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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 Jupiter 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 conse quence, 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 Legendre 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
* Nouvelles Méthodes pour la Détermination des Orbites des Comètes. Paris, 1806.
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 Théorie Analytique of 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 number, the mean error is √ a2 + b2 + c2 + &c.
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 connection with the mat
ters in hand, should enter 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. Bernoulli was the first who found the quantity enter into the expressions of probability; and he appears to have thought it very remarkable.
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 the coefficients 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 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 timekeeper. 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 ar ticle 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 human 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