GENERAL- AND GROUP-FACTOR METHODS 309

second : and this relation it demonstrates to be essentially a
relation between a set of positive group-factors, on the one hand)
and a corresponding set of bipolar factors, on the other; obtained
by taking weighted differences between the former.

Our concrete example will make this clearer. To find p^s
measurement in the new first factor, T instructs us to take a large
positive amount of the old first factor (e general intelligence ') and
add small positive amounts of all the others. To find his measure-
ments for the new second factor, we take none of the old first
(' general intelligence ') because the new factor is highly specific ;
but we take a large positive amount of the old second factor (esti-
mated from the verbal tests of reading and spelling) and smaller
negative amounts of the other three factors (which were based on
the non-verbal tests). In the case we are imagining, therefore, the
new second factor will depend on the difference between the man's
measurement in the verbal tests, on the one hand, and his measure-
ments in the non-verbal tests, on the other. Similarly with the
other factors. The bipolar character of all of them is thus at once
explained : their saturation coefficients will now range, not from

0 to + i, but from + r (denoting, e.g., complete verbality) to — I
(denoting complete non-verbality), and, between these limits, will
express the difference between the two contrasted tendencies.

Although bipolar factors are generally supposed to be devoid of
intelligible meaning when they are obtained by correlating cognitive
tests, nevertheless, if we were to correlate precisely the same set of
measurements by persons instead of by tests, we should accept such
bipolar factors as quite natural: we continually class children into
6 verbal' and ' non-verbal' types, ' practical' and * non-practical'
types, * mathematical' and * non-mathematical' types, and so on,
and such antithetical pairs logically imply a bipolar principle. Now

1 have endeavoured to show that, whether our analysis starts by
correlating persons or by correlating traits, the same set of factors
will be discernible in the same set of measurements. It follows that
the bipolar factors must be equally intelligible when we start with
the tests instead of with the persons tested. As I have argued
elsewhere ([128], p. 69), if we grant that factors are essentially
principles of classification, then bipolar factors denote the correspond-
ing principles of' dichotomous ' classification, while the corresponding
group-factors deducible from them denote the corresponding principles
of ' manifold' or co-ordinate classification. And no one would contend
that the dichotomous classification of, say, c Porphyry's tree 9 was
necessarily devoid of meaning, because half the subaltern genera