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without danger, those great changes are avoided which cannot happen without incalculable mischief.
One of the most important applications of the doctrine of Probability, is to determine the most probable mean,`or average, among a number of observations. The most accurate experiments and observations are liable to errors, which must affect the truth of the results obtained from them. To make these disappear as much as possible, observations must be greatly multiplied, in order that the errors in defect and in excess may destroy one another, and the mean, of consequence, become nearly correct. Still, however, the manner of striking this mean to the greatest advantage, remains to be examined, as also the degree of error to which, after all, it must be liable.
For a long time mathematicians were contented with taking the arithmetical mean as the true result of the observations; that is, they added them altogether, and divided the sum by the number of observations. This was sufficient when the observations appeared to be all equally good, and entitled to equal weight in the determination of the result. This, however, was far from being always the case; and COTES was the first, as M. LAPLACE remarks, who thought of a method by which each observation should have an influence in the determination of the results proportioned to its real value. Suppose that it is the position of an object that is required to be found by astronomical observation; let the place given by each individual observation be found, and at each of these conceive a weight to be placed proportional to the accuracy, or inversely as the error which it is reasonable to assign to that particular observation; the centre of gravity of all these weights is the true, or the most probable place of the object. This was in fact a generalization of the common method of taking an arithmetical mean; for it is only conceiving, that if one observation A, was twice as good as another observation B, then, instead of A, there should be accounted two observations of the same value with B, and giving the same result with A, and so on in any other proportion, even if the proportion were expressed by a fraction. The principle here is, that after a great number of observations, the errors in opposite directions (the positive and negative errors) must be equal. This is true, if the number were infinitely great; and, in all cases, affords a probable approximation to the truth.
The above theorem, which COTES has given at the end of his Estimatio Errorum, admits of a simple analytical expression, but does not appear, as is remarked by LAPLACE, to have been made use of till EULER, in his tract on the Inequalities of Jupi,
ter and Saturn, employed equations of condition, for the first time, in determining the elements of the orbits of these two planets. Much about the same time, TOBIAS MAYER employed similar methods in his Inquiry into the Libration of the Moon, and afterwards in his Lunar Tables.
The method of COTES, when there is but one result to be determined, is of most easy application; but when there are more than one, and, of consequence, as many equations as there are observations, it is not obvious how it can be applied, and how the equations are to be combined to the best advantage. The idea occurred to LE GENDRE to introduce another equation, by supposing the sums of the squares of the errors of the observations to be a minimum. This is a very happy generalization of the method of the centre of gravity, and applicable to cases to which it could not easily be accommodated. The same idea occurred to M. GAUSS about the same time. It was not demonstrated, however, till it was done in the THEORIE ANALYTIQUE of M. LAPLACE, that the result thus obtained is the best of all, that which leaves the least probable error, the limits of which are assigned at the same time.
The mean result being determined, the following rule for the limit of the accuracy is given. Take the difference between the mean result of all the observations, and the result of each particular observation. The mean error, or the greatest that is to be feared, (and it may be either positive or negative), is a fraction, having for its numerator the square root of the sum of the squares of the differences above obtained, and for its denominator the number of observations multiplied into the square root of the number which denotes the ratio of the circumference to the diameter.
Thus, if the differences between the mean of the observations and the observations themselves be a, b, c, d, and if n be their √ a2 + b2 + c2 + &c. number, the mean error is n√x
It would be unsafe to wager that the error was less than this quantity.
It will no doubt appear singular, that a quantity having apparently no connexion with the matters in hand, should en ter into the above expression. It is introduced there by the operation of integration; by means of which, it is often brought into expressions, where it was not expected. BERNOUILLI was the first who found the quantity enter into the expressions of probability; and he appears to have thought it very remarkable.
* Nouvelles Méthodes pour la Determination des Orbites des Comètes. Paris, 1806.
The preceding conclusion may be useful in many cases of practical astronomy, and in other parts of natural philosophy; or indeed, when any thing is to be determined in quantity or position from a great number of observations; and especially when the things to be found are represented by the co-efficients of the terms of an algebraic formula.
As an instance :-Suppose it were required having two sorts of lunar tables; and, having compared them with observations, to determine which is the best. The common way is to add together the errors of observation, and to take the arithmetical mean: the tables to which the least mean error belongs, are accounted the best. This, however, is not the way in which the question ought to be decided. The sums of the squares of the differences between the observed and the calculated places should be added together that set in which the square root of the sum divided by the number of observations is least, is the most exact. If the number of the terms be the same, the mere comparison of the sums of the squares decides on which side the preference lies. This instance of the utility of the method of finding the mean, is given by M. LAPLACE himself. Another of the same kind may be added.-Suppose that two chronometers have been compared with the sun at noon, for a certain number of days running, and from the register kept of their errors it is required to find which of them is the best. This ought to be done by taking the squares of the differences of the errors of the chronometer for every day that in which the sum of these squares is the least, is the preferable time-keeper. If it is required to compute the error that might be found, if either of them were applied to find the longitude, it will be determined by the formula above, and will be very considerably different from the result that would arise from a mere arithmetical mean.
We have here an instance of a problem, to which, in this country, very frequent recourse has been had in the trials of chronometers for the longitude. The only method of resolving it, has hitherto been by finding the arithmetical mean, which, however, the late Astronomer-Royal did in a particular way, which, though not the same with this, was probably the best then known. It is, however, certain, that the true going of a clock, or the measure of its merit, cannot be accurately determined, but by means of the rule which has just been explained.
We shall conclude our extracts from this small, but comprehensive volume, with one from the article on Population, which we have great pleasure in laying before our readers.
The ratio of the population to the number of births would be
increased if we could diminish or destroy any disease that is dangerous and common. This has been done, happily, in the case of the small-pox, first by the common inoculation for the disease itself, and afterwards in a much more complete manner by the vaccine inoculation, the inestimable discovery of JENNER, who has rendered himself, by that means, one of the greatest benefactors of the hu man race. '
The most simple way of calculating the advantage which the extinction of a disease would produce, consists in determining from observation the number of individuals of a given age who die of it yearly, and in subtracting the amount from the total number of deaths of persons of that same age. The ratio of the difference to the total number alive at the same age would be the probability of dying at that age if the disease did not exist. By summing up all these probabilities from the beginning of life to a given age, and taking the sum from unity, the remainder will be the probability of living to that age, on the hypothesis of the disease in question being extinguished. From the series of these probabilities, the mean duration of life on the same supposition may be computed, according to rules that are well known. M. DUVILARD has found that the mean duration of human life is increased at least three years by the vaccine inoculation.' p. 69.
But as this review is now in danger of becoming longer than the book reviewed, we shall conclude, with recommending to our readers the perusal of the work itself; and with assuring them, that they will find in it much valuable and important matter, which has not fallen within the scope of this analysis.
ART. IV. A Voyage round the World, in the Years 1803, 4, 5, &6 Performed by Order of his Imperial Majesty Alexander the First, Emperor of Russia, in the Ship Neva. By UREY LISIANSKY, Captain in the Russian Navy, and Knight of the Orders of St George and St Vladimer. London. Booth, Longman & Co. 4to. pp. 388. 1814.
A COUNTRY butcher makes his customers take a certain proportion of gravy beef, when he serves them with what are denominated the prime parts. In vain the carnivorous purchaser may plead, that he wants only to roast, and has not the most distant thought of stewing: the cunning slaughterer of horned cattle holds him fast in the chains of sensuality, and loads him with lean, and useless flank, before he allows him to enjoy the flavour of the rib, or to pasture on the obesity of the rump. Travellers are as bad as butchers. Instead of coming at once to
the spot for which the book was written, and in which its interest really consists, they make you purchase their voyage through the Chops of the Channel;-they speak of Falmouth-give chapter on the Island of Madeira-stay for 10 pages at Rio Janeiro-and seldom double the Cape of Good Hope, or Cape Horn, before the middle of the first volume.
Our friend Urey Lisiansky has been initiated into this mystery of the literary shambles, and has added to the important parts of his book no small number of offal chapters, both at the beginning and the end. His voyage (such as it is) was undertaken, first of all, at the suggestion of the Russian American Company. They had experienced great difficulty in supplying their colonies on the north-west coast of America with provisions and necessaries; and, on account of the length of the journey by land to Ochotsk, resolved to try if the voyage by sea would not be more expedient. For the purpose of ascertaining whether or not this project was practicable, a plan was formed of an expedition from Cronstadt round Cape Horn. The Emperor of Russia expanded this commercial scheme into a voyage of discovery and circumnavigation; and the ships engaged in it were at the same time directed to carry out a Russian ambassador to Japan. This was the first voyage round the globe carried into effect by the Russian Government. On the return of the expedition to Cronstadt, a separate account of the voyage of each vessel was ordered to be printed, at the expense of the Emperor; and Captain Krusenstern's voyage has already appeared in an English translation.
Lisiansky visited, without his companion, the Easter and Sandwich Islands; passed a year on the island of Cadiack and at Sitca; and discovered an island and a shoal, of importance to the navigation of the South Seas.
The track of Lisiansky is as follows:-He sailed from Cronstadt in July 1803-made the island of Teneriffe by the middle of October, and that of St Catherine by the Christmas of the same year-doubled Cape Horn by the middle of March 1804— touched at Easter Island, and from thence to the Marquesas and Sandwich Islands-and so on to Cadiack and Sitca on the north-western coast of America, where the Russians have settlements. At Cadiack he wintered; and in the spring took the liquid high road to Canton, and to Europe.
It strikes us as somewhat singular, that his Imperial Majesty should think it of importance to patronize this voyage round the world, and not worth while to render it a little more subservient to the general interests of science. Why not a chemist, a botanist, an astronomer, a mineralogist, in an expedition which