tions should follow the same law," 1 and the special Laws of Error which will apply to certain instruments, as for instance the repeating circle, have been investigated by Bravais. He concludes that every distinct cause of error gives rise to a curve of possibility of errors, which may have any form,-a curve which we may either be able or unable to discover, and which in the first case may be determined by à priori considerations on the peculiar nature of this cause, or which may be determined d posteriori by observation. Whenever it is practicable and worth the labour, we ought to investigate these special conditions of error; nevertheless, when there are a great number of different sources of minute error, the general resultant will always tend to obey that general law which we are about to consider. Establishment of the Law of Error. Mathematicians agree far better as to the form of the Law of Error than they do as to the manner in which it can be deduced and proved. They agree that among a number of discrepant results of observation, that mean quantity is probably the best approximation to the truth which makes the sum of the squares of the errors as small as possible. But there are three principal ways in which this law has been arrived at respectively by Gauss, by Laplace and Quetelet, and by Sir John Herschel. Gauss proceeds much upon assumption; Herschel rests upon geometrical considerations; while Laplace and Quetelet regard the Law of Error as a development of the doctrine of combinations. A number of other mathematicians, such as Adrain, of New Brunswick, Bessel, Ivory, Donkin, Leslie Ellis, Tait, and Crofton have either attempted independent proofs or have modified or commented on those here to be described. For full accounts of the literature of the subject the reader should refer either to Mr. Todhunter's History of the Theory of Probability or to the able memoir of Mr. J. W. L. Glaisher.3 1 Philosophical Magazine, 3rd Series, vol. xxxvii. p. 324. 2 Letters on the Theory of Probabilities, by Quetelet, translated by O. G. Downes, Notes to Letter XXVI. pp. 286-295. 3 On the Law of Facility of Errors of Observations, and on the Method of Least Squares, Memoirs of the Royal Astronomical Society, vol. xxxix. p. 75. According to Gauss the Law of Error expresses the comparative probability of errors of various magnitude, and partly from experience, partly from à priori considerations, we may readily lay down certain conditions 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 a, a, a, &c., are always intrinsically positive, whether a be positive or negative. There is no a 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; 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, in which a 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 observa 1 Méthode des Moindres Carrés. Mémoires sur la Combinaison des Observations, par Ch. Fr. Gauss. Traduit en Français par J. Bertrand, Paris, 1855, pp. 6, 133, &c. .... tions 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." 1 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," "the probability of that error is the unknown function of its square, i.c. 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 1 De Morgan, Penny Cyclopædia, art. Least Squares. 2 Edinburgh Review, July 1850, vol. xcii. p. 17. Reprinted Essays, P. 399. This method of demonstration is discussed by Boole, Trans. actions of Royal Society of Edinburgh, vol. xxi. pp. 627--630. 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. 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 arguments are satisfactory. The law adopted is chosen rather on the grounds of convenience and plausibility, than because it can be seen to be the 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. 182-188) we learn that no error at all can happen only in one way; an error of one inch can happen in 4 ways; and the ways of happening of errors of 2, 3 and 4 inches respectively, will be 6, 4 and I in number. 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 rare occurrence. 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 : Negative error of 2 inches. No error at all.. Positive error of 1 inch. Positive error of 2 inches I way. 4 ways. 6 ways. 4 ways. I way. We may now imagine the number of causes of error increased and the amount of each error decreased, and the arithmetical triangle will give us the frequency of the resulting errors. Thus if there be five positive causes of error and five negative causes, the following table shows the numbers of errors of various amount which will be the result: 210 1024 It is plain that from such numbers I can ascertain the probability of any particular amount of error under the conditions supposed. The probability of a positive error of exactly one inch is in which fraction the numerator is the 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 1002 1024 We may see at once that, according to these principles, the probability of small errors is far greater than of large ones: the odds are 1002 to 22, or more than 45 to that the error will not |