to which the law will certainly conform. It may fairly be assumed as a first principle to guide us in the selection of the law, that large errors will be far less frequent and probable than small ones. We know that very large errors are almost impossible, so that the probability must rapidly decrease as the amount of the error increases. A second principle is that positive and negative errors shall be equally probable, which may certainly be assumed, because we are supposed to be devoid of any knowledge as to the causes of the residual errors. It follows that the probability of the error must be a function of an even power of the magnitude, that is of the square, or the fourth power, or the sixth power, otherwise the probability of the same amount of error would vary according as the error was positive or negative. The even powers x2, x4, x6, &c., are always intrinsically positive, whether x be positive or negative. There is no à priori reason why one rather than another of these even powers should be selected. Gauss himself allows that the fourth or sixth power would fulfil the conditions as well as the second;4 but in the absence of any theoretical reasons we should prefer the second power, because it leads to formulæ of great comparative simplicity. Did the Law of Error necessitate the use of the higher powers of the error, the complexity of the necessary calculations would much reduce the utility of the theory.
By mathematical reasoning which it would be undesirable to attempt to follow in this book, it is shown that under these conditions, the facility of occurrence, or in other, words, the probability of error is expressed by a function of the general form ε–h2 x2, in which x represents the variable amount of errors. From this law, to be more fully described in the following sections, it at once follows that the most probable result of any observations is that which makes the sum of the squares of the consequent 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 + (c - 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 Mathesi.”5
Herschel’s Geometrical Proof.
A second way of arriving at the Law of Error was proposed by Herschel, and although only applicable to geometrical cases, 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 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 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 Herschel,6 “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