways — the one theoretical and the other practical. When we see an event happen several times in succession in some particular manner, there arises, in the first place, a primâ facie probability that it will happen once more in that manner; which, if the 1850.
Study of Causes.
Elimination of Chance.
number of repetitions be large, forms of itself a very cogent ground of expectation. But the probability that such repetition has not been merely fortuitous, but has resulted from a determining, or at least a biassing cause, increases with each repetition in a far higher ratio, than the simple probability of the once more happening of the event itself. The distinction is that between a geometrical and an arithmetical progression. Thus, for example, the expectation that the sun will rise to-morrow, grounded on the sole observation of the fact of its having risen a million times in unbroken succession, has a million to one in its favour. But to estimate the probability, drawn from that observation, of the existence of an influential cause for the phenomenon of a daily sunrise, we have to raise the number 2 to the millionth power-thus producing a number inexpressible in words and inconceivable in thought, and the ratio of this enormous number to unity, is that of the probability of the phenomenon having happened by cause, to that of its having happened by chance. The theorem on which depends this curious application of the doctrine of probabilities to the expulsion from philosophy of the idea of chance, is known to Geometers by the name of its first promulgator, Bayes. It must be observed, that as to the nature of the cause thus insisted on, the calculus says nothing. There may be opposing causes, and a daily struggle between them for the mastery. In this case we are simply forced to admit that the arrangements of Nature are highly favourable to the successful exertion of the one, and highly unfavourable to the other.
It is however as a practical auxiliary of the inductive philosophy that we have chiefly to contemplate this theory. Its use as such depends on that mutual destruction of accidental deviations from the regular results of permanent causes which always takes place when very numerous instances are brought into comparison. Examples of this sort have been already adduced and might be multiplied indefinitely in every department of practical inquiry. Indeed every phenomenon which Nature offers on the great scale may be regarded as such. Nothing can be more irregular and uncertain than the action of the wind on the waters, yet, in the most violent storms, the general surface of the ocean preserves its level. What more fortuitous than the fall of a drop of rain in a shower or the growth of a blade of grass? Yet the soil is uniformly irrigated, and the unbroken sheet of verdure testifies to the resultant equilibrium of that and a thousand other causes of inequality. These things, it will perhaps be said, are the results of Providential arrangement. No doubt they are so; but it is an arrange
ment working through a complication of secondary causes and contingencies, -on which man, if he will, philosophise at all, is obliged to do it by reference to the laws of probability. Still there is no one who is not astonished, in cases where what we are obliged to call contingency enters largely, to find not only that the mean results of several series of trials agree in a wonderfully exact manner with each other, but that the very errors of individual trials — precisely those portions of the special results which are purely attributable to that which is contingent in the process — group themselves around the mean with a regularity which would appear to be the effect of deliberate intention.
This singular result? (says M. Quetelet) “always astonishes persons unfamiliar with this kind of research. How, in fact, can it be believed that errors and inaccuracies are committed with the same regularity as a series of events whose order is calculated in advance ? There is something mysterious, which however ceases to surprise when we examine things more closely.'
The rationale of this mystery is this. Where the number of accidental causes of deviation is great, and the maximum effect of each separately minute in comparison of the result we seek to determine, - great total deviations can only arise from the conspiring of many of these small causes in one direction — the more that so conspire, the greater the deviation. Now all combinations being equally possible individually, and those combinations which can alone give rise to the extremes of error being necessarily very much fewer in number than those which result in moderate amounts of deviation, we easily perceive that the opportunities for the occurrence of great errors are much rarer than for small ones. And this is in fact the reasoning, which, carried out by exact analysis (assimilating the causes of plus and minus error to black and white balls in an urn), takes the form of that demonstration of the law of probability, which we have above spoken of as devised by Laplace and simplified to the utmost by M. Quetelet.
There still remains behind, however, this inquiry, — which we have known to occur as a difficulty to intellects of the first order, — Why do events, on the long run, conform to the laws of probability ? What is the cause of this phenomenon as a matter of fact? We reply (and the reply is no mere verbal subtlety), that events do not so conform themselves, the fact to the imagination, the real to the ideal,- but that the laws of probability, as acknowledged by us, are framed in hypothetical accordance with events. To take the simplest case, that of a single contingency,- the drawing of one of two balls, a black and a white. We suppose the chances equal, in theory;
Causes disclosed by preponderant Tendencies.
but, in practice, what is to assure us that they are so ? The perfect similarity of the balls ? But they need not be similar in any one quality but such as may influence their coming to hand. And, on the other side, the most perfect similarity in all visible, tangible, or other physical qualities cognisable to our tests is not such a similarity as we contemplate in theory, if there remain inherent in them, but undiscernible by us, any such difference as shall tend to bring one more readily to hand than the other. The ultimate test, then, of their similarity in that sense is not their general resemblance, but their verification of the rule of coming equally often to hand in an immense number of trials: and the observed fact, that events do happen according to their calculated chances, only shows that apparent similarities are very often real ones.
The application of this calculus to the detection of causes turns essentially upon this view of the conformity in question, and of the nature and delicacy of this test by indefinite multiplication of trials which we are enabled, in many cases, to apply to mixed phenomena. All experience tells us, that where efficient causes are known, but from the complication of circumstances cannot be followed out into their specific results, we may yet often discern plainly enough their tendencies, and that these tendencies do result, in the long run, in producing a preponderance of events in their favour. Were it asked, Why do the strong men, in a general scramble, carry off the spoil, and the weak get nothing? the reply would be, that such is not the fact in every instance; that, although we cannot go fully into the dynamics of the matter, we can clearly see the mode of action in some individual struggles, and that in the whole affair there is a visible enough tendency to the defeat of the weaker party. Again, when we reverse this process of reasoning, and declare our conviction that success in the long run is a proof of ability, we give this name to some personal quality or assemblage of qualities which, acting as an efficient cause through a complication of events we do not pretend to penetrate, has a tendency in that direction which issues in success. Here the tendency becomes known by observation, and the nature of the cause is concluded from the nature of the tendency, hy appeal to experience, which, in some instances, has shown us the cause in action, and informed us of its direct effect. But it may happen that observation may plainly enough indicate the direction of a tendency which yet experience has not enabled us to connect with any known cause. And it may further happen that this tendency, which we are driven to substitute in our language for its efficient cause, may be so feeble — whether owing to the feebleness of the unknown cause, its counteraction by others, or the few and disadvantageous opportunities afforded for its efficacious action (general words, framed to convey the indistinctness of our view of the matter)—as not to become known to us but by long and careful observation, and by noting a preponderance of results in one direction rather than another.
And thus we are led to perceive the true, and, we may add, the only office of this theory in the research of causes. Properly speaking, it discloses, not causes, but tendencies, working through opportunities, — which it is the business of an ulterior philosophy to connect with efficient or formal causes; and having disclosed them, it enables us to pronounce with decision, on the evidence of the numbers adduced, respecting the reliance to be placed on such indications, — the degree of assurance they afford us that we have come upon the traces of some deeplyseated cause, — and the precision with which the intensity of the tendency itself may be appreciated.
Such tendencies are often apparent enough, without any refined considerations, or reference to any calculus. Thus, on the consideration of thirteen instances of coincidence between the direction of circular polarisation in rock crystal, with that of certain oblique faces in its crystalline form, - it was asserted that the phenomena were connected in that invariable manner which is one of the characters of efficient causation. The chances against such a coincidence happening thirteen times in succession by mere accident are more than 8000 to l; and this, therefore, was the probability that some law of nature, some cause, was concerned. Subsequent observation has brought forward no exception; but, on the contrary, other cases of a similar character have arisen, which go to place the observed tendency in uncounteracted connexion with the efficient cause which, however, still remains concealed. So again, an examination of the elements of all known cometary orbits has disclosed a tendency to direct or easterly motion, increasing in the degree of its prominence with the approach to coincidence of the orbit with the plane of the ecliptic, - and especially marked in the cases where calculation has assigned elliptic elements to the orbit. Here we have a tendency pointing to a cause, still unknown, but with whose effects we are so far familiar that we can trace its action throughout the planetary system, with only two known exceptions among its most remote and insignificant constituents.
It is, however, the extreme delicacy of the test above spoken of — that property it possesses of bringing out into salience and placing in indisputable evidence, by sufficient multiplication of