CHAPTER XIV. 



UNITS AND STANDARDS OF MEASUREMENT. 



As we have seen, instruments of measurement are 

 only means of comparison between one magnitude and 

 another, and as a general rule we must assume some 

 one arbitrary magnitude, in terms of which all results 

 of measurement are to be expressed. Mere ratios be- 

 tween any series of objects will never tell us their 

 absolute magnitudes ; we must have at least one ratio 

 for each, and we must have one absolute magnitude. The 

 number of ratios n are expressible in n equations, which 

 will contain at least n + i quantities, so that if we 

 employ them to make known n magnitudes, we must 

 have one magnitude known. Hence, whether we are 

 measuring time, space, density, mass, weight, energy, or 

 any other physical quantity, we must refer to some con- 

 crete standard, some actual object, which if once lost and 

 irrecoverable, all our measures lose their absolute mean- 

 ing. This concrete standard is in all cases arbitrary in 

 point of theory, and its selection a question of practical 

 convenience. 



There are two kinds of magnitude, indeed, which do not 

 need to be expressed in terms of arbitrary concrete units, 

 since they pre-suppose the existence of natural standard 

 units. One case is that of abstract number itself, which 

 needs no special unit, because any object which exists or 

 is thought of as separate from other objects (p. 157) fur- 

 nishes us with a unit, and is the only standard required. 



Angular magnitude is the second case in which 

 we have a natural unit of reference, namely, the whole 



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