ORDER, NUMBER, QUANTITY 275 



of such techniques is far beyond the scope of this book. 

 In addition to what has already been said x a few remarks 

 may be made. 



In the first place, measurement should be understood in 

 its true light as an operation which is essentially abstractive. 

 The attempt is made to consider only one property of an 

 event, and to portray this only in its quantitative aspect. 

 This is the foundation for the claim that quantitative meth- 

 ods in science lose the qualitative features of events and 

 reduce them to abstract structures. It is well to note that 

 although the measured values are numbers, and are manip- 

 ulated according to the rules of mathematical operations, 

 they are not purely such. A measured value is always a 

 qualitative entity, for it is a value of something, e.g., a 

 length, a mass, a duration, or a velocity. Thus quality is 

 not completely lost in the measurement, for there must 

 always be reference to the unit in terms of which the meas- 

 urement is made. But it is true that the measuring tech- 

 niques serve to preserve not the individual features of the 

 quality but only its abstract quantitative aspects, such as 

 its position in a series of similar quantities, and its reduci- 

 bility or irreducibility to other quantities. 



In the second place, brief recognition should be made of 

 the essential measurement techniques. They may be roughly 

 divided into three types. The first and most common sort 

 is the direct measurement by scales and clocks of extensive 

 quantities, such as lengths, areas, volumes, angular spreads, 

 and durations. This may require the actual physical ma- 

 nipulation of certain devices such as tapelines and pro- 

 tractors. Or the operations may be incorporated into re- 

 cording machines which require nothing more of the observer 

 than the reading off of a number. The second kind of 

 measurement is applicable to quantities which are not ex- 

 tensive but which have convenient extensive correlates; in 

 cases of this kind the intensive variations of the quantity 

 to be measured are associated through physical laws with 



1 Chapter VI, pp. 112 et seq. 



