n8	THE  FACTORS OF THE MIND
ments." ^l A man with good auditory imagery could con-
struct a whole-tone scale without any apparatus at all.
Taking the notion of class as fundamental,^2 it would seem
that the properties the psychometrist has to consider are of
two kinds; the elements of the class must be subject (i) to
certain relations and (2) to certain operations. The former
chiefly determine the requirements of intensive magnitude ;
the latter those of extensive magnitude.
(i) Relations.-—-To construct a * unidimensional variable '
what the f actorist primarily needs is not a * set of quantities,'
but merely a linear series. To transform a classification into
a grading, all that is needed is to convert each of the^c classes'
into an * ordered class.' This can be done by the aid of
exclusively logical notions', without invoking any * arith-
metical ' concepts or the ' mathematical methods of the
physical sciences^? as ordinarily understood. And no one,
I presume, will criticize the factorist for trying to be more
logical.
To arrange traits, personalities, or anything else in order,
it is necessary and sufficient to find a relation that is (i) con-
' nexive, (ii) asymmetrical, and (iii) transitive, and to demon-
strate by empirical observation that this relation holds good
of the members of the class. Thus, if #, y, and z denote
^1	H. Poincare, La science et Pbypotkese, p. 36*   The tendency of the
materialistic physics of the nineteenth century was to assume that absolute
space, absolute time, and quantity of matter were alone directly measurable,
and to suppose that measurement consisted essentially in the physical division
of lengths, durations, and masses into additive unit parts.   More recent
developments alike in physics and in mathematics would seem to render this
view rather difficult to sustain ; yet it appears to persist whenever the critic
discusses the possibility of mental measurement.
^2	The modern notions of measurement, like those of mathematics, are
ultimately derived from Cantor's theory of classes (Mengenlehre) :  Math.
Ann., Vol. XV-XLIX, 1872, et seq.   In this country his views have chiefly
been developed by Russell,   A most important contribution, which seems far
better known to mathematicians and logicians than to psychologists, is
Meinong's paper Uber Me Bedeutung d*s Weler'schen Gesetzes, 1896 (re-
printed Ges. Abhandlungen, 1913).   In the paragraphs above I have borrowed
my postulates mainly from Russell (e.g. Introduction to Mathematical Phil-
osophyv 1919, pp* 29 et seq.) with some slight modifications suggested by later
writers (ci N. F. Campbell, Physics: The Elements, 1920, and Measurement
and- Calculation, 1929; also Johnson, Logic> Pt. II, chap, vii).