stated and explained, are calculated to interest the student, and to satisfy his inquiries. In contemplating the gradual advancement of science and the certain and increased benefits which it is conferring upon man, multiplying his social comforts, diminishing his labours, and affording both leisure and inducements to moral and intellectual culture, great cause certainly exists for high hope and expectation. An indirect influence which this state of things exerts, is to equalize and to bring in more intimate brotherhood, the members of the society in which it exists. The humblest cottager now enjoys the pleasures of life, to a degree that kings barbaric, whose pearls and gold are but a mere display of petty superiority of power, could not effect. The infant may lisp truths more profound than sage philosophers afore-time ever imagined. Mankind thus generally advance; the barriers which factitious and fortuitous merit may have interposed between individuals, have given way; and the dependence of every one upon others, in a greater or less degree, being thus secured, the bonds of social feeling have been drawn closer and more close around the great human family. REVIEW OF THE DOCTRINE OF PROBABILITIES. 1. From the American Quarterly Review for June 1832. Library of Useful Knowledge. On Probability. Parts I. and II. London: 1830. 2. Report from the Select Committee on Life Annuities. Ordered, by the House of Commons, to be printed; 4th June, 1829. 3. Report of John Finlaison, Actuary of the National Debt, on the Evidence and Elementary Facts on which the Tables of Life Annuities are founded. Ordered, by the House of Commons, to be printed; 31st March, 1829. 4. A comparative view of the various Institutions for the Assurance of Lives. By CHARLES BABBAge Esq. M. A. etc., etc., etc. London, 1826. EVER since Pascal devoted the powers of his original mind to the subject of probabilities, it has been growing into a science of the greatest exactness. The keen shaft of ridicule which he hurled with such admirable skill, in his Lettres Provinciales, against the casuistic dogma, that the opinion of a theological doctor determined the existence of a probability, will ever be remembered as among the happiest efforts of his satirical pen; while the satisfactory answers which he gave to the questions propounded to him by de Meré, will fully justify the use which he made of this powerful instrument in the ascertainment of truth; as they will also rank him as the father of this science. Rarely indeed has it happened, that one mind has combined the perceptions of original truth, with the power to exhibit the deformities of error and prejudice, in so masterly a manner! he, at once, discovered and established the true principle of probability. It was not so with Copernicus, when he gave to the world his theory of the celestial motions, in opposition to those who would 66 build, unbuild, contrive To save appearances, and gird the sphere Nor yet even with Newton, when he published his immortal work. The current of error and superstition set strong against both; to divert which, it required the lapse of time and the aid of other intellects: so firm a hold does received opinion retain upon the mind, despite of the dictates of reason. And even at this late day, an author who treats of the science of probability, is liable to have his motives impugned, and his efforts decried, by those who forget the important applications of its principles to the great concerns of life, in the lurking danger which they may perceive in its use in games of hazard. It is unfortunate for many of the modes of increasing the comforts of life, that their use is compounded with their abuse; and it is especially so for the science of chances, where sentiments which have their birth in religious feeling, exert a peculiarly unfavour. able influence. We may, then, observe, en passant, that the prejudice against the doctrine of probabilities is equally unfounded, and repugnant even to the common processes of reasoning and to the ordinary rules of action: for, in fact, the greater part of our knowledge is but probable, derived by analogy and induction; and thus, though it obtain full faith and credit with the many, it is received with care, and with a due regard to opposing probabilities, by the philosopher. Our belief that the sun will rise on the morrow, is but a probability founded upon the observance of the regularity with which it has heretofore risen; we assume it as a law deduced from past experience, and apply it to the conduct of the future: and it is evident that we have no higher foundation for our credence, than the hypothetical conclusion that nature will be uniform in her operations. It is easy to perceive, that our faith in the rising of the sun does not depend upon its being morally impossible that it should not rise. So, also, the statesman, when he determines upon any measure of prospective effect, forms his resolution upon the history of the past, which is nothing more than a series of probabilities, and which leads him to suppose that its experience will apply for the future. The whole groundwork of his proceeding, is a reliance upon the testimony of others; and, therefore, may possibly be false, though, from the strength of the possibility that it is not, he is induced to consider it as certain. The same rules are adopted by every one, in the most ordinary duties of life. But We regard the doctrine of probabilities, in one point of view, at least, as inculcating a great moral truth; and as opening an avenue for the admission of light upon our benighted minds. It serves to show that we are ignorant of the causes or laws of the most ordinary events; to put before us a true mirror, wherein we may see how pompously and vainly we have arrayed ourselves; to sound for us the shallowness of human knowledge; in a word, to lead us to a more correct estimate of ourselves. Take the simple and unlettered man; and the most striking moral difference between him and his more intelligent neighbour, will be observed to consist in his greater credulity; the degree of which will be found to be the measure of intelligence in all men. mark the operation of this principle. He who believes readily and firmly, in all cases, takes probable truths with the same confidence that he does certain truths; and possible, with the same faith that he does probable, and thus doing, he is ignorant of the sources and founts of knowledge, as he must also consequently be of its real nature. It has become as a thrice told tale, which is wearisome to the ear, that advancement in human knowledge teaches us how empty are its pretensions; and, we may say, that he who studies the science of probabilities with attention, must arrive at this same conclusion, and terminate his meditations with subdued and humbled feelings. This science, however, derives its importance from the assistance which it renders to the judgment in reducing all events of one description to a certain number of equally possible cases; by means of which, a balance may easily be struck, between those which are favourable, and those which are not. It provides certain and determinate rules for our aid in all probable events, and enables us to give them the greatest degree of certitude to which they can be entitled. It does not pretend to discover new facts, but to prevent deception in our reasoning, by divesting the subject of our inquiry of numerous and confusing, but unimportant circumstances. It affords us a square by which we may quadrate our opinions with the utmost correctness of which the event may be capable. Much of the opposition with which the doctrine of probabilities has had to contend, has arisen from an obscurity in relation to the meaning of the word chance. By many it is considered as applying to events which have no cause other than a direct interpositon of the Deity; and, therefore, (and very justly were this true) that it is impious to resort to any operation which may seem to depend upon this principle. It was, however, long ago, observed by James Bernoulli, in his treatise Ars Conjectandi, that chance has reference only to the state of our knowledge. We refer an event to this explanation, only when we cannot discover the law which controls it, and not in respect of its having no cause. Thus, when a die is thrown, it may fall six or an ace, according as the player may have taken it up in a peculiar manner, or have shaken it a certain number of times, or thrown it out with singular force. We are unable to detect the laws which govern these various movements, though we cannot doubt that such laws exist; and we, therefore, act as if they did not decide the event. This inclination of the mind was observed by a distinguished writer of the last age: "though there be," says Hume, "no such thing as a chance in the world, our ignorance of the real cause of any event has the same influence on the understanding, and begets a like species of belief or opinion." Though we are unable to decide positively upon the happening or failure of an event which depends upon chance, we may, at least, give the probability which belongs to either or both contingencies; and, consequently, lay down positive rules for our guidance in all cases, as we shall presently see, which are to be classed as probable. The chances possible to decide any event being supposed equal, the mathematical probability may be defined to be that which results from dividing the number of favourable chances by the whole number of chances; and may be represented by a proper fraction, unity being the representative of certainty, and zero of no chance; and if we denote the number of chances unfavourable to an event by y, and the number of chances favourable by x, we shall have the simple expression as the expression of the probability; and if we further denote this expression by m, the contrary probability will be represented by I-m; the sum of the opposing probabilities, (one that an event will happen, and the other that it will not,) being equivalent to unity. We may further illustrate this idea by supposing an urn to contain a certain number of balls, say twenty-five, and that five of these are coloured red, and the remaining twenty white. The probability of drawing out a red ball will be represented by the fraction 5/25 or 5, and the probability of drawing out a white one will be represented by 2025 or 4%; and the sum of these probabilities by 25/25 or 5/5 or 1. In deciding upon an event, individuals will differ according to the different data upon which they form their opinions; some having more, and some less, they will of course arrive at dif ferent conclusions. The same fact related to many individuals, will obtain different degrees of belief, according to the extent of their knowledge in regard to it; and it is on account of this cir See his Inquiry concerning Human Understanding, Section vi. 4 |