26. _Representation of Geometrical Magnitude by Number._--The
application of arithmetical methods to geometrical measurement presents
some difficulty. In reality there is a transition from a cardinal to an
ordinal system, but to an ordinal system which does not agree with the
original ordinal system from which the cardinal system was derived. To
see this, we may represent ordinal numbers by the ordinary numerals 1,
2, 3, ... and cardinal numbers by the Roman I, II, III, ... Then in the
earliest stage each object counted is indivisible; either we are
counting it as a whole, or we are not counting it at all. The symbols 1,
2, 3, ... then refer to the individual objects, as in fig. 1; this is
the primary ordinal stage. Figs. 2 and 3 represent the cardinal stage;
fig. 2 showing how the I, II, III, ... denote the successively larger
groups of objects, while fig. 3 shows how the name II of the whole is
determined by the name 2 of the last one counted.
When now we pass to geometrical measurement, each "one" is a thing which
is itself divisible, and it cannot be said that at any moment we are
counting it; it is only when one is completed that we can count it. The
names 1, 2, 3, ... for the individual objects cease to have an
intelligible meaning, and measurement is effected by the cardinal
numbers I, II, III, ..., as in fig. 4. These cardinal numbers have now,
however, come to denote individual points in the line of measurement,
i.e. the points of separation of the individual units of length. The
point III in fig. 4 does not include the point II in the same way that
the number III includes the number II in fig. 2, and the points must
therefore be denoted by the ordinal numbers 1, 2, 3, ... as in fig. 5,
the zero 0 falling into its natural place immediately before the
commencement of the first unit.
1 2 3 _____ _____ _____
_____ _____ _____ ... 0 1 2 3 ...
I II III
\____/ / / FIG. 5
\____________/ /
\_____________________/