de several points ing ones on the taken as the true od entirely depends e curve, so that of a on one side of the er side. The method rail. alty is encountered in * because the rate at lling is almost impertherefore, to note the As a fixed point somewhat sing and falling, and take water. But this mode of es not give a correct result, rent laws in rising and in again in selecting the highest much importance in tidology. tide of the second day prethe sun and moon is nearly day following; and, believing pose a devase of the tides proceeded in a ICPY Crater, he decided that the highest we would we set thirty-six hours after the consanction, that is "hafway between the second day before d the ith day uhert This method is also employed in determining the time of the middle or densest point of a stream of The earth takes two or three days in passing hough the November stream; but astroor their calculations to have some definite few minutes if possible. When near 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 moreover th may be de other side, place of t upon the two equa maxim fails wh In t fixing which יור NOR might seem beyond ely diverges from law, ess to suppose that out of the most remarkable lect is the establishment aly enables us among disco the truth, but to assign fairly attaches to this conmisapprehension indeed to sarily the best guide under string instrument and every ve its own special law of error; cut be a tendency in one direcopposite direction. Every proAs to mistake and disturbance, i thom the necessity of vigilantly special difficulties. The general guide only when we have exof approximation, and still find te to entirely unknown causes. Crow and dual differences in some way or ow to wait in all accurate experiments, Accordingly the ultimate " must be a grm 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 4 '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. |