ESSAYS 



A PSYCHOLOGICAL GEOMETRY (F. R. Hoare, B.A., F.R.A.I.) 



It is at first sight somewhat strange that the quantitative and statistical 

 methods from which so much was lioped in biology a generation ago should 

 so far have proved more fruitful and suggestive when applied to the appar- 

 ently much less measurable phenomena of the mind. Yet the hopes that 

 the early biometricians rested on them have certainly not been realised and 

 their present exponents in biology have had to make practically a fresh 

 start with new groups of phenomena and different objectives. In psy- 

 chology, on the other hand, their recent developments derive in unbroken 

 succession from the work of such pioneers as Binet, who first employed 

 systematic mental tests and measured the results. 



One of the most purely mathematical of these developments was origin- 

 ated by Professor Spearman in a paper published in 1904 in the Atnerican 

 Journal of Psychology. Its latest and most curious extensions are described 

 in two papers by Mr. Maxwell Garnett published in the Proceedings of the 

 Royal Society (A, 96, 1919) and the British Journal of Psychology (May 1919) 

 respectively. What follows Is largely an account in non-mathematical 

 language of as much of Mr. Garnett's description as can be so expressed. 



It is necessary to note the preliminary mathematical assumptions. In 

 the first place, any mathematical analysis of the numerical values obtained 

 by the measurement of mental phenomena must start by expressing any such 

 series of values in terms of a system of factors. These factors may, if we 

 please, be taken to represent mental qualities underlying and accounting 

 for the values. In the absence of further evidence, at least as many factors 

 must be allowed for as there are values measured, and allowance must also 

 be made for the possibility that each value is influenced to some extent by 

 every factor. 



A definite test of interdependence is given by the mathematical concept 

 of correlation. Two factors (or qualities) are said to be correlated when 

 their numerical values vary in some sort of relationship with each other 

 (for example, that of simple proportion, though it is commonly a much 

 more complex one) ; moreover, their degree of correlation can be measured. 

 The aim of mathematical analysis is to find the irreducible minimum of 

 independent factors (factors whose correlation is zero) in terms of which 

 all the values can be expressed. These factors, if they are not by nature 

 or definition constant in value, are called independent variables. 



It was argued by Spearman in the paper referred to that, if a series of 

 values fulfils certain mathematical conditions, a first step towards reducing 

 the number of factors can be taken by expressing each value in terms of one 

 special factor (representing a quality contributing to that value and to that 

 alone) and an independent factor (representing some quality of the indi- 

 vidual that contributes to all the values). It is said that measurements 

 previously recorded and others specially made {e.g. by Burt and Webb) 

 do, in fact, fulfil these conditions, and Dr. Webb is quoted as asserting that, 

 on the basis of a vast mass of material, the existence of the " general factor " 



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