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MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



only oiiglit to answer right oftener tban wrong, but we ouglit to do so in a predictible ratio of 

 cases.* 



We have experimented with the pressure sense, observing the proportion of errors among 

 judgments as to which is the greater of two pressures, when it is known tliat the two are two 

 stated pressures, and the question presente<l for the decisiou of the observer is, which is wliich ? 

 From the probability, tlius ascertained, of committing an error of a given magnitude, the probable 

 error of a judgment can be calculated according to the mathematical theory of errors. If, now, 

 we find that when the ratio of the two ])ressures is smaller than a certain ratio, the erroneous 

 judgments number ouehaJf of the whole, while the mathematical theory requires them to be sen- 

 sibly fewer, then this theory is plainly disproved, and the maximum ratio at which this phenom- 

 enon is observed the so-called UnttrHchu'dHnchwcUe. If, on the other hand, the values obtained for 

 the probable error are the same for errors varying from three times to one-fourth of the i)robable 

 error (the smallest for which it is easy to collect sufficient observations), then the theory of the 

 method of least squares is shown to hold good within those limits, the presumi)tiou will be that 

 it extends still further, and it is possible that it holds for tiie smallest diffei'ences of excitation. 

 But, further, if this law is shown to hold good for differeuce so slight that the observer is not 

 conscious of being able to discriminate between them at all, all reason for believing in an Uuttr- 

 schicdsschwdh' is destroyed. The mathematical theory has the advantage of yielding conceptions 

 of greater detiniteness than that of the physiologists, and will thus tend to improve methods of 

 observation. Moreover, it affords a ready method for determining the sensibility or fineness of 

 perception and allows of a comparison with the results of others; for, knowing the number of 

 errors iu a certain number of experiments, and accepting the conclusions of this paper, the calcu- 

 lated ratio to the total excitation of that variation of excitation, in judging which we should err 

 one time out of four, measures the sensibility. Incidentally our experiments will afford additional 

 information upon the value of the normal avei-age sensibility for the pressure sense, which they 

 seem to make a finer sense than it has hitherto been believed to be. But in this regaixl two things 

 have to be noted : (1) Our value relates to the probable error or the value for the point at which 

 an error is committed half the time; (2) in our experiments there were two opportunities forjudg- 

 ing, for the initial weight was either first increased and then diminished, or riec versa, the sub- 

 ject having to say which of these two double changes was made. It would seem at first blush 

 that the value thus obtained ought to be multiplied by -/^ (I.41I) to get the error of a single judg- 

 ment. Yet this would hardly be correct, because the judgment, in point of fact, depended almost 

 exclusively on the sensation of increase of pressure, the decrease being felt very much less. The 

 ratio ^2 (1.414) would therefore be too great, and 1.2 would perha])s l)e about correct. The 

 advantage of having two changes in one experiment consists iu this: If only one change were 

 employed, then some of the experiments would have an increase of excitation only and the others 

 a decrease only ; and since the former would yield a far greater amount of sensation than the latter, 

 the nature of the results would be greatly complicated; but when each experiment embraces a 



* The rule for tiuding this ratio is as follows: Divide the logarithm of the ratio of excitatiixis by the probable 

 error and multiply the quotient by 0.477. Call this jnoduct t. Euter it iu the table of the integral Ot, given in most 

 works on probabilities; St is the proportion of cases iu which the error will be less than the ditt'erence between the 

 given excitations. In all these cases, of cour.4e, we shall answer correctly, and also by chance in one-half of the 

 remaining cases. The proportion of erroneous answers is therefore (1— Ot)— 2. In the following table the first col- 

 umn gives the i|niitient of the logarithm of the ratio of excitation, divided by the iirobable error, and the second 

 column sliows the proportion of erroneous judgments : 



To guess the correct canl out of a ])ack of tifty-two once in eleven times it would bo necessary to have a sensation 

 amounting to 0.:17 of the i>roliabIc error. This would be a .-eu.sation of wliicii we sliouhl i>robably never become 

 aware, as will appear below. 



