the necessary calculations would much reduce the utility of the theory.

By a process of reasoning, which it would be undesirable to attempt to follow in detail in this place, it is shown that, under these conditions, the most probable result of any series of recorded observations is that which makes the sum of the squares of the errors the least possible. Let a, b, c, &c., be the results of observation, and x the quantity selected as the most probable, that is the most free from unknown errors: then we must determine x so that (a − x)2 + (b − x)2 + ( − x)2 + .... shall be the least possible quantity. Thus we arrive at the celebrated Method of Least Squares, as it is usually called, which appears to have been first distinctly put in practice by Gauss in 1795, while Legendre first published in 1806 an account of the process in his work, entitled, 'Nouvelles Méthodes pour la détermination des Orbites des Comètes.' It is worthy of notice, however, that Roger Cotes had long previously recommended a method of equivalent nature in his tract, Estimatio Erroris in Mixta Mathesid.'

6

Herschel's Geometrical Proof.

A second method of demonstrating the Principle of Least Squares was proposed by Sir John Herschel, and although only applicable to geometrical notions, it is remarkable as showing that from whatever point of view we regard the subject, the same principle will be detected. After assuming that some general law must exist, and that it is subject to the general principles of probability, he supposes that a ball is dropped from a high point with the intention that it shall strike a given mark on a horizontal plane. In the absence of any known causes of deviation it will either strike that mark, or, as

d De Morgan, 'Penny Cyclopædia,' art. Least Squares.

is infinitely more probable, diverge from it by an amount which we must regard as error of unknown origin. Now, to quote the words of Sir J. Herschel, 'the probability of that error is the unknown function of its square, i. e. of the sum of the squares of its deviations in any two rectangular directions. Now, the probability of any deviation depending solely on its magnitude, and not on its direction, it follows that the probability of each of these rectangular deviations must be the same function of its square. And since the observed oblique deviation is equivalent to the two rectangular ones, supposed concurrent, and which are essentially independent of one another, and is, therefore, a compound event of which they are the simple independent constituents, therefore its probability will be the product of their separate probabilities. Thus the form of our unknown function comes to be determined from this condition, viz., that the product of such functions of two independent elements is equal to the same function of their sum. But it is shown in every work on algebra that this property is the peculiar characteristic of, and belongs only to, the exponential or antilogarithmic function. This, then, is the function of the square of the error, which expresses the probability of committing that error. That probability decreases, therefore, in geometrical progression, as the square of the error increases in arithmetical.'

Laplace's and Quetelet's Proof of the Law
of Error.

However much presumption the modes of determining the Law of Error, already described, may give in favour of the law usually adopted, it is difficult to feel that the

e Edinburgh Review,' July 1850, vol. xcii. p. 17. Reprinted Essays,' 399. This method of demonstration is discussed by Boole, Transus of Royal Society of Edinburgh,' vol. xxi. pp. 627-630.

arguments are satisfactory and conclusive. The law adopted is chosen rather on the grounds of convenience and plausibility, than because it can be seen to be the true and necessary law. We can however approach the subject from an entirely different point of view, and yet get to the same result.

Let us assume that a particular observation is subject to four chances of error, each of which will increase the result one inch if it occurs. Each of these errors is to be regarded as an event independent of the rest and we can therefore assign, by the theory of probability, the comparative probability and frequency of each conjunction of errors. From the Arithmetical Triangle (pp. 208, 213) we learn that the ways of happening are as follows:

We may infer that the error of two inches is the most likely to occur, and will occur in the long run in six cases out of sixteen. Errors of one and three inches will be equally likely, but will occur less frequently; while no error at all, or one of four inches will be a comparatively If we now suppose the errors to act as often in one direction as the other, the effect will be to alter the average error by the amount of two inches and we shall have the following results:

rare occurrence.

Negative error of 2 inches.

Negative error of 1 inch

No error at all

I

Positive error of 1 inch

Positive error of 2 inches

We may now imagine the number

4

increased and the amount of each error decreased, and the arithmetical triangle will always give us the proportional frequency of the resulting errors. Thus if there be five positive causes of error and five negative causes, the following table shows the comparative numbers of aggregate errors of various amount which will be the result:

210

It is plain that from such numbers I can ascertain the probability of any particular amount of error under the conditions supposed. Thus the probability of a positive error of exactly one inch is 1024' in which fraction the numerator is the exact number of combinations giving one inch positive error, and the denominator the whole number of possible errors of all magnitudes. I can also, by adding together the appropriate numbers, get the probability of an error not exceeding a certain amount. Thus the probability of an error of three inches or less, positive or negative, is a fraction whose numerator is the sum of 45+120 + 210 +252 + 210+ 120 + 45, and the denominator, as before, giving the result 1024

1002

We may see at once that, according to these principles, the probability of small errors is far greater than of large ones; thus the odds are 1002 to 22, or more than 45 to 1, that the error will not exceed three inches; and the odds are 1022 to 2 against the occurrence of the greatest possible error of five inches. The existence of no error at all is the most likely event; but a small error, such as that of one inch positive, is little less likely.

If any case should arise in which the observer knows the number and magnitude of the independent errors which may occur, he ought certainly to calculate from the Arithmetical Triangle the special Law of Error which would apply. But the general law, of which we are in search, is to be used in the dark, when we have no knowledge whatever of the sources of error. To assume any special number of causes of error is then an arbitrary proceeding, and mathematicians have chosen the least arbitrary course of imagining the existence of an infinite number of infinitely small errors, just as, in the inverse method of probabilities, an infinite number of infinitely improbable hypotheses were submitted to calculation (p. 296).

The reasons in favour of this choice are of several different kinds.

1. It cannot be denied that there may exist infinitely numerous causes of error in any act of observation.

2. The resulting law on the hypothesis of a large finite, or even a moderate finite number of causes of error, does not appreciably differ from that given by the hypothesis of infinity.

3. We gain by the hypothesis of infinity a general law capable of ready calculation, and applicable by uniform rules to all problems.

4. This law, when tested by comparison with extensive series of observations, is strikingly verified, as will be

shown in a later section.

When we imagine the existence of any large number of causes of error, for instance one hundred, the numbers of combinations become impracticably large, as may be seen to be the case from a glance at the Arithmetical Triangle (p. 208), which proceeds only up to the seventeenth line. M. Quetelet, by suitable abbreviating processes, succeeded in calculating out a table of probability of errors on the