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This book is the first volume of a two-volume work on the Foundations of Quantum Mechanics, and is intended as a new edition of the author's book Die Grundlagen der Quantenmechanik 37] which was published in 1954. In this two-volume work we will seek to obtain an improved formulation of the interpretation of quantum mechanics based on experiments. The second volume will appear shortly. Since the publication of 37] there have been several attempts to develop a basis for quantum mechanics which is, in the large part, based upon the work of J. von Neumann 38]. In particular, we mention the books ofG. W. Mackey 39], J. Jauch 40], C. Piron 41], M. Drieschner 9], and the original work ofS. P. Gudder 42], D. J. Foulis and C. H. Randall 43], and N. Zierler 44]. Here we do not seek to compare these different formulations of the foundations of quantum mechanics. We refer interested readers to 45] for such comparisons.