THE LOGICAL  STATUS OF MENTAL FACTORS    109
minimizing the number of factors, but by maximizing the amount of
variation that each in turn will account for. If, therefore, the psycho-
logist has a dozen tests for a dozen different abilities, which differ in
a way that is at once significant and relevant to his problem, he must
be prepared to admit a dozen different factors.
In practice, of course, he will probably be unable to show that
they are all genuinely significant, and will usually confine his
calculations to three or four factors at the most. But he must not
start by limiting the amount of discoverable variance on purely a
priori grounds at the very outset. If a solid object has three
dimensions, it is not a ^c fundamental postulate of science' that the
axes of reference should be reduced to two ; the dictates of economy
are satisfied if the axes chosen are independent. Similarly, if a
correlation matrix is of order n X n, we must not exploit our ignor-
ance of the diagonal entries to insist on reducing it to a rank^x of
r = n — *\/2n. Indeed, if the correlation matrix were a covariance
matrix, such a reduction would in general prove impossible.
The foregoing requirements will become clearer still, if we turn
for a moment from a study of the resemblances between traits to a
study of the resemblances between persons.
The Four-factor Theory : (b) in Correlating Persons.—
Most psychologists have started by correlating traits or
tests. But is there any reason why we should not start
by correlating persons ? With this approach coefficients
of correlation can be calculated which will serve to measure
resemblances, not between traits, but between persons.
Fact or-analysis can be applied on the same lines as before ;
and factors of the same four types will presumably emerge.
The student who is not yet familiar with factorial research
finds it actually easier (that at least has been my own
^1 With this approximate formula (accurate enough for most purposes), if the
value is not integral, we take the integer next above the value given. The
exact formula for determining the minimum rank is r =s= n — -JV %^n + !+•£>
where, if the value is not integral, we take the integer next below.
With the simple summation method, the saturation coefficients are
virtually deviations about an average. After the first, each column of
coefficients must add up to zero ; after the second column, each section of a
column (cf. p. 466 and Tables VIIx ii, x!a). Thus the number of degrees
of freedom is one less with each factor, i.e. n, n—i, n—2, .,., until all
the \n (n—i) degrees are used up. (Weighted summation has the same
effect.) On summing and solving for r, the above formula follows at once.