such functions of two independent elements is equal to the same function of their sum. But it is shown in every work on algebra that this property is the peculiar characteristic of, and belongs only to, the exponential or antilogarithmic function. This, then, is the function of the square of the error, which expresses the probability of committing that error. That probability decreases, therefore, in geometrical progression, as the square of the error increases in arithmetical.”
Laplace’s and Quetelet’s Proof of the Law.
However much presumption the modes of determining the Law of Error, already described, may give in favour of the law usually adopted, it is difficult to feel that the arguments are satisfactory. The law adopted is chosen rather on the grounds of convenience and plausibility, than because it can be seen to be the necessary law. We can however approach the subject from an entirely different point of view, and yet get to the same result.
Let us assume that a particular observation is subject to four chances of error, each of which will increase the result one inch if it occurs. Each of these errors is to be regarded as an event independent of the rest and we can therefore assign, by the theory of probability, the comparative probability and frequency of each conjunction of errors. From the Arithmetical Triangle (pp. 182–188) we learn that no error at all can happen only in one way; an error of one inch can happen in 4 ways; and the ways of happening of errors of 2, 3 and 4 inches respectively, will be 6, 4 and 1 in number.
We may infer that the error of two inches is the most likely to occur, and will occur in the long run in six cases out of sixteen. Errors of one and three inches will be equally likely, but will occur less frequently; while no error at all, or one of four inches will be a comparatively rare occurrence. If we now suppose the errors to act as often in one direction as the other, the effect will be to alter the average error by the amount of two inches, and we shall have the following results:—
| Negative error of 2 inches | 1 | way. |
|---|---|---|
| Negative error of 1 inch | 4 | ways. |
| No error at all | 6 | ways. |
| Positive error of 1 inch | 4 | ways. |
| Positive error of 2 inches | 1 | way. |
We may now imagine the number of causes of error increased and the amount of each error decreased, and the arithmetical triangle will give us the frequency of the resulting errors. Thus if there be five positive causes of error and five negative causes, the following table shows the numbers of errors of various amount which will be the result:—
| Direction of Error. | Positive Error. | Negative Error. | |
|---|---|---|---|
| Amount of Error. | 5, 4, 3, 2, 1 | 0 | 1, 2, 3, 4, 5 |
| Number of such Errors. | 1, 10, 45, 120, 210 | 252 | 210 120, 45, 10, 1 |
It is plain that from such numbers I can ascertain the probability of any particular amount of error under the conditions supposed. The probability of a positive error of exactly one inch is 210/1024, in which fraction the numerator is the number of combinations giving one inch positive error, and the denominator the whole number of possible errors of all magnitudes. I can also, by adding together the appropriate numbers get the