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The theory of Probabilities has been characterised by Laplace, one of those who have contributed most largely to its advance, as 'good sense reduced to a system of calculation;' and such, no doubt, it is. But it must be especially noticed that there is hardly any subject to which thought can be applied, which calls for so continuous an application of that excellent quality, or in which it is easier to make mistakes from simple want of circumspection. And, moreover, that its reduction to calculation is attended with difficulties of a very peculiar nature, such as occur in no other application of mathematical analysis to practical subjects, arising out of the great magnitudes of the numbers concerned, which defeat the ordinary processes of arithmetical and logarithmic calculation, by exhausting the patience of the computer, and require special methods of approximate evaluation to bring them within the compass of human industry, These methods form a conspicuous feature of the general subject, and have furnished scope for very extraordinary displays of mathematical talent and invention. That very large numbers will inevitably be concerned in questions where numerous and independent contingencies may take place, and in any order or mode of combination, will be apparent to any one who considers the astonishing fecundity of such combinations numerically estimated, when the combining elements are many. For example, the number of possible hands’ at whist (regard being had to the trump) is 1,270,027,119,200.
The calculus of Probabilities, under the less creditable name of the doctrine of Chances, originated at the gaming table; and was for a long time confined to estimating the chances of success and failure in throws of dice, combinations of cards, and drawings of lotteries. It has since effectually obliterated the stain on its cradle, as there is no monitor more severe, no lecture which can be delivered on the certain ruin which attends habitual gambling more emphatic than may be found in its demonstrations. Questions of this kind, it is true, are still retained in treatises on the subject; nor indeed can they be conveniently dispensed with, since they furnish the simplest and readiest illustrations of the combination of independent events, and the superposition of contingencies arising out of them, which belong essentially to its principles. They, however, form a very insignificant part of its applications, in comparison with the problems which its scope at present takes in, and which its modern developements have enabled it to handle.
Its first advances towards the dignity of a distinct branch of Mathematics are attributable to the celebrated Blaise Pascal, and his no less celebrated contemporary and correspondent
Fermat, - both reasoners of extraordinary acuteness, and who seem to have been specially attracted (like many of their fol. lowers) by the close reasoning and careful analysis its problems demand for their successful issue. Subsequent to these, but still among its earlier contributors, we find the distinguished names of Huyghens (to whom we owe the first treatise on the subject), those of the Grand Pensionary De Witt, Hudde, and Halley (with whom originated its application to the probabilities of life and the construction of tables of mortality), and that of James Bernouilli, who may be considered the first philosophical writer on the subject. To him we owe the demonstration of two great fundamental theorems or laws of Probability, as applied to the results of very numerous trials of any proposed species of contingency: viz. lst, that in any vast number of trials there is a demonstrably greater probability that the events will happen in numbers proportional to their respective chances in a single trial, than in any other specified proportion ; and, 2dly, That a number of trials may always be assigned so great, as to make the probability of the events happening in numbers falling within any assigned limits of deviation from that proportion, however narrow, approach to certainty as nearly as we please. The first of these propositions has the air of a truism, when the meaning of its terms is not nicely weighed. But the second is obviously of paramount importance; since it goes to take the totality of results obtained in any sufficiently extensive series of trials, almost out of the domain of chance, and to place in evidence the influence of any 'cause' or circumstantial condition common to the whole series, which may give even a trifling preponderance of facility to any one of the classes of events contemplated over the rest.
Common sense, it may perhaps be said, would tell us as much as this. No doubt it might suggest some such propositions as likely enough to be true; and the usual course of inductive reasoning up to causes tacitly assumes their truth. But when we come to demand what number of trials may reasonably be expected to bring out into prominence a very small given preponderance of facility ? or to declare within what limits of accuracy such preponderance may reasonably be expected to be represented on the upshot or final average of a given number of trials ? — or, lastly, what is the probability that on a given number of trials such an average will represent the preponderant facility in question within given limits of exactness ? — all of them, and especially the last, evidently practical questions of much interest; we find ourselves forced to appeal from the unaided judgment of simple good sense, to strict numerical calculation, — taking for its basis not a mere aperçu, but a rigorous demonstration of the truth of the propositions above stated. This is very much the case with all the more important conclusions of this theory ; when generally enunciated, they are almost universally seen to be pretty plainly conformable to ordinary clear judging apprehension of their relations. Even the apparently paradoxical conclusions by which we are occasionally startled, lose that aspect when their exact wording is duly attended to, and all the conditions implied in it clearly apprehended. It is their applicability to exact computation, and the means they afford thereby to precise determinations useful in practice, which give them all their value.
Problems of the class abovementioned were first successfully treated by De Moivre, to whom also we owe the happy idea of applying Stirling's theorem to approximate to the ratio of the high numbers which enter into such calculations, without which they would be impracticable. From these it would appear but a small step to pass to what may be deemed the inverse calculus of Probabilities, which applies the knowledge gained by the observation of past events to the prediction of future, by concluding from the succession of facts observed the respective degrees of probability of the existence of each out of several equipossible determining conditions, and thence starting as it were anew, and ascertaining from the knowledge thus acquired the probability of an event or events similarly determined in futuro. It was reserved, however, for another member of the gifted family of Bernouilli to make this step, which has in some respects changed the whole aspect of the subject, and given to it that degree of importance it possesses as an auxiliary of the inductive philosophy.
It may perhaps be doubted whether subsequent writers have added very materially to the intrinsic philosophy of the subject, though there can be no hesitation as to the value of the improvements they have made in its methods of procedure, whether in point of elegance or power; the extension given to its formulæ; or the numerous and important applications made of its principles, especially in those cases (which comprise almost all the really interesting ones) where the transition has to be made from the finite to the infinite, from the limited though often large number of possible combinations which its simple and more elementary problems offer, to the literally infinite multitude which the gradation of natural causes and influences obliges us to consider, and which call for the perpetual employment of the most refined theories, and the most delicate and abstruse applications of the integral calculus. In all these 1850.
History of the Subject.
respects the great work of Laplace (* Théorie Analytique des • Probabilités”) stands deservedly preeminent; occupying in this department of science the same rank and position which the
Mécanique Analytique' of his illustrious rival Lagrange holds in that of force and motion, and marking (we had almost said) the ne plus ultra of mathematical skill and power. So completely has this sublime work been held to embody the subject in its utmost extent, and to satisfy every want of the theorist, that an interval of a quarter of a century elapsed from the date of its appearance (1812) before any further original contribution of moment was made to the theory. The valuable memoir of Poisson, published in 1837, on the probability of judicial decisions * (which contains a resumé of the whole theory of Probabilities), though admirable for its clear exposition of principles and elegant analysis, can hardly be said to have carried the general subject much beyond the point where Laplace left it.
It may easily be imagined that a work like this of Laplace, followed at a short interval by an admirable exposé of its contents by himself (* Essai Philosophique sur les Prob.'), could not fail to make a lively impression and to excite general attention. Laplace possessed in an eminent degree the talent of stating the most profound results of his own geometry in a style at once philosophical, luminous, and pleasing. Few works have been more extensively read or more generally appreciated than this Essay and that on the Système du Monde' by the same author. There is in both a breadth and simple dignity corresponding to the greatness of the subjects treated of, a loftiness of style, the direct result of generality of conception, and which is felt as adding to rather than detracting from clearness of statement, and a masterly treatment which fascinates the attention of every reader. Nowhere can be found so great a body of important discoveries, so consecutively enchained, and so distinctly and impressively announced. It is not, perhaps, too much to say, that were all the literature of Europe, these two Essays excepted, to perish, they would suffice to convey to the latest posterity an impression of the intellectual greatness of the age which could produce them, surpassing that afforded by all the monuments antiquity has left us.
Previous to the publicatiou of the Essai Philosophique,' few except professed mathematicians or persons conversant with insurances and similar commercial risks, possessed any know
* Recherches sur la Probabilité des Jugemens en Matière Criminelle et en Matière Civile; précédées des Règles Générales du Calcul des Probabilités. Paris, 1837.
ledge of the principles of this calculus, or troubled themselves about its conclusions, — regarding them as merely curious, and perhaps not altogether harmless speculations. Thenceforward, however, apathy was speedily exchanged for a lively and increasing desire to know something of a system of reasoning which for the first time seemed to afford a handle for some kind of exact inquiry into matters no one had ever expected to see reduced to calculation and bearing on the most important concerns of life. Men began to hear with surprise, not unmingled with some vague hope of ultimate benefit, that not only births, deaths, and marriages, but the decisions of tribunals, the results of popular elections, the influence of punishments in checking crime — the comparative value of medical remedies, and different modes of treatment of diseases — the probable limits of error in numerical results in every department of physical inquiry — the detection of causes physical, social, and moral, nay, even the weight of evidence, and the validity of logical argument — might come to be surveyed with that lynx-eyed scrutiny of a dispassionate analysis, which, if not at once leading to the discovery of positive truth, would at least secure the detection and proscription of many mischievous and besetting fallacies. Hence a demand for elementary treatises and popular exposition of principles, which has been liberally answered.
Among the valuable works of this kind in the French and English languages which have appeared since the epoch in question, we may notice more especially Lacroix's · Traité
Elementaire du Calcul des Probabilités; Paris, 1822,' and the several encyclopædic essays and articles on the subject by Sir John Lubbock and Mr. Drinkwater (Bethune), in the Library of Useful Kowledge, by Mr. Galloway in the Encyclopædia Britannica, (since published separately in a small and compendious form - a work of great merit and utility), and by Mr. De Morgan, in the Encyclopædia Metropolitana. To the last-mentioned treatise, as well as to two admirable chapters on the subject in the recent elaborate work by the same author on the Formal Logic, we may refer as containing, par excellence, the clearest views of the métaphysique of the subject, and the most satisfactory analysis of the state of the mind as to belief or disbelief, and the degree of assurance afforded by the conclusions of the calculus in cases where the data themselves are vague and uncertain, which can any where be found. All or any of these works will afford the English student a perfect insight into the mathematical treatment and reasonings of the subject, and consequently serve as an abundant preparation for the study and mastery of Laplace's great work ;- but we would
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