88 THE FACTORS OF THE MIND

of the terms) that work is done against resistance—i.e. by
overcoming the difficulty of the task,

The differences between the various tests, however, are
qualitative differences. Consequently, we must devise
some conventional rule for expressing qualitative differences
quantitatively. Let us assume that a similarity in the
direction of two lines denotes a corresponding similarity
in the nature of two tests. We can then use the calculated
correlation between those tests to measure the agreement in
direction: coincidence of direction will represent perfect
correlation; orthogonal directions will represent zero
correlations.1 With these assumptions we may now legiti-
mately enlarge the notion of directed forces, and, if we like,
speak of * mental forces' as responsible for these mental
changes of state. On this basis we "can regard any given
test-performance as the resultant of hypothetical compon-
ent forces or ' factors,' so chosen as to be mutually inde-
pendent ; and the correlation between any two tests will
then be deducible from the correlations between those
two tests and the components or factors in accordance with
the familiar cosine law.2

1 In the theory of factor-analysis, the idea of expressing correlations as
angular functions was, I think, first mooted in connexion with emotional
tendencies [30], The notion was based on Pearson's interpretation [10] of
partial correlation in terms of spherical trigonometry, referred to above:
taking the coincidence of directions (0 = o°) to represent perfect agreement,
reversal of direction (6 = 180°) to represent perfect disagreement, and there-
fore 0 = 90° to represent zero agreement, the natural functions are r =
•fa (90 —- 0) or sin (90 — 6). On this basis it was suggested that correlations
could be " represented inversely by distances of arc " ([30], p. 696 and dia-
gram). In re-examining some of Webb's data and my own, Maxwell Garnett
gave a better and more formal expression to these vague suggestions: extend-
ing the familiar correlation diagram to n dimensions, he formally deduced
what he called the applicability of the cosine law: putting 6 » cos~l f,
we may write cos 6 = cos a^ cos b1 + ... 4- c°s »i cos n2, where cos n^
denotes the correlation of the test i with the nth. factor [37].

An equally legitimate, but entirely different procedure (or so it might
seem at first sight) consists in expressing the correlations, not as cosines of
angles, but as ratios of the axes of the frequency-ellipse or ellipsoids; the
relation between the two alternative modes of representation is obvious
enough on considering the ordinary correlation diagram for two variables
only, and was given a general formulation in [93], pp. 253 et seq.

2 See below, p, 91.