iz8 THE FACTORS OF THE MIND

interchange is impossible ; we cannot move the intervals
about as we transport the interval on a foot-rule from one
object to another. Theoretically, we may define equality
as the limit that is reached when a difference is reduced
indefinitely (e.g. when in tuning a violin we diminish the
intervals between A17 and A or between AS and A as far
as we can). Practically, we shall assume x = y, when we
can no longer decide that x > y or that y > x. Such a
decision should be based, not on one observation, but on
many, and will lead to statistically defined criteria, like
those adopted in the ordinary psychophysical methods.1

From a formal standpoint the central problem will be
to show that the intervals or * distances? form a c group '
for the operation of addition. If possible, therefore, we
have to demonstrate, not only that they form an asym-
metrical and transitive series, but also that they conform to
the further postulates that addition logically presupposes.
Such a demonstration can only be carried out by actual
experiment. Hence the result can only be that the pos-
tulates hold approximately, i.e. after due allowance has been
made for a certain amount of inevitable error. The fact
that an experiment is necessary, and as such will generally
involve a physical operation, does not mean that the process

1 Some of the more obvious devices, mainly based on the traditional
psychophysical methods, were described in my earlier reports; they are more
fully and systematically set out in such works as Guilford's Psychometric
Methods* The conversions proposed themselves rest on additional assump^
tions for which both the empirical evidence and the a -priori arguments are
often far from convincing, e.g. that time-measures form a geometrical
rather than arithmetical scale, or that frequencies are distributed in corre-
spondence with the normal curve. My own view is that in theory we should
start with a more general form of conversion (logistic rather than logarithmic
in the first case, hypergeometric rather than normal in the second), and seek
experimental data for the requisite constants. In practice an empirical
procedure is usually sufficient, if checked by the results obtained with other
types of scale. Let me add that in the early controversy between Prof.
Karl Pearson and myself over the non-linear character of the Binet age-scale
[27], I did not mean to imply that such scales could never be made linear (or
sufficiently linear for practical purposes), but merely that the assumption of
linearity in the original unstandardized version required preliminary testing
and (probably) considerable readjustment within the scale itself (cf. [41],
P. 139, * diagrammatic representation of the test-series as a linear scale').