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hold both the previous doctrine of "intel- 

 ligence," and the contradictory doctrine 

 of "faculties." We talk glibly of our 

 friends' "judgment," "concentration," 

 "imagination," or what not, just as we 

 speak of their intelligence. As soon as 

 we attempt to make the conception of 

 faculties more definite, however, we find 

 ourselves involved in difficulties. There 

 is no general agreement as to what facul- 

 ties should go on our list; no two psychol- 

 ogists would agree; and as soon as we 

 have invented a list, some other worker 

 can show that some of the faculties which 

 we have considered as functional units 

 ought really to be analyzed further. 



The logical conclusion to which this 

 process of analysis leads us is that there 

 exist innumerable elementary abilities, 

 which may be either independent or inter- 

 correlated. Those who adopt this view 

 attempt to justify their mental testing 

 procedure on the ground that they thus 

 secure an average sample of a person's 

 abilities. Spearman points out here that 

 for our average to have any meaning it 

 must be based on a really representative 

 selection, in which all abilities are repre- 

 sented, and in which duplication is 

 avoided. This, however, is just what the 

 present tests do not do. There is not 

 really representative selection . There can- 

 not be, for there is no agreement as to the 

 elementary abilities which are to be tested. 

 Further, there is the question as to whether 

 such a sample would really have any 

 great meaning if we obtained it. 



That there is some meaning in the cur- 

 rent mental testing is, however, obvious. 

 The curious fact is that tests based on 

 widely different theories, and apparently of 

 widely different forms, do show substan- 

 tial correlations with each other. Some 

 theory is required to unify the facts. 

 Having disposed, to his own satisfaction, 

 of all the rival theories, Spearman ad- 

 vances his own. 



Stated in its simplest terms, the theory 

 is that the achievement of any individual 

 x in any ability aha linear function of 

 two factors g and s, g being common to 

 most if not all abilities, and s being specific 

 to each ability. Put otherwise, the ob- 

 served correlation between activities is 

 usually due to the g, which is common to 

 all of them, and only rarely to the fact 

 that they share an s. Or, mathematically, 

 we may write 



max = r aB gx + r aSa Sax ) 



m bx = n g g x + n S i sbx \ for any individual x (1) 



m a y — Tag gy ~T fas a Sal 



for any individual y 



Such a system we shall refer to as a 

 (g> -0 system. 



Now if we accept this theory, certain 

 results can be derived from such a set of 

 equations. We assume that the correla- 

 tions of g with any s, and of one j- with 

 another, are all zero. Then the correla- 

 tion of one ability with another will be 

 due entirely to g, and we can show, with 

 a little algebra, that 



r r, a l r r * 



and that 



= 



(1) 



The quantity on the left of equation 

 (z) is termed by Spearman the "tetrad 

 difference" of the correlations involved. 

 We shall write (ab, cd) for this quantity. 

 •It is, of course, evident that four abilities 

 will (by permutation of the subscripts) 

 give us six tetrad differences, three of 

 which will be numerically equal but of 

 opposite sign to the other three. 



The importance of the tetrad difference 

 in Spearman's theory is that he makes it 

 the criterion by which the whole theory is 

 to be judged. It has been shown that if 



