282 



SCIENCE. 



[N. S. Vol. XI. No. 269. 



ward the middle of tlie century, Dumas 

 made his classic experiments on the com- 

 position of water. The probable error of 

 a single experiment was, in round num- 

 bers, 1 part in 400, so that the probable 

 error of the average of the 19 famous ex- 

 periments was 1 part in 2250. ISTow, this 

 means that his final value was not likely to 

 differ more than a certain small quantity 

 from the result of the repetition of even a 

 very large number of experiments made in 

 the same way, with the same skill and care. 

 But as to the difference between this result 

 of the 19 experiments and the unknown 

 true value, we are told absolutely nothing 

 by the proposition that the probable error 

 of Dumas' result was 1 part in 2250. It is 

 a commonplace to say, that the calculation 

 of the probable error of a series of experi- 

 ments does not show how nearly the result 

 approaches the truth, but how near it is to 

 the result of a greater number of similar 

 experiments. It decides, not how nearly 

 we approach the desired goal, but whether 

 it is useful to persevere by the present 

 method of approach. Dumas made 19 ob- 

 servations, and got the value, 15.96, mth 

 a probable error of 0.007 ; that is, if he had 

 made 100 or 1000 experiments, it is un- 

 likely that the final result would not have 

 been between 15.95 and 15.97, and very 

 unlikely indeed that it would not have 

 been between 15.94 and 15.98. But he 

 would never have obtained a value near 

 that which now commands confidence. 



It is interesting to recall that there is 

 hardly any instance on record where the 

 judgment of an experimenter as to the de- 

 gree of approximation to the truth attained 

 in his work has been better justified than 

 in the case of Dumas' classic experiments. 

 As we all remember, towards the end of his 

 work, there was discovered in his own 

 laboratory a source of error, not easy to 

 eliminate, which had affected all his deter- 

 minations. The amount of the error was 



not a fixed quantity, and no numerical cor- 

 rection could be applied to the results of 

 observation. Dumas accordingly gave to 

 the public the uncorrected and unmodified 

 results of experiment. But he also stated 

 his opinion as to the degree in which his 

 results approximated, not to the mean of a 

 larger number of experiments of the same 

 kind, but to the unknown and unattainable 

 true value. He expressed the hope that 

 his value would be found not more than one 

 part in 200 from the result of those subse- 

 quent experiments which should be thought 

 satisfactory ; and it is by just 1 part in 200 

 that his value differs from that which is 

 now accepted. 



So the concordance of Dumas' experi- 

 ments did not prove that his result was 

 right ; neither did the agreement of experi- 

 ments by different observers. Erdmann 

 and Marchand made eight experiments by 

 a method like that of Dumas, with some 

 modifications. Their result was 15.973, 

 with a probable eiTor of 0.011. This value 

 differs from that of Dumas by less than the 

 sum of the probable errors, so that that 

 agreement is perfectly satisfactory. So, 

 also, Eegnault determined the ratio of the 

 densities of oxygen and hydi'ogen, from 

 which was computed the atomic weight of 

 oxygen as 15.963, with a probable error of 

 0.004. The results of Dumas, of Erdmann 

 and Marchand, and of Eegnault, show a 

 very good agreement. But all of them, 

 and the mean of all of them, we now know 

 to be in error by 1 part in 200. 



I adduce this example, somewhat in de- 

 tail, to enforce the proposition that we must 

 not excuse ourselves from looking for error 

 because observations agree. We have ex- 

 periments which give the atomic weight of 

 oxygen with a probable error of 1 part in 

 50,000, but do we know it within 1 part 

 in 1000? Each individual experimenter 

 whose work would now be regarded as free 

 from known and tangible error, agrees 



