QUANTITATIVE ESTIMATES OF SENSORY EVENTS 299 



Strictly speaking, therefore, the only measurable magnitudes are A magni- 

 tudes. Density is only ' measurable ' in the sense that we can arrive at an 

 evaluation of it by processes of measurement — measurement of the A magni- 

 tudes volume and mass ; and any relation involving density is primarily a 

 relation involving volume and mass. Similarly what we call the ' measure- 

 ment ' of any other B magnitude is really only the measurement of two or 

 more A magnitudes and the combination of the results by some appropriate 

 formula to give a single number which we term the ' value ' of the B magni- 

 tude in question. Confusion is sometimes caused by the fact that in practice 

 we frequently evaluate A magnitudes by processes which are more appro- 

 priate to B magnitudes. After scales of measurement have been established 

 for many magnitudes, and instruments of various kinds developed 

 for their practical evaluation, it is often easier with the apparatus at one's 

 disposal to evaluate an A magnitude indirectly from measured amounts of 

 other magnitudes than to measure it directly. For example, electrical 

 resistance, which is an A magnitude, can be measured as such with a 

 Wheatstone bridge system which does not involve measurement of any 

 other magnitude. It can also be determined indirectly from measurement 

 of the potential difference required to drive a measured current through it, 

 and this quite often is the most convenient method. Similarly length, 

 which is pre-eminently an A magnitude, is often measured indirectly by 

 methods involving measurements of the angles of a triangle. But these 

 indirect methods are only possible because we have previously established 

 the scales of measurement of the A magnitudes by direct methods involving 

 only the criteria of equality and addition appropriate to each, and have, then, 

 from measurements made possible by the existence of these scales, deter- 

 mined the various quantitative relations among phenomena which we must 

 know before we can deduce the value of an A magnitude from measurements 

 of other magnitudes. 



We must clearly distinguish between indirect methods of measuring A 

 magnitudes which we may adopt as a matter of choice ajter we have accumu- 

 lated various experimental data involving previous direct measurement of 

 the magnitudes, and the indirect methods that are inevitable in the case of 

 B magnitudes, which, because there is no criterion of addition applicable 

 to them, cannot be measured except as the numerical constants in laws 

 relating the quantities of two or more A magnitudes which are found to be 

 associated together in some important class of circumstances. Density, 

 for example, is simply the constant in the experimental law that for any 

 lump of a given substance under specified conditions if we determine the 

 number which is the measure of its mass on the scale appropriate to mass 

 and the number which is the measure of its volume on the scale appropriate 

 to volume the ratio of the former number to the latter is constant. 



The significance of all measurement is therefore derived from the principles 

 of measurement of A magnitudes. These, very briefly, are as follows : 

 We prepare a large number of samples of some physical entity which 

 exhibits the magnitude in question. In the case of length or mass, for ex- 

 ample, these samples will be material rods or lumps. In the case of a 

 magnitude like intensity of radiant energy the samples of the physical 

 entity exhibiting the magnitude may consist of lamps or other sources of 

 radiation operating under constant conditions. 



By some method appropriate to the particular case we adjust these samples 

 until they fulfil our experimental criterion of equality. In this way we 

 obtain any required number of ' equal ' quantities of the magnitude. We 

 attach the same numeral to each of these quantities, thereby associating 

 them with one number. We then produce other quantities in increasing 



