moreover this advantage in the method that several points may be determined with the corresponding ones on the other side, and the mean of the whole taken as the true place of the maximum. But this method entirely depends upon the existence of symmetry in the curve, so that of two equal ordinates one shall be as far on one side of the maximum as the other is on the other side. The method fails when other laws of variation prevail.

In tidal observations great difficulty is encountered in fixing the moment of high water, because the rate at which the water is then rising or falling is almost imperceptible. Dr. Whewell proposed, therefore, to note the time at which the water passes a fixed point somewhat below the maximum both in rising and falling, and take the mean time as that of high water. But this mode of proceeding unfortunately does not give a correct result, because the tide follows different laws in rising and in falling. There is a difficulty again in selecting the highest spring tide, another object of much importance in tidology. Laplace discovered that the tide of the second day preceding the conjunction of the sun and moon is nearly equal to that of the fifth day following; and, believing that the increase and decrease of the tides proceeded in a nearly symmetrical manner, he decided that the highest tide would occur about thirty-six hours after the conjunction, that is half-way between the second day before and the fifth day after.

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

a Airy On Tides and Waves,' Encycl. Metrop. pp. 364*-366*.

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 whole body 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 used to be, and perhaps still are, of importance 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 excellently explained by Sir John Herschel, 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 a certain time in entering the shadow. Different observers using different telescopes would usually select different moments for that of the eclipse. But it is evident that the increase of light in the emersion will proceed according to a law exactly 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. The personal 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.

r 'Outlines of Astronomy,' 4th edition, § 538.

Ff

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 well be deemed hopeless to suppose that out of error we can 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 gross misapprehension 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 mistake and disturbance, and we are never relieved from the necessity of vigilantly providing against such 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 entirely unknown causes. We must treat such residual differences in some way or other, since they will occur in all accurate experiments, and as their peculiar nature and 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 special case a special Law of Error may apply, and should be discovered and adopted if possible. Nothing can be more unlikely than that the errors committed in all classes of observations should follow the same lawa,' and the special Laws of Error which will apply to certain instruments, as for instance the repeating circle, have been investigated by M. Bravais. He concludes that every partial and distinct cause of error gives rise to a curve of possibility of errors, which may have any form whatever,— a curve which we may either be able or unable to discover, and which in the first case may be determined by considerations à priori, 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 nature of the ultimate 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 most nearly approximate to the truth which makes the sum of the squares of the errors as small as possible. But there are at least three different ways in which this principle has been arrived at respectively by Gauss, by Laplace, by Quetelet and by Sir John Herschel. Gauss proceeds much upon assump

a Philosophical Magazine,' 3rd Series, vol. xxxvii. p. 324.

b Letters on the Theory of Probabilities,' by Quetelet, transl. by O. G. Downes, Notes to Letter XXVI. pp. 286-295.

tion; Herschel rests upon geometrical considerations; while Laplace and Quetelet regard the Law of Error as a development of the doctrine of combinations; that of Gauss may be first noticed.

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 accordingly as the error was positive or negative. The even powers x2, x, xo, &c., are always intrinsically positive, whether a 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 powers would fulfil the conditions as well as the seconde, 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

c Méthode des Moindres Carrés.' 'Mémoires sur la Combinaison des Observations, par Ch. Fr. Gauss.

Paris, 1855, pp. 6, 133, &c.

Traduit en Français par J. Bertrand,'