132 SCIENCE PROGRESS 



has been established with an exactitude " such as to rival the niceties which 

 physical measurements reveal." This factor has been named " general 

 ability " and is referred to as g. 



It should be noted here that in handling the numerical element in all 

 these calculations, allowance has to be made (systematically) for the inevit- 

 able margin of inaccuracy in the observations on which they are based, 

 and, when a result is said to be exact, it is meant that by the usual statis- 

 tical calculus it has been shown to fall within an assigned limit of probable 

 error. 



To obtain these results a considerable variety of tests is required, for if 

 the tests are confined to a group of similar qualities a discrepancy appears 

 and it becomes clear that some tertium quid has ceased to be negligible. It 

 is said that by combining suitable qualities (choosing those whose inter- 

 dependence is shown by their " high correlations "), and analysing the 

 resulting discrepancy, it is possible to isolate and define the discrepant 

 element. This proves to be a second independent factor, a " group-factor," 

 important within the group but without appreciable effect on other groups 

 of values. 



Now, it has been shown that, if certain mathematical conditions are ful- 

 filled, any three values can be expressed in terms of two independent factors 

 and two only. Thus, if within a group of mental qualities three could be 

 found whose values fulfilled these conditions, they could be expressed in 

 terms of the general factor and the group factor alone, without any specific 

 quality entering as a factor into any of them. This provides a more rigorous 

 test for membership of the group. Qualities can be taken and tested in 

 threes in this way until all qualities suspected of belonging to the group 

 have been subjected to the test. 



So far it has been possible to speak only in the most general terms of the 

 mathematical side of the discussion. At this point, however, the argu- 

 ment, in Mr. Garnett's hands, receives geometrical treatment, exceedingly 

 interesting and to some extent capable of verbal description. By adopting 

 a certain mathematical artifice (the conversion of measure of correlation 

 into direction cosines) it is possible to represent geometrically both the 

 correlations of qualities and the conditions for the dependence of three 

 qualities on two factors (or qualities) only. Thus, the correlation between 

 two qualities can be represented by the angle between two radii drawn 

 from a point, two qualities with no correlation being represented by radii 

 at right angles to each other. A third quality can be represented by a 

 third radius drawn from the same point but not necessarily in the same 

 plane. It most conveniently happens, however, that the geometrical equiva- 

 lent of the conditions for three qualities depending on two factors only is 

 that the three radii representing them shall lie in one plane ; in this case 

 the correlations between the three qualities taken in pairs are represented 

 by the angles between the first and second, the second and third, and the 

 first and third respectively. 



Thus, let us take three correlated qualities, two being specific qualities 

 within a group and one being g. Let us suppose that we know mathe- 

 matically that they depend on two independent factors only, the general 

 factor, or g, being one of them and the group-factor being the other, 

 and we wish to find out the nature of the group-factor. If we represent 

 the general factor by one radius we can represent the unknown group-factor 

 by a radius at right angles to it, since by hypothesis they are independent 

 (that is to say, their correlation is zero). We can represent the two specific 

 qualities by two other radii whose angles with the g radius will repre- 

 sent their correlations with g. Then, if one of these radii lie close to the 

 unnamed radius, there is a strong presumption that the quality it represents 

 is akin to the unknown factor. Such a diagram has been drawn, and on 



