To bring error itself under law might seem beyond human power. He who errs surely diverges from law, and it might be deemed hopeless out of error to draw truth. One of the most remarkable achievements of the human intellect is the establishment of a general theory which not only enables us among discrepant results to approximate to the truth, but to assign the degree of probability which fairly attaches to this conclusion. It would be a mistake indeed to suppose that this law is necessarily the best guide under all circumstances. Every measuring instrument and every form of experiment may have its own special law of error; there may in one instrument be a tendency in one direction and in another in the opposite direction. Every process has its peculiar liabilities to disturbance, and we are never relieved from the necessity of providing against special difficulties. The general Law of Error is the best guide only when we have exhausted all other means of approximation, and still find discrepancies, which are due to unknown causes. We must treat such residual differences in some way or other, since they will occur in all accurate experiments, and as their origin is assumed to be unknown, there is no reason why we should treat them differently in different cases. Accordingly the ultimate Law of Error must be a uniform and general one.
It is perfectly recognised by mathematicians that in each case a special Law of Error may exist, and should be discovered if possible. “Nothing can be more unlikely than that the errors committed in all classes of observations 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.2 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 à 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
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