hypothesis of one thousand distinct causes f; but mathematicians have generally proceeded on the hypothesis of infinity, and then, by some of the beautiful devices of analysis, have substituted a general law of easy treatment. In mathematical works upon the subject, it is shown that the standard Law of Error is expressed in the formula

y = Ye-cx2,

in which x is the amount of the error, Y the maximum ordinate of the curve of error, and c a number constant for each series of observations, and expressing the general amount of the tendency to error, but varying between one series of observations and another, while e is the peculiar constant, 271828 ..... the base of the Naperian logarithms. To show the close correspondence of this general law with the special law which might be derived from the supposition of any moderate number of causes of error, I have in the accompanying figure drawn a

curved line representing accurately the variation of y when x in the above formula is taken equal to 0, 1, 1, 3, 2,

I

2'

&c., positive or negative, the arbitrary quantities Y and c

f 'Letters on the Theory of Probabilities,' Letter XV. and Appendix, note pp. 256-266.

being both assumed equal to unity, in order to simplify the calculations. In the same figure are inserted eleven dots, whose heights above the base line are proportional to the numbers in the eleventh line of the Arithmetical Triangle, thus representing the comparative probabilities of errors of various amounts arising from ten equal causes of error. It is apparent that the correspondence of the general and the special Law of Error is almost as close as can be exhibited in the figure, and the assumption of a greater number of equal causes of error would render the correspondence far more close.

It may be explained that the ordinates, for instance. NM, nm, n'm', represent values of y in the equation expressing the Law of Error. The occurrence of any one definite amount of error is infinitely improbable, because an infinite number of such ordinates might be drawn. But the probability of an error occurring between certain definite limits is finite, and is represented by a portion of the area of the curve. Thus the probability that an error, positive or negative, not exceeding unity will occur, is represented by the area Mmnn'm', in short, by the area standing upon the line nn'. Since every observation must either have some definite error or none at all, it follows that the whole area of the curve should be considered as the unit expressing certainty, and the probability of an error falling between particular limits will then be expressed by the ratio which the area of the curve between those limits bears to the whole area of the curve.

Derivation of the Law of Error from Simple

Logical Principles.

It is worthy of notice that this Law of Error, abstruse though the subject may seem, is really founded upon the simplest principles. It arises entirely out of the difference

between permutations and combinations, a subject upon which I may seem to have dwelt with unnecessary prolixity in previous pages (pp. 200-216). The order in which we add quantities together does not affect the amount of the sum, so that if there be three positive and five negative causes of error in operation, it does not matter in which order they are considered as acting. They may be indifferently intermixed in any arrangement, and yet the result will be the same. The reader should not fail to notice how laws or principles which appeared to be absurdly simple and evident when first noticed, reappear in the most complicated and mysterious processes of scientific method. The fundamental Laws of Identity and Difference gave rise to the Logical Abecedarium, which, after abstracting the character of the differences, led to the Arithmetical Triangle (p. 214). The Law of Error is defined by an infinitely high line of that triangle, and the law proves that the mean is the most probable result, and that divergencies from the mean become much less probable as they increase in amount. Now the comparative greatness of the numbers towards the middle of each line of the Arithmetical Triangle is entirely due to the indifference of order in space or time, which was first prominently pointed out as a condition of logical relations, and the symbols indicating them (pp. 40-42), and which was afterwards shown to attach equally to numerical symbols, the derivatives of logical terms (pp. 180, 181).

Verification of the Law of Error.

The theory of error which we have been considering rests entirely upon an assumption, namely that when known sources of disturbances are allowed for, there yet remain an indefinite, possibly an infinite number of other

minute sources of error, which will as often produce excess as deficiency. Granting this assumption, the Law of Error must be as it is usually taken to be, and there is no more need to verify empirically than to test the truth of one of Euclid's propositions mechanically, after we have proved it theoretically. Nevertheless, it is an interesting occupation to verify even the propositions of geometry in an approximate manner, and it is still more instructive to inquire whether a large number of observations will be found to justify our assumption of the Law of Error.

Encke has given an excellent instance of the correspondence of theory with experience, in the case of certain observations of the difference of Right Ascension of the sun and two stars, namely a Aquila and a Canis minoris. The observations were 470 in number, and were made by Bradley and reduced by Bessel, who found the probable error of the final result to be only about onefourth part of a second (o"2637). He then compared the number of errors of each magnitude from th part of a second upwards, as actually given by the observations, with what should occur according to the Law of Error. The results were as follows:

I

ΙΟ

Encke, 'On the Method of Least Squares,' Taylor's Scientific Me

moirs,' vol. ii. pp. 338, 339.

The reader will remark that the correspondence is remarkably close, except as regards larger errors, which are excessive in practice. It is one objection, indeed, to the theory of error, that, being expressed in a continuous mathematical function, it contemplates the possible existence of errors of every magnitude, such indeed as could not practically occur; yet in this case the theory seems to under-estimate the number of large errors.

Another excellent comparison of the law with observation has been made by Quetelet, who has investigated the errors of 487 determinations in time of the Right Ascension of the Pole-star, made at Greenwich during the four years 1836-39. These observations, although carefully corrected for all known causes of error, as well as for nutation, precession, &c., are yet of course found to differ, and being classified as regards intervals of one-half second of time, and then proportionately increased in number, so that their sum may be one thousand, give the following results as compared with what theory would lead us to expect h:

In this instance the correspondence is also satisfactory, but the divergence between theory and fact is in the opposite direction to that discovered in the former com

h Quetelet, Letters on the Theory of Probabilities,' translated by Downes, Letter XIX. p. 88. See also Galton's Hereditary Genius,' P. 379.