THE LOGICAL STATUS OF MENTAL FACTORS 99

mining variables is far more complex than in astronomy or physics,
if only because of their greater number. But with the development
of statistical method during the last half century, formulae have
become available whereby at almost every stage we can calculate
suitable constants ; and, it may be added, whether we are dealing
with what the statistician would call the ' theory of attributes ' or
with the * theory of variables '—with qualitative classes or with
quantitative grades—his equations and his demonstrations rest at
bottom on much the same principles.

Now, when we are analysing not qualitative attributes
but the quantitative variables that replace them, the
factorial components, as we have seen, take the form of
vectors; that is, they can be represented by a line which
in its turn is specified by a length and an angle ; or, in
other words, they vary (i) by a definite amount, and (ii) in
a definite direction. Omit the quantitative element—
namely, thec amount'—and we are left with the qualitative
classification according to * direction.' Once again, there-
fore, we see that our factors still remain in their essential
nature principles of classification, though rendered more
discriminative and exact by being cast into quantitative
form. Where the plain man classifies children into ( good '
and c bad,' * bright' and c dull,' the administrator has to
lay down sharper lines of demarcation, and talks of * guilty *
and * not guilty,' c certifiably defective ? and * capable of
profiting by a secondary education,' and the like ; and the
theoretical psychologist tries to improve the classification
still further by thinking of the population as varying in
certain directions, each more or less independent of the
other and each carrying with it a number of inferable traits
—as varying in what he calls,f innate general intelligence,'

* innate specific abilities,' * innate general emotionality,5

* acquired moral character,' and the like—and then seeks to
graduate these variations on a linear scale. Thus, what
appear to be the measurements of a variable are merely
minute subdivisions by manifold classification as distinct
from the first broad differentiation reached by twofold
classification ; and the factorization of such variables is
simply a mode of reclassifying the varying individuals
according to a more refined logical procedure which has