A CRITICAL EXAMINATION 29 



on the nature of the data and, perhaps even more 

 on the nature of the person handling them, this 

 hypothesis may take a directly mathematical 

 form, as say the equation of a curve, or it may be 

 seemingly quite unrelated to anything mathemati- 

 cal — as for example chemical, cytological, psy- 

 chological, or what not. What the hypothesis is 

 does not matter, except in this respect that it always 

 somewhere involves a statement or implication as 

 to the qualitative cause of the quantitative phenomena 

 observed. The next step in the investigation is to 

 calculate out for each particular observed case the 

 numerical results to be expected on the basis of 

 the hypothesis. These "expectations" are com- 

 pared with the observations. If the agreement 

 is good, the investigator is likely, and here lies 

 the fallacy, to draw the conclusion that this 

 agreement proves that the qualitative assumptions 

 made in the hypothesis are correct. Of course 

 the agreement logically proves nothing of the 

 sort. The reason why it does not is found in the 

 lack of uniqueness in the quantitative relations of 

 qualitatively distinct natural phenomena. 1 Be- 

 cause two series of events follow the same curve it 

 by no means follows that they are due to the same 

 cause. A reasonable accordance of the numerical 



1 It should be pointed out here that in the present development 

 of this argument I am drawing freely from a previous paper (Amer. 

 Nat., Vol. XLIII, pp. 302-315, 1909), in which the same point was 

 discussed in relation to investigations on growth. 



