50. Given any relation of positions (directions, distances, intervals),
it may be turned upon a center and so made to take an indefinite number
and variety of attitudes. It may be inverted and the inversion may be
turned upon a center, producing another series of attitudes. Except
in cases of axial balance, the attitudes of the second series will be
different from those of the first.
[Illustration: Fig. 50]
In this case the relation of positions being turned upon a center
changes its attitude, while the positions within the group remain
relatively unchanged. There is no change of shape.
[Illustration: Fig. 51]
In this case the same group has been inverted, and a second series of
attitudes is shown, differing from the first series.
[Illustration: Fig. 52]
In this case, however, which is a case of axial balance, the inversion
of the group and the turning of the inversion on a center gives no
additional attitudes.
51. Among all possible attitudes there are four which are principal or
fundamental, which we may distinguish as follows:—
[Illustration: Fig. 53]
These principal attitudes are: First, I, the original attitude,
whatever it is; second, II, the single inversion of that attitude, to
the right on a vertical axis; third, III, the double inversion of the
original attitude, first to the right then down; and, fourth, IV, the
single inversion of the original position, down across the horizontal
axis.