THE LOGICAL STATUS OF MENTAL FACTORS 123

Secondly, suppose the master has successfully ranked the
20 pupils in his form in order : what is he to do if six new
boys arrive ? Two perhaps may require to be placed
between the former 2nd and 3rd, and three between the
former 8th and gth ; and the brightest may be better than
any he had before. Yet a simple order of merit, based on
consecutive integral numbers, makes no allowance for pos-
sible gaps. On the other hand, if he merely renumbers
them, the figures for most of them will be altered, and the
last boy will be called no longer 2Oth but 26th.

Thirdly, the existence of such gaps, and of gaps differing
in size, is a subject of constant comment from the teacher.
His task is like that of the earlier chemists, who sought to rank
all the elements according to their atomic numbers: at
certain points they felt compelled to leave extra wide spaces,
though at the time there was no known element to insert
between the neighbours. In the same way, simply to
number the pupils in order does not convey all the relevant
knowledge that an observant teacher could infuse into his
grading. Nearly always, for example, the intervals between
the first boy and the second, and again between the last and
the last but one, are more glaring than the tiny differences
between pupils near the middle of the list, whom he is
tempted to bracket as £ ties.'

It seems evident, therefore, that we require some rule
whereby (i) the order of pupils can be described regardless
of the number of pupils in each particular batch and (ii) the
numerals used to indicate the order shall also indicate the
distance between the successive members. Even if we
grade by consecutive relations, we surely need cardinal
numbers rather than ordinal to express the amount of
difference.

The plain teacher, who knows nothing of ' percentile ranks ' or
fractional' grades,' thinks first of a simple solution. Could we only
assume that every mental quality—intelligence, intensity of sensa-
tion, amount of pleasure, or impression of beauty—was formed each
out of separable parts coalescing into a continuous whole, as rain-
drops coalesce to form a pool of water, then his task would be easy.
If intelligence, for example, consisted of atomic elements, and if
Tom's sample contained % elements and Dick's contained y elements,