out as a condition of logical relations, and the symbols indicating them (pp. 32–35), and which was afterwards shown to attach equally to numerical symbols, the derivatives of logical
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 it empirically than to test the truth of one of Euclid’s propositions mechanically. Nevertheless, it is an interesting occupation to verify even the propositions of geometry, and it is still more instructive to try whether a large number of observations will 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 observations of the differences of Right Ascension of the sun and two stars, namely α Aquilæ and α 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 one-fourth part of a second (0·2637). He then compared the numbers of errors of each magnitude from 0·1 second upwards, as actually given by the observations, with what should occur according to the Law of Error.
The results were as follow:—8
| Magnitude of the errors in parts of a second. | Number of errors of each magnitude according to | |||
|---|---|---|---|---|
| Observation. | Theory. | |||
| 0·0 | to | 0·1 | 94 | 95 |
| ·1 | " | ·2 | 88 | 89 |
| ·2 | " | ·3 | 78 | 78 |
| ·3 | " | ·4 | 58 | 64 |
| ·4 | " | ·5 | 51 | 50 |
| ·5 | " | ·6 | 36 | 36 |
| ·6 | " | ·7 | 26 | 24 |
| ·7 | " | ·8 | 14 | 15 |
| ·8 | " | ·9 | 10 | 9 |
| ·9 | " | 1·0 | 7 | 5 |
| above | 1·0 | 8 | 5 |