300 REPORTS ON THE STATE OF SCIENCE, ETC. 



order of magnitude by the successive performance of our practical operation 

 of addition on the equal quantities at our disposal. In this way we obtain 

 a series of discrete samples of the magnitude having a relation structure 

 similar to that of an arithmetical progression. Though not essential, it is 

 convenient for identification purposes that the same numerals should be 

 used as names for the corresponding members of the phenomenal and 

 numerical series. The particular arithmetical progression which fulfils 

 this condition is that in which the first term and constant difference is the 

 number which we have arbitrarily associated with each of the equal samples 

 used to build up the series of magnitudes. If for convenience we take unity 

 for this number, our series of magnitudes consists of members, associated 

 by virtue of our practical criteria of equality and addition, with the cardinal 

 numbers i, 2, 3, 4, . . . etc., the number associated with each member 

 being simply the number of our original equal samples of the magnitude 

 which have been added together to obtain it. This series constitutes a 

 scale of measurement of the magnitude in terms of the original quantity 

 as the ' unit.' 



A scale of measurement would, however, be of little use if its significance 

 were confined to the members of the initial series built up in this way. We 

 want to be able to use the scale to ' measure ' any sample of the magnitude 

 which we chance to encounter. We do this by comparing the unknown 

 sample with the various members of our built-up series to find if it fulfils 

 our practical criterion of equality with any one of them. If we find it to 

 be equal to the member associated with the number n, we say that n is the 

 measure of the unknown sample. There is an obvious difficulty if we 

 cannot find a member in our standard series to which the unknown magni- 

 tude is equal, for we have not defined a practical criterion for any numerical 

 relation other than equality. All we can say in such a case is that the 

 sample measures more than n and less than « + i. In principle no scale 

 of measurement can be continuous because it involves an association with 

 number, which is essentially discontinuous. A scale of measurement can 

 only define and identify a discrete series of quantities. In practice of course 

 we can reduce the gap between successive members of the measurable 

 series by taking smaller samples of the magnitude for our initial collection 

 of equal quantities. If we take them so small that the gaps in our scale are 

 of the same order of magnitude as the uncertainty in determining whether 

 or not our practical test of equality is fulfilled we shall always be able to find 

 some member which seems to be equal to any given sample of the magnitude. 

 In other words we can measure any sample to within the accuracy of our 

 practical tests, though in principle any scale we can construct, however 

 fine-grained, has exactly the same kind of discontinuity as one constructed 

 with large steps. It would clearly be tedious to build up a complete scale 

 of any magnitude from very small samples of the magnitude, and in practice 

 various short cuts based on the numerical relations imposed by our initial 

 definitions of equality and addition are employed, but the fundamental 

 principle is not aflfected. 



It is obvious that to state the number which measures any given sample 

 of a magnitude tells us nothing unless we know what quantity has been taken 

 as the ' unit.' This is not necessarily the quantity of the members of the 

 initial collection of equal samples used to build up the scale : as already 

 mentioned we are free to associate any number we like with this quantity, 

 though in the absence of any reason to the contrary it would be natural to. 

 associate it with the number i . Usually, however, there are reasons to the 

 contrary, but these are extraneous to the principles of measurement and are 

 mere reasons of convenience which we need not go into here. There is no 



