annually out of a given number of inhabitants fixt with great precision, as well as of those that are born, and that have reached to the different periods of life. In the first case, the irregularities bear a great proportion to the whole: in the second, they compensate for one another; and a rule emerges, from which the deviations on opposite sides appear almost equal. This is true not only of natural events, but of those that arise from the institutions of society, and the transactions of men with one another -Hence insurance against fire, and the dangers of the sea. Nothing is less subject to calculation, than the fate of a particular ship, or a particular house, though under given circumstances. But let a vast number of ships, in these circumstances, or of houses, be included, and the chance of their perishing, to that of their being preserved, is matter of calculation founded on experience, and reduced to such certainty, that men daily stake their fortunes on the accuracy of the results. This is true, even where chance might be supposed to predominate the most; and where the causes that produce particular effects, are the most independent of one another. LAPLACE observes, that at Paris, in ordinary times, the number of letters returned to the Post Office, the persons to whom they were directed not being found, was nearly the same from one year to another. We have heard the same remark stated of the Dead Letter Office, as it is called, in London. Such is the consequence of the multiplication of the events least under the controul of fixt causes: And the instances just given, are sufficient to illustrate the truth of the general proposition; which LAPLACE has thus stated. The recurrences of events that depend on chance, approach to fxt ratios as the events become more numerous, in such a manner that the probability of the mean results not differing from those ratios by any given quantity, may come nearer to certainty than the smallest limit that can be assigned.' Thus, if in an urn, the number of white balls to that of black, have the ratio of p to q, the number of white balls brought out if the whole number drawn be n, will approach to P Xn, the more nearly the greater that the number ʼn is taken. This proposition is deducible a priori from the theory of Probability. It was first demonstrated by BERNOUILLI, in the Ars Conjectandi, by a method that is very elaborate, and confessedly the work of much thought and study. A more simple demonstration was given by DEMOIVRE, in his doctrine of Chances. Our author, in his Theorie Analytique, has given one much pre ferable to either, deduced from his theory of Generating Func tions. The solution of another curious problem which LAPLACE has given, is closely connected with the preceding. An event having happened a certain number of times in succession, what is the probability that it will happen once more? When the number of times the event has happened is small, the formula that contains the answer to this question is considerably complicated; when the number is very great, it is extremely simple. Suppose the number to be n, the chance that the same event will again occur, is which, if n be great, is very near to unity, and may express a probability not sensibly inferior to certainty. 1826214 n+1 Thus, supposing with M. LAPLACE, that the greatest antiquity to which history goes back is 5000 years, or 1826213 days, the probability that the sun will rise again to-morrow, is, according to this rule, ; or there is 1826214 to 1, to wager in favour of that event. This, therefore, may be considered as affording a measure of the probability that the course of nature will continue the same in future that it has been in time past. It is not however on the refined principles of this calculus, that the universal belief of mankind in such continuance is founded. The above theorem was first given by BERNOUILLI Our author's demonstration of it in the Essai Analytique, we believe to be new and more simple than any other. The same multiplication of events enables us to employ the theory of probability in the discovery of causes. On this subject LAPLACE has made a number of very important observations. The phenomena of nature are for the most part enveloped in such a number of extraneous circumstances, and so many disturbing causes unite their influence, that it is very difficult, when they are small, to eparate them from one another. The best way to discover them is to multiply observations, that the accidental effects may destroy one another, and leave a mean result containing only what is essential to the phenomenon. The entire removal of the accidental part is not to be expected, as has just appeared, without an infinite number of observations: the greater the number of observations, however, the more nearly is this mean result approximated. Of this application of the doctrine of Probabilities, a number of examples are then given. The first relates to the diurnal variation of the barometer, as found from the observations of that instrument made at the Equator, where it is least subject to the action of irregular causes. From these, there appears to be a small diurnal oscillation, of which one maximum takes place about 9 in the morning, and a minimum about 4 in the evening: a second maximum at 11 at night, and a second minimum about 4 in the morning. The oscillations of the day are greater than those of the night, and amount to about th of an inch. The inconstancy of the weather does not allow this variation to be immediately observeable without the tropics, or within the range of the variable winds. Nevertheless, by applying the calculus of Probabilities to a great number of accurate barometrical observations made by RAMOND during several successive years, M. LAPLACE has found such indications of the same oscillation, as to leave no doubt of its existence, though concealed under the irregular action of many accidental causes. This oscillation having its period equal to a solar day, must arise from the sun's action, most probably, in the heating and cooling of the atmosphere. To the same calculus, in what regards the irregularities of the planetary system, our author professes to be greatly indebted. The difficulty in such cases is, often, to know whether a certain small irregularity, combined as it is with many other irregularities, has an existence or not. If it has an existence, it will give a certain determination to all the results one way more than another; and by comparing a great number of results, the reality of the determination may be discovered. It is just as if a die were thrown a great number of times, and it was required to find whether it had a bias to a certain side or not. Áfter a vast number of throws, if there is no bias, each face must have turned up nearly the same number of times, If this is not found to hold; if there be one face which has turn ed up considerably oftener than the rest, it may safely be concluded that there is a bias to that side; and from the calculus of probabilities, the amount of the bias may be estimated. In this way, the calculus may be applied to several astronomical phenomena, and may be considered as a means of discovering from induction, some conclusions that could hardly be otherwise obtained. M. LAPLACE gives an instance of this in his own researches, concerning the diminution of a certain inequality in the precession of the equinoxes, relatively to the moon only, which was suspected by MAYER, but rejected by most astronomers as not being explained on the principle of gravitation. A scrupulous examination of observations, and the application of the calculus, convinced M. LAPLACE, that the existence of the inequality was highly probable; so that he began to look out for the cause of it. It was not long before he perceived that it must arise from the spheroidal figure of the earth, which must change a little the laws of gravity towards that body, and produce of consequence an inequality in the lunar motions. This cause had hitherto been neglected by astronomers; but, when taken into account, it explained with precision the irregularity in question, and the magnitude which, by the rules of probability, he had been led to assign to it. Other instances are given in the irregularities of Jupiter and Saturn, the satellites of Jupiter, &c. We shall only mention one result, and it is a very remarkable one, deduced from the motions of the planets being all in the same direction. One of the most remarkable phenomena in the solar system, is, that the motions of rotation and of revolution in the planets and satellites are all in the same direction, viz. in that of the sun's rota tion, and not far from the plane of his equator. A phenomenon so remarkable cannot be the effect of chance; and it obviously indicates one general cause, which has determined all these motions. To estimate the probability with which this cause is pointed out, it must be considered, that the planetary system, such as we now see it, is composed of eleven planets and eighteen satellites; and that the rotation of the sun, of six planets, of the satellites of Jupiter, of the ring of Saturn, and of one of his satellites, are all known. These movements, taken in conjunction with those of revolution, make a total of forty-three-all in the same direction. Now, by the calculation of probabilities, it will be found that there are more than 4 millions of millions to wager against one, that this disposition is not the effect of chance; a probability much superior to that of the historical events about which we entertain the least doubt. We must therefore believe at least with equal confidence, that ONE Primitive Cause has directed all the planetary motions; especially when we consider, that the greater part of these motions are also nearly in the same plane.' Our Author proceeds, then, to offer some conjectures concerning the physical cause to which these motions are to be ascribed. He brings together a great number of facts, from Dr HERSCHELL'S observations concerning the nebule which, combined with the preceding, seem to point out the solar atmosphere as the most probable cause. But where the facts lie so far out of the reach of accurate observation as many of these do, and when the supposed cause has ceased so entirely to act, the evidence we can have is so slight, and the difficulties so many, that even the AUTHOR of the Mécanique Céleste must fail in giving weight and durability to his system. In those sciences which are in a great measure conjectural, such as medicine, agriculture and politics, the calculus of probabili ties may be employed for discovering the value of the different methods that are had recourse to. Thus, to find out the best of the treatments in use in the cure of a particular disease, the comparison of a number of cases, where the circumstances have been as much alike as possible, will enable us to judge of the accidental causes that in each particular case assisted or impeded the cure these last will make a compensation for one another; and if the number of cases is sufficiently great, will leave the efficacy or inefficacy of the remedies distinctly visible. The same he adds, may be applied to political economy; with respect to which, the operations of governments are so many experiments, made on a great scale, and calculated to throw light on the conduct to be pursued on similar occasions. So many unforeseen, concealed, and inappreciable causes, have an influence on human institutions, that it is impossible to judge a priori of their effects.— Nothing but a long series of experiments can unfold these effects, and point out the means of counteracting those that are hurtful. It would conduce much to this object, if, in every branch of the administration, an exact register were kept of the trials made of different measures; and of the results, whether good or bad, to which they have led.' He concludes with a maxim, which the circumstances of the times in which he has lived, must have but too deeply engraven on the mind of every Frenchman. 'Ne changeons qu'avec une circonspection extrême nos anciennes institutions et usages auxquels nos opinions et nos habitudes se sont depuis long-tems pliées. Nous connaissons bien par l'expérience du passé les inconveniens qu'ils nous presentent; mais nous ignorons quelle est l'etendue des maux que leur changement peut produire.' These are safe and just maxims; and we are glad to think that he who expresses them holds a high situation in the government of his country. There is, however, another maxim grounded also on the doctrine of Probability, which we should think hardly less necessary than this, viz. that the rulers of mankind, in order to remove as much as possible all chance of sudden and great revolutions, would strike at the roots of the causes which so often render them inevitable, by taking care that all political institutions are gradually and slowly corrected, as their errors are found out, or as new circumstances in the situation of the world render them inapplicable. The negative precept, of not changing things but slowly, is not alone sufficient; it is necessary to add the affirmative precept, of changing them slowly, but readily, when reason for such change appears. In this way the causes that tend to disturb the public order are prevented from Accumulating, so as to create, or even to justify, the spirit of revolution; and by gradual reformations, which may be made |