QUANTITATIVE ESTIMATES OF SENSORY EVENTS 301 



reason inherent in the nature of things why the number i should be associ- 

 ated with any particular quantity of any magnitude. What is essential to 

 measurement is that this number shall be uniquely associated with some 

 quantity, and that standard samples shall be available by means of which 

 we can reproduce this association at any time. These standard samples 

 need not be, though they often are, of the exact quantity constituting the 

 unit. They may have any value on the scale defined by the unit. Their 

 function is to make possible the co-ordination of the scales of the various 

 measuring instruments which have to be used in practice. Every measuring 

 instrument has a scale of its own. If it is constructed on. correct principles 

 its scale will be similar to the standard scale but will not in general be 

 identical with it. By measurement on the scale of the instrument of a 

 standard sample whose value on the standard scale is specified we ' calibrate ' 

 the instrument so that values on the standard scale can be deduced from 

 its readings. 



It is clear that standards are only necessary for A magnitudes. No 

 standards are required for B magnitudes, whose numerical values are 

 entirely determined by the scales used for measuring A magnitudes. 



It is obvious that the practical criterion of equality and the practical 

 operation of addition which together define an A magnitude must be 

 applicable at all parts of the scale of the magnitude. In arithmetic it is 

 not only true that 1 + 1 + 1 + 1 + 1=5, but also that 1+4 = 5 ^i^d 

 that 2 -(- 3 = 5 and also that 5 + 5 = 10, 5 + 8 = 13, and so on. Once 

 we set up a scale of measurement for a magnitude, and begin to apply 

 arithmetic to the results of measurement, the laws of arithmetic will predict 

 that phenomenal relations ' similar ' to the numerical relations just quoted 

 will exist among the members of the magnitude series, and these predictions 

 are meaningless unless the practical criteria of equality and addition are 

 applicable irrespective of the ' size ' of the samples to be compared or 

 combined. Thus it is not enough to have a practical criterion of equality 

 for, say, length which can only be applied to samples of i mm., or a criterion 

 of addition of lengths which can only be applied to samples i mm. long. 

 From such limited criteria we could construct any number of equal samples 

 I mm. long and could add them to form a series in ascending order of magni- 

 tude, but the process would be of no use whatever for measurement. The 

 laws of arithmetic would predict that one group of five of our equal samples, 

 added by our criterion of addition, should be equal to any other group of 

 five similarly added ; but this arithmetical prediction has no phenomenal 

 equivalent if the practical criterion of equality by which our i mm. samples 

 are selected is for some reason inapplicable to the comparison of 5 mm. 

 samples. Further, arithmetic predicts that one 5 mm. sample added to 

 another 5 mm. sample is equal to a sample obtained by adding ten of our 

 I mm. samples. This prediction is again meaningless if our process for 

 adding the i mm. samples is not also available for adding 5 mm. samples. 

 There is, in fact, an important principle of measurement, which as far as 

 I am aware has never been explicitly stated because it is so obviously ful- 

 filled in most of the measurements carried out by physicists that to state it 

 seems superfluous. This principle is that the phenomenal significance of 

 equality and addition as applied to any magnitude must remain the same to 

 whatever samples of the magnitude they are applied. 



If this principle is violated by changing the significance either of addition 

 or equality as we ascend the series of magnitudes, we destroy the cross- 

 relation between the series of magnitudes and the series of numbers on 

 which their ' similarity ' depends, and the result, whatever it may be, is not 

 measurement. 



