This method is also employed in determining the time of passage of the middle or densest point of a stream of meteors. The earth takes two or three days in passing completely through the November stream; but astronomers need for their calculations to have some definite point fixed within a few minutes if possible. When near to the middle they observe the numbers of meteors which come within the sphere of vision in each half hour, or quarter hour, and then, assuming that the law of variation is symmetrical, they select a moment for the passage of the centre equidistant between times of equal frequency.
The eclipses of Jupiter’s satellites are not only of great interest as regards the motions of the satellites themselves, but were, and perhaps still are, of use in determining longitudes, because they are events occurring at fixed moments of absolute time, and visible in all parts of the planetary system at the same time, allowance being made for the interval occupied by the light in travelling. But, as is explained by Herschel,[275] the moment of the event is wanting in definiteness, partly because the long cone of Jupiter’s shadow is surrounded by a penumbra, and partly because the satellite has itself a sensible disc, and takes time in entering the shadow. Different observers using different telescopes would usually select different moments for that of the eclipse. But the increase of light in the emersion will proceed according to a law the reverse of that observed in the immersion, so that if an observer notes the time of both events with the same telescope, he will be as much too soon in one observation as he is too late in the other, and the mean moment of the two observations will represent with considerable accuracy the time when the satellite is in the middle of the shadow. Error of judgment of the observer is thus eliminated, provided that he takes care to act at the emersion as he did at the immersion.
CHAPTER XVII.
THE LAW OF ERROR.
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,”[276] and the special Laws of Error which will apply to certain instruments, as for instance the repeating circle, have been investigated by Bravais.[277] 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.[278]
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 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;[279] 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.”[280]