298 REPORTS ON THE STATE OF SCIENCE, ETC. 



a connection we must artificially associate a phenomenal criterion with 

 numerical equality and a phenomenal operation with numerical addition. 

 When we have done this, but not before, we can predict by arithmetical 

 calculation those phenomenal relations which involve only the stipulated 

 practical criteria of equality and addition. A phenomenal class defined by 

 two such practical criteria constitutes a measurable magnitude of the type 

 which Dr. Norman Campbell, in his well-known text-book on the principles 

 of measurement, has termed an A magnitude. Any such magnitude can 

 be measured by processes which do not imply the measurability of any other 

 magnitude. It is true that the practical criterion of equality for any A 

 magnitude will always involve the observation of some phenomenal state or 

 condition which may (and usually does) involve other magnitudes ; but the 

 observational criterion is always of the null type — no difference, or no 

 observable change, in the prescribed state or condition : no numerical 

 relations have to be determined or even be assumed to exist for these other 

 magnitudes. Familiar examples oi A magnitudes are length, volume, mass, 

 electrical resistance, and many others which need not be detailed. The 

 practical criteria of equality and addition which define these magnitudes for 

 purposes of measurement are sufficiently familiar to require no description. 

 It is their significance which is not so widely recognised as it might be. It 

 is probably usual to regard the experimental processes of determining 

 equality and of adding as something which we have just found to be a con- 

 venient method of determining quantitative relations inherent in the nature 

 of the magnitudes, whereas these processes are the necessary connecting 

 links between phenomena and number without which there would be no 

 basis of comparison between the laws of the former and those of the latter. 

 The experimental criteria do not merely enable us to measure a magnitude, 

 they create the magnitude by defining the fundamental relations which 

 are to be the basis of classification. 



In Physics the general term measurement is not confined to A magnitudes. 

 By suitable experimental processes we affix numerals to many aspects of 

 phenomena to which no operation can be performed having any similarity 

 in relation structure to the operation of addition. Familiar examples are 

 density, specific heat, electrical resistivity, etc. All those things which we 

 ordinarily regard as properties of substances are magnitudes of this type. 

 They are the B magnitudes of Campbell's classification. We can usually 

 formulate a practical criterion of equality for a B magnitude, but not of 

 addition. Nevertheless the numerals affixed to these magnitudes by our 

 experimental processes associate the members of the magnitude series with 

 the members of the series of numbers in such a way that predictions based 

 on arithmetic will always come out right. This does not, however, mean 

 that measurement without a practical criterion of addition is possible. It 

 results from the fact that B magnitudes are evaluated simply as a function 

 of the measured quantities of two or more A magnitudes. The density 

 of a substance, for example, is nothing more nor less than the ratio of the 

 numbers which measure the two A magnitudes, volume and mass, associated 

 with any lump of the substance. Its importance is simply that this ratio 

 is found to be approximately constant for all lumps of what we call the 

 ' same ' substance and so is worth noting as a property of the substance. 

 But it would be impossible to do what, for brevity, we call measuring density 

 unless we were already able to measure volume and mass. The associations 

 between phenomena and number required for the measurement of density 

 are supplied, not in practical criteria applicable to density as such, but in' 

 the practical criteria of equality and addition on which the scales of volume 

 and mass are based. 



