parison, the larger errors being less frequent than theory would indicate.

We may also regard the experiments enumerated in the chapter on Probabilities (p. 238), as forming an empirical verification of the theory of error.

Remarks on the General Law of Error.

The mere fact that the Law of Error allows of the possible existence of errors of every assignable amount shows that it is only approximately true. We may fairly say that in measuring a mile it would be impossible to commit an error of a hundred miles, and the length of life would never allow of our committing an error of one million miles. Nevertheless the general Law of Error would assign a probability for an error of that amount or more, but so small a probability as to be utterly inconsiderable, and almost inconceivable. All that can, or in fact need, be said in defence of the law is, that it may be made to represent the errors in any special case to a very close approximation, and that the probability of large and practically impossible errors, as given by the law, will be so small as to be entirely inconsiderable. And as we are

dealing with error itself, and our results pretend to nothing more than approximation and probability, an indefinitely small error in our process of approximation is of no importance whatever.

The Probable Mean Result as defined by the Law

of Error.

One immediate result of the Law of Error, as thus stated, is that the mean result is the most probable one; and when there is only a single variable this mean is found by the familiar arithmetical process. An unfortunate error

has crept into several works which allude to this subject. Mr. Mill, in treating of the Elimination of Chance,' remarks in a note that the mean is spoken of as if it were exactly the same thing with the average. But the mean, for purposes of inductive inquiry, is not the average, or arithmetical mean, though in a familiar illustration of the theory the difference may be disregarded.' He goes on to say that, according to mathematical principles, the most probable result is that for which the sums of the squares of the deviations is the least possible. In Bowen's Treatise on Logic' (p. 439), we find the Method of Least Squares mentioned as a mode of finding the most probable result in those cases in which the arithmetical mean is not an applicable expedient for determining the probability.' It seems probable that these and other writers were misled by Dr. Whewell's remarks on the subject; for he says k that The Method of Least Squares is in fact a Method of Means, but with some peculiar characters. . The method proceeds upon this supposition; that all errors are not equally probable, but that small errors are more probable than large ones.' He adds that this method ' removes much that is arbitrary in the Method of Means.' It is strange to find a mathematician like Dr. Whewell making such remarks, when there is no doubt whatever that the Method of Means is only an application of the Method of Least Squares. They are, in fact, the same method, except that the latter method may be applied to cases where two or more quantities have to be determined at the same time. Many authorities might be quoted to this effect, but it will be sufficient to mention Lubbock and Drink water, who say, 'If only one quantity has to be

6

iSystem of Logic,' bk. iii. chap. 17, § 3. 5th ed. vol. ii. p. 56.

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k Philosophy of the Inductive Sciences,' 2nd ed. vol. ii. pp. 408, 409.

Essay on Probability,' by J. W. Lubbock and J. E. Drink wat er, Useful Knowledge Society, 1833, p. 41.

determined, this method evidently resolves itself into taking the mean of all the values given by observation.' Encke, again, distinctly says m, that the expression for the probability of an error 'not only contains in itself the principle of the arithmetical mean, but depends so immediately upon it, that for all those magnitudes for which the arithmetical mean holds good in the simple cases in which it is principally applied, no other law of probability can be assumed than that which is expressed by this formula.'

It can be shown, too, in a moment that the mean is the result which gives the least sum of squares of errors. For if a, b, c, &c., be the results of observation and x the selected mean result, the sum of squares of the errors is (α −x)2 + (b − x)2 + (c−x)2 + &c., which is at a minimum when its differential coefficient 2(a-x+b−x+c−x+ &c.) = o. From this equation we immediately obtain, denoting by n the number of separate results, a, b, c, &c., x = (a + b + c + ... ) ;;; or the ordinary arithmetic mean.

Weighted Observations.

It is to be distinctly understood that when we take the mean of certain numerical results as the most probable number aimed at, we regard all the different results as equally good and probable in themselves. The theory of error expresses no preference for any one number over any other. If, then, an observer has reason to suppose that some results are not so trustworthy as others, he must take account of this difference in drawing the mean. By the method of weighting observations this difference of value is easily allowed for. Astronomers are in the habit

Ι

of recording with an observation the value or degree of confidence with which they regard it, freely estimated according to the impression of success or failure in accuracy immediately after the observation. This value is usually expressed in a decimal scale, so that 10 denotes the highest degree of satisfaction with the result, and I the least degree. Before taking the mean of the observations each number is multiplied by its weight or value, and the sum of the products is divided by the sum of the weights. Thus if a, b, c, &c., be the observed numbers, and w, w', w", &c., the weights, then the most probable mean is aw+bw'+cw" +

This formula, it will be observed, is w+w+w" + identical in form with that for finding the centre of gravity of particles of different weights arranged in a straight line. When we regard w, w', w", &c., as all equal, it becomes identical with the formula for the ordinary This method of weighting observations, now of much importance in astronomical and other very exactly quantitative investigations, appears to have been first proposed by Roger Cotes, the editor of the Principia,' as pointed out by De Morgan".

mean.

The practice of giving weights would open the way to much error and abuse, if the weights were assigned when the mean was being drawn, and when the divergence of some results from the others would be likely to become the guide. As a general rule the weights must be assigned at the moment of observation, and afterwards. rigidly maintained, and they must be assigned not from regard to the apparent intrinsic accuracy of the result, but the extrinsic circumstances which seem to render it valuable. An observed result, in short, must be discredited, not because it is divergent, but because there were other reasons to suppose that it would be divergent.

n Penny Cyclopædia,' art. Least Squares.

The Probable Error of Mean Results.

When we draw any conclusion from the numerical results of observations we ought not to consider it sufficient, in cases of importance, to content ourselves with finding the simple mean and treating it as true. We ought also to ascertain what is the degree of confidence we may place in this mean, and our confidence should be measured by the degree of concurrence of the observations from which it is derived. In some cases the mean may be so close to the correct result that we may consider it as approximately certain and accurate. In other cases it may really be worth little or nothing. The Law of Error enables us to give exact expression to the degree of confidence proper in any case; for it shows how to calculate the probability of a divergence of any amount from the mean, and we can thence ascertain the probability that the mean in question is within a certain distance from the true number. The probable error is taken by mathematicians to mean the limits within which it is as likely as not that the truth will fall. Thus if 5'45 be the mean of all the determinations of the density of the earth, and 20 be approximately the probable error, the meaning is that the probability of the real density of the earth falling between 525 and 565 is. Any other limits might have been selected at will. We might readily calculate the limits within which it was one hundred or one thousand to one that the truth would fall; but there is a general convention to take the even odds, one to one, as the quantity of probability of which the limits are to be estimated.

Many books on the subject of probability give rules for making the calculations, but as, in the gradual progress of science, all persons ought to be more familiar with these processes, I propose to repeat the rules here and illustrate their use. The calculations, when made in strict accordance with the directions, involve none but arithmetic operations.