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The first direct method, by means of which structures were solved, was the
symbolic addition method. This method originates from Gillis (1948), however,
due to the work of Karle and Karle (1966) it developed to a standard method.
The problem can be defined as how do we find *m* phases, provided there are *n*
phase relationships ()? In the first place a few ( 3) phases can
be chosen to fix the origin and then, using phase relationships, new phases can
be derived from these three. In general it will not be possible to phase all
reflections in this way and hence a suitable reflection (large |*E*|, many
relationships with large *E _{3}*) is given a symbolic phase and again the
relationships are used to find new phases in terms of the already known ones.
Usually it will be necessary to choose several symbols in order to phase most of
the strong reflections. Finally the numerical values of the symbols are
determined (e.g. by using negative quartet relations) and from the known phases
a Fourier map can be calculated. This process is known as the symbolic addition
method. Most structures are now solved by multi-solution tangent refinement
procedures, which use many starting sets of numerical phases and the tangent
formula (31) to extend and refine the phases. The correct solution may then be
selected by using figures of merit, based e.g. on the internal consistency of
the triplet-relations, or on the negative quartets.

**Copyright © 1984, 1998 International Union of
Crystallography**