NATURE  OF THE  FACTORIAL TECHNIQUE        81
of three axes at right angles to one another drawn through the centre
of the earth—a central axis through the poles, which would repre-
sent the appropriate g, and two supplementary or * specific' axes
orthogonal to this, through Greenwich and through Galapagos.
An astronomer of the Thurstone persuasion would reply that it was
uneconomical to use three sets of co-ordinates where two would
suffice, and further that the last two axes had * no astronomical
significance ' and so must be rotated. He would doubtless elect to
substitute the plane of the earth's orbit (the ecliptic) and the plane
of its equator, thus rotating the results of the initial analysis and
referring the entire data to a couple of oblique planes or axes only.
But throughout all the arguments and calculations, the general
formula which the navigator employs are almost identical with
those which the psychologist would use, if the heavenly bodies
were a set of persons to be tested, and if their measurements by
different observers at arbitrary spots represented the performances
of the persons with so many sets of arbitrary tests. Thus, every
day of his voyage, the navigator is carrying out a * factor-analysis.' ^x
We may push the analogy further. So far we have spoken of our
observers as interested primarily in the positions of the stars ; but
they will adopt exactly the same methods to determine the positions
of their ships. Similarly the psychologist can use the same
observations and the same mode of analysis to compare either the
characteristics of the individuals he is testing or the characteristics
of the tests he employs. With either mode of approach the navi-
gator's task is doubtless far simpler : for he is concerned with two
dimensions only—latitude and longitude. But evidently the
^1 The analogy leads me to point out in passing the occasional advantages
of using spherical co-ordinates instead of the more usual Cartesian co-
ordinates. As I have elsewhere shown ([93], pp. 247, 300), the formulae
used in multiple factor-analysis are merely special applications of the ordinary
formulae for multiple and partial correlation; and for three variables the
latter were given a suggestive interpretation in terms of spherical tri-
gonometry by Karl Pearson [10] : the procedure thus indicated can be
generalized to any number of dimensions. For teaching purposes, three
factors and their rotations can be represented very vividly to the eye by
' plotting coefficients on a black globe, such as is used in classes on geography
and navigation. Even non-mathematical students have usually picked up
some elementary notion of spherical geometry from their early lessons on
geography ; and for them the rudiments of factor-analysis can be expounded
quite simply in this concrete form. The approach by spherical trigonometry
was used in one or two early factorial papers, and will be found to lend itself
to interesting theoretical investigations, suggestive alike of new formulae and
new working methods.
6