known nothing previously about either A or B, we discover on trial that A can just distinguish between two weights such as those bearing the register-marks 5 and 7, and that B can just distinguish between another pair, say, bearing the register-marks 2 and 6; then since the difference between the marks in the latter case is twice as great as in the former, we know that the dulness of the muscular sense of B is exactly twice that of A. Their relative dulness, or if we prefer to speak in inverse terms, and say their relative sensitivity, is determined quite independently of the particular pair of weights used in testing them.

It will be noted that the conversion of results obtained by the use of one series of test-weights into what would have been given by another series, is a piece of simple arithmetic, the fact ultimately obtained by any apparatus of this kind being the “just distinguishable” fraction of real weight.  In my own apparatus the unit of weight is 2 per cent.; that is, the register-mark 1 means 2 per cent.; but I introduce weights in the earlier part of the scale that deal with half units; that is, with differences of 1 per cent.  In another apparatus the unit of weight might be 3 per cent., then three grades of mine would be equal to two of the other, and mine would be converted to that scale by multiplying them by 2/3.  Thus the results obtained by different apparatus are strictly comparable.

A sufficient number of test-weights must be used, or trials made, to eliminate the influence of chance.  It might perhaps be thought that by using a series of only five weights, and requiring them to be sorted into their proper order by the sense of touch alone, the chance of accidental success would be too small to be worth consideration.  It might be said that there are 5 x 4 x 3 x 2, or 120 different ways in which five weights can be arranged, and as only one is right, it must be 120 to 1 against a lucky hit.  But this is many fold too high an estimate, because the 119 possible mistakes are by no means equally probable.  When a person is tested, an approximate value for his grade of sensitivity is rapidly found, and the inquiry becomes narrowed to finding out whether he can surely pass a particular mistake.  He is little likely to make a mistake of double the amount in question, and it is almost certain that he will not make a mistake of treble the amount.  In other words, he would never be likely to put one of the test-weights more than one step out of its proper place.  If he had three weights to arrange in their consecutive order, 1, 2, 3, there are 3x2 = 6 ways of arranging them; of these, he would be liable to the errors of 1, 3, 2, and of 2, 1, 3, but he would hardly be liable to such gross errors as 2, 3, 1, or 3, 2, 1, or 3, 1, 2.  Therefore of the six permutations in which three weights may be arranged three have to be dismissed from consideration, leaving three cases only to be dealt with, of which two are wrong and one is right.  For the same reason there are only four reasonable chances of error in arranging four weights, and only six in arranging five weights, instead of the 119 that were originally supposed.  These are—­