APPENDIX. 



MEASUREMENT AND UNITS. 



292. Measurement. The mathematical theory of measurement rests 

 on the assumed possibility of dividing an object into an integral number of 

 parts which are identical in respect of some property. Thus, to measure 

 the length of a segment of a line, we must suppose the segment divided into 

 a number of equal segments, where the test of equality of length is con- 

 gruence ; to measure the mass of a body we must suppose it capable of 

 division into a number of bodies of equal mass, where equality of mass is 

 tested by weighing ; to measure an interval of time we measure the angle 

 turned through by the Earth in the interval ; this requires the division of 

 an angle into a number of equal angles, and the test of equality of angles 

 is congruence. 



The measurement of an object in respect of any property requires (1) a 

 unit or standard of comparison, and (2) a mode of referring to the standard. 

 The standard must be an object which possesses the property in question. 

 The mode of referring to the standard must be such that it determines a 

 positive number (integral, rational but not integral, or irrational) which is 

 the measure of the object in respect of the property. The number is deter- 

 mined by the following rules : 



(a) When the object can be divided into an integral number n of parts, 

 each of which is identical with the standard in respect of the property in 

 question, the measure of the object in respect of that property is n. 



(b) When the object and the standard can be divided into p and q parts 

 respectively (p and q being integers), such that all the parts are identical in 

 respect of the property in question, the measure of the object in respect of 

 that property is the rational fraction pjq. 



Here it is to be noted (1) that the rule (a) is the case of the rule (6) for 

 which <?=!, and (2) that in practice the integer q may be taken so large that 

 an integer p may be found for which the fraction p[q measures the object 

 within the limits of experimental error. 



In the mathematical theory of measurement the case where no rational 

 fraction p/q can measure the object may not be so simply dismissed. It may 

 happen that however great q is taken there is no corresponding number p, 

 but that, while the fraction plq would measure an object somewhat smaller 



