730 SCIENCE AND HYPOTHESIS 



that there is a law of the probability of errors. Is there a law of 

 errors? The law to which all calculators assent is Gauss's law, that 

 is represented by a certain transcendental curve known as the " bell." 



But it is first of all necessary to recall the classic distinction be- 

 tween systematic and accidental errors. If the metre with which we 

 measure a length is too long, the number we get will be too small, 

 and it will be no use to measure several times that is a systematic 

 error. If we measure with an accurate metre, we may make a mis- 

 take, and find the length sometimes too large and sometimes too 

 small, and when we take the mean of a large number of measurements, 

 the error will tend to grow small. These are accidental errors. 



It is clear that systematic errors do not satisfy Gauss's 

 law, but do accidental errors satisfy it? Numerous proofs have 

 been attempted, almost all of them crude paralogisms. But start- 

 ing from the following hypotheses we may prove Gauss's law: 

 the error is the result of a very large number of partial and inde- 

 pendent errors; each partial error is very small and obeys any law 

 of probability whatever, provided the probability of a positive error 

 is the same as that of an equal negative error. It is clear that these 

 conditions will be often, but not always, fulfilled, and we may reserve 

 the name of accidental for errors which satisfy them. 



We see that the method of least squares is not legitimate in every 

 case; in general, physicists are more distrustful of it than astrono- 

 mers. This is no doubt because the latter, apart from the systematic 

 errors to which they and the physicists are subject alike, have to con- 

 tend with an extremely important source of error which is entirely 

 accidental I mean atmospheric undulations. So it is very curi- 

 ous to hear a discussion between a physicist and an astronomer about 

 a method of observation. The physicist, persuaded that one good 

 measurement is worth more than many bad ones, is pre-eminently 

 concerned with the elimination by means of every precaution of the 

 final systematic errors ; the astronomer retorts : " But you can only 

 observe a small number of stars, and accidental errors will not dis- 

 appear." 



What conclusion must we draw ? Must we continue to use the 

 method of least squares ? We must distinguish. We have eliminated 

 all the systematic errors of which we have any suspicion ; we are quite 

 certain that there are others still, but we cannot detect them ; and yet 

 we must make up our minds and adopt a definitive value which will 

 be regarded as the probable value; and for that purpose it is clear 

 that the best thing we can do is to apply Gauss's law. We have only 

 applied a practical rule referring to subjective probability. And there 

 is no more to be said. 



Yet we want to go farther and say that not only the probable 

 value is so much, but that the probable error in the re- 



