82 THE FACTORS ,OF THE MIND

principle may be generalized. If, for example, we were piloting,
not a surface vessel but a submarine or an aeroplane, he would
require to work with three dimensions. And Jules Verne, who
imagined a vessel steering through a universe of n dimensions, had
merely to set his navigator the same problem in w-dimensional form.
As a matter of fact, numerous questions in aeronautics are nowadays
solved by expressing the spatial data as matrices, and then reducing
the matrices to terms of simpler factors by precisely the same iterative
devices as are used for finding factor saturations in psychology,1

These analogies suggest a further conclusion. They create a
strong presumption that the factors in terms of which the psycho-
logist ultimately expresses his results can at most claim only the
same kind of existence as the lines or points to which the navigator
refers his measurements. The ecliptic and the equator, the poles
and the first point of Aries, are not concrete objects like the stars
themselves : they are simply items in an abstract jrame of reference.
As such they are naturally presumed to be constant; but they are
wanted merely for descriptive purposes ; and no one would be tempt-
ed to assign them an actual physical existence. No doubt, where
selection is possible and knowledge permits, it will be convenient to
choose such points and lines as possess some simple relation to
extraneous bodies.2 When, however, such extraneous existents,
and relations to such existents, are to be established, something
more than a mere mathematical analysis is employed.

1 The interested reader will find it instructive to compare some of the
methods employed by the Aerodynamics Department of the National
Physical Laboratory (see, for example, the recent volume by Frazer, Duncan,
and Collar on Elementary Matrices and Some Applications to Dynamics,
Cambridge University Press, 1938, especially the worked examples of
* iterative numerical solutions of linear dynamical problems,' pp. 133-154
and 308 et seq?).

2 Thus, as I have argued elsewhere, it might be of advantage to define
the leading factors in psychology by some provisional convention that is more
or less arbitrary, much as we define the zero meridian or the celestial north
pole. Until we know more of the functional relationships that obtain be-
tween different mental characteristics, this would probably be the most
convenient procedure in actual practice. Yet it is not strictly necessary in
theory. As more advanced students will be aware, a calculus has been
devised in pure mathematics which enables the physicist to perform his
analysis of space or of space-time in a way which leaves the axes of co-
ordinates entirely undetermined: he is thus able to describe a complex
pattern of relations without having to specify what are the relata. Prima
facie, therefore, there is no real need for the psychologist to seek stability
for his factors by forthwith endeavouring to identify each one with some
concrete psychological reality, as is so commonly supposed.