Prepared for publication with the assistance of J. C. C. McKinsey This work has a long history. Its main results were found in 1930 and first mentioned in print a year later. It took nine years, however, before the material in its full development was prepared for publication. In fact, a monograph embodying those results was scheduled to appear in 1939 under the title The completeness of elementary algebm and geometry in the collection Actualites scientifiques et industrielles, Hermann & Cie, Paris. As a result of the war, the publication did not materialize; two existing sets of page proofs are probably the only trace of this venture.Naturally, I did not abandon the hope of seeing the results in print. However, as often happens with authors, the original version ceased to satisfy me. The specific character of the results seemed to call for a more formal presentation, and the addition of new observations and conclusions seemed to be desirable. The perplexity of the war and postwar periods resulted in postponing the revision from year to year. Hence I was happy when, in the beginning of 1948, the RAND Corporation, Santa Monica, California, became interested in my results and offered to publish them. It was especially fortunate that Professor J. C. C. McKinsey (Stanford University), who was at the time working with the RAND Corporation, was entrusted with the task of preparing the work for publication. With my collaboration, he actually prepared a new draft of the monograph and, aside from his competent editorial work, he contributed to a simplification of the development. Within a few months the monograph was published. As was to be expected, it reflected the specific interests which
25the RAND Corporation found in the results. The decision method for elementary algebra and geometry -which is one of the main results of the workwas presented in a systematic and detailed way, thus bringing to the fore the possibility of constructing an actual decision machine. Other, more theoretical aspects of the problems discussed were treated less thoroughly, and only in notes.The present edition is a photographic reprint of that published by the RAND Corporation, and hence no extensive changes could be introduced in the text. However, all known errors have been corrected, the introduction has been partly changed, and some supplementary notes have been added. In addition to new bibliographical references (mostly to recent literature), these supplementary notes contain elaborations of the original text which, in at least one case, result in enriching the content of the monograph by new theoretical material.I should like to thank the readers and reviewers who drew my attention to misprints and minor
