MEASUREMENT 129 



orange, yellow, and so on ; but in this order there would 

 be no room for reds of different lightness. Colours 

 cannot be arranged in a single order, and it is for this 

 reason that colour is not measurable as is density. 



NUMERICAL LAWS 



But though arrangement in this manner in an order 

 and the assignment of numerals in the order of the pro- 

 perties are to some extent measurement and represent 

 something physically significant, there is still a large 

 arbitrary element involved. If the properties A, B, C, 

 D, are naturally arranged in that order, then in assigning 

 numerals to represent the properties I must not assign 

 to A 10, to B 3, to C 25, to D 18 ; for if I did so the order 

 of the numerals would not be that of the properties. 

 But I have an endless number of alternatives left ; I 

 might put A i, B 2, C 3, D 4,; or A 10, B 100, C 1,000, 

 D 10,000 ; or A 3, B 9, C 27, D 81 ; and so on. In the 

 true and fundamental measurement of the first part of 

 the chapter there was no such latitude. When I had 

 fixed the numeral to be assigned to one property, there 

 was no choice at all of the numerals to be assigned to the 

 others ; they were all fixed. Can I remove this latitude 

 here too and find a way of fixing definitely what numeral 

 is to be assigned to represent each property ? 



In some cases, I can ; and one of these cases is density. 

 The procedure is this. I find that by combining the 

 numerals representing other properties of the bodies, which 

 can be measured definitely according to the fundamental 

 process, I can obtain' a numeral for each body, and that 

 these numerals lie in the order of the property I want to 

 measure. % If I take these numerals as representing the 

 property, then I still get numerals in the right order, but 

 the numeral for each property is definitely fixed. An 

 example will be clearer than this general statement. In 



