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LAPLACE, SUR LES PROBABILITÉS.
It is to the imperfection of the human mind, and not to any irregularity in the nature of things, that our ideas of chance and probability are to be referred. Events which to one man seem accidental and precarious, to another, who is better informed, or who has more power of generalization, appear to be regular and certain. Contingency and verisimilitude are therefore the offspring of human ignorance, and, with an intellect of the highest order, cannot be supposed to have any existence. In fact, the laws of the material world have the same infallible operation on the minute and the great bodies of the universe ; and the motions of the former are as determinate as those of the latter. There is not a particle of water or of air, of which the condition is not defined by rules as certain as that of the sun or the planets, and that has not described from the beginning a trajectory determined by mechanical principles, subjected to the law of continuity, and capable of being mathematically defined. This trajectory is therefore in itself a thing knowable, and would be an object of science to a mind informed of all the original conditions, and possessing an analysis that could follow them through their various combinations. The same is true of every atom of the material world ; so that nothing but information sufficiently extensive, and a calculus sufficiently powerful, is wanting to reduce all things to certainty, and, from the condition of the world, at any one instant to deduce its condition at the next; nay, to integrate the formula in which those momentary actions are included, and to express all the phenomena that ever have happened, or ever will happen, in a function of duration reckoned from any given instant. This is in truth the nearest approach that we can make to the idea of Omniscience ; of the Wisdom which presides over the least as well as the greatest things ; over the falling of a stone as well as the revolution of a planet ; and which not only numbers and names the stars, but even the atoms that
* From the Edinburgh Review, Vol. XXIII. (1814.)—Ep.
The farther, accordingly, that our knowledge has extended, the more phenomena have been brought from the dominion of Chance, and placed
under the government of physical causes ; and the farther off have the boundaries of darkness been carried. It was, says Laplace, of the phenomeva not supposed to be subjected to the regulation of fixed laws, that superstition took hold, for the purpose of awakening the fears and enslaving the minds of men. The time, adds he, is not far distant, when unusual rains, or unusual drought, the appearance of a comet, of an eclipse, of an aurora borealis, and, in general, of any extraordinary phenomenon, was regarded as a sign of the anger of heaven ; and prayers were put up to avert its dangerous consequences. Men never prayed to change the course of the sun or of the planets, as experience would have soon taught them the inefficacy of such supplications. But those phenomena of which the order was not clearly perceived, were thought to be a part of the system of nature which the Divinity had not subjected to fixed laws, but had left free, for the purpose of punishing the sins of the world, and warning men of their danger. The great comet of 1456 spread terror over all Europe, at that time alarmed by the rapid successes of the Turks, and the fall of the Greek empire; and the Pope directed public prayers to be said on account of the appearance of the comet, no less than the progress of Mahomet. .
It is curious to remark how different the sensations have been which, after four revolutions, this same comet has excited in the world. Halley having recognized its identity with the comets of 1531, 1607, 1682, showed it to be a body revolving round the sun in 75 years nearly; he foretold its return in 1758, or the beginning of 1759, and the event has verified the most remarkable prediction in science. Comets have since ceased to be regarded as signs of the Divine displeasure; and every body must have remarked, with satisfaction, how far the comet of 1811 was from being viewed with terror, (in this country at least,) even by the least instructed of the people, and from exciting any sentiment but admiration of its extraordinary beauty. The dominion of Chance is thus suffering constant diminution; and the Anarch Old
; may still complain, as in Milton, of the encroachments that are continually making on his empire.
Probability and chance are thus ideas relative to human ignorance. The latter means a series of events not regulated by any law that we perceive. Not perceiving the existence of a law, we reason as if there were none, or no principle by which one state of things determines that which is to follow. The axiom, or, as it may be called, the definition, on which the doctrine of probability is founded, is, that when any event may fall out a certain number of ways, all of which, to our apprehension, are
, equally possible, the probability that the event will happen with certain conditions accompanying it, is
expressed by a fraction, of which the numerator is the number of the instances favourable to those conditions, and the denominator the number of possible instances. Thus, the probability of throwing an ace with one die is denoted by t, as there are six ways that the event may turn out, and only one in which it can be an ace. With two dice, the chance of throwing 2 aces is lo; as each face of the one ace may be combined with any face of the other. Certainty is denoted here by unity ; it is what happens when all the cases are favourable to the condition expected, and when the numerator and denominator of the fraction are the same. It were absurd to say, that the sentiment of belief produced by any probability, is proportional to the fraction which expresses that probability; but it is so related, or ought to be so related to it, as to increase when it increases, and to diminish when it diminishes.
The calculation of Probability is therefore a very ingenious application of mathematical reasoning, in order to substitute for that certainty which is quite beyond our reach, the degree of evidence that the case admits of, and to reduce this to a system of accurate reasoning. The thing obtained is only probability; but we have a certainty as to the degree in which it exists,
The invention of this calculus does not go far back. It is true, that wherever there have been