first important contribution that Karle and I made was the recognition that it would be necessary to exploit prior structural knowledge to transform the phase problem from an unsolvable one to one that was solvable, at least in principle. Our first step in this direction was to exploit the nonnegativity of the electron density function p(r). Before our analysis was complete, however, David Harker and John Kasper [1948] published their famous paper . . . in which they derived inequalities in which the measured intensities restrict the possible values of the phases. This was a very mysterious paper, because nowhere in it does there appear any explicit mention of the basis for the inequality relations, and indeed the most important fact is conspicuous by its absence. It is simply that the electron density function is nonnegative everywhere. This fact is, however, implicit in Harker and Kasper's work. In very short order Karle and I completed our own analysis and derived the complete set of inequality relationships based on the nonnegativity of the electron density function [Karle and Hauptman, 1950] . . . . It includes the Harker-Kasper inequalities as a special case, and many others besides. Although the complete set of inequalities greatly restricts the values of the phases, the relations appear to be too intractable to be useful in applications, except for the simplest structures, and their potential has never been fully exploited . . . .

"Beyond any doubt our most important contribution during the early 1950's was the introduction of probabilistic techniques—in particular, use of the joint probability distribution of several diffraction intensities and the corresponding phases—as the central tool in the solution of the phase problem [Hauptman and Karle, 1953]. . . . We assumed to begin with that all positions of the atoms in the unit cell of the crystal were equally likely, or, in the language of mathematical probability, that the atomic position vectors were random variables, uniformly and independently distributed. With this assumption the intensities and phases of the scattered X-rays, as functions of the atomic position vectors, are also random variables, and one can use the methods of modern mathematical probability theory to calculate the joint probability distribution of any collection of intensities and phases. A suitably chosen joint probability distribution leads directly to the conditional probability distribution of a specified structure invariant, assuming again an appropriately chosen set of diffraction intensities. The conditional distribution in turn leads to the structure invariant, an estimate of which is given, for example, by its most probable value. Once one has a sufficiently large number of sufficiently reliable estimates of structure invariants, one can use standard techniques to calculate the values of the individual phases, provided that the process incorporates a recipe for specifying the origin.

"Although probabilistic methods played an essential rule in the development of the direct method and provided it with its logical foundation, it must also be pointed out that nonprobabilistic methods also played an important part . . . . In particular the well known Sayre equation, a relationship of fundamental importance among measured magnitudes and unknown phases, continues to be useful to the present day and lies at the heart of some of the more successful computer programs for solving crystal structures.

"I cannot conclude this brief account of the early history of the direct methods of X-ray crystallography without also describing the reception this work received at the hands of the crystallographic community. This was, simply, extreme skepticism, if not outright hostility. . . .

"Today some 100,000 molecular structures are known, most determined by the direct methods, and about 5,000 new structures are added to the list every year. It is no exaggeration to say that modern structural chemistry owes its existence to this development . . . .

"Work on the phase problem continues to this day and applications to structures of ever increasing complexity continue to be made. It still appears that progress is made only in proportion to our ability to bring more powerful mathematical techniques to bear on this fascinating problem."