New York, NY: Springer. xvi, 399 p. DM 68.00; öS 496.40; sFr. 62.00; £ 26.00; $ 35.00 (1998).

This is the history of the International Mathematical Union (IMU) from its foundation in 1920 through approximately 1990. The author of this fascinating account, the Finnish function theorist Olli Lehto (born 1925), served as a member of the Executive Committee of the IMU since 1975 and as its Secretary from 1983 until 1990. Although it becomes clear from the account that Lehto personally was instrumental in persuading China to joint the IMU and in the creation of the Nevanlinna prize for mathematical aspects of computer science, the author does not unduly exaggerate his own rôle in the organization.
One of the aims of the book is to destroy misrepresentations of the IMU as explained in the preface: “The Union is often blamed for its poor visibility, accused of being an institution that in secretive ways steers the fate of the international mathematical community” (viii). The history of the IMU splits into two clearly separate parts. The first is from its foundation to its gradual dissolution after 1932. The second part is from the re-foundation of the IMU in 1952 until today. There was, according to Lehto, not a single document relating to the earlier IMU in the archives of the IMU in Zürich, which were transferred to Helsinki in 1994. So it fell upon Lehto -- apparently without much assistance from historians or fellow-mathematicians -- to reconstruct the early history of the IMU by drawing on scattered archival findings. Lehto gives a detailed and accurate report and extensive appendices on the chronological order of events esp. on the policies of the IMU with respect to the International Congresses of Mathematicians, the foundations and work of various subcommissions (for Mathematical Instruction, for Development and Exchange, for History of Mathematics, for Fields Medals etc.). Extensive listings on Fields medallists, the leading representatives of the various commissions, and photographs of the IMU-functionaries are provided. The author succeeds in giving a vivid picture of the complicated “historical triangle” constituted by the IMU, the International Mathematical Congresses (which were increasingly prepared by the IMU), and the International Council of Scientific Unions (formerly the International Research Council), to which the IMU belongs.
Throughout the book the “apolitical character” of the IMU is stressed as a precondition for its service to mathematical communication. However, Lehto leaves no doubt that eminent political and diplomatic skills were needed to keep international contacts in mathematics going during the Cold War, to manage conflicts between East- and West-German mathematics, between the People’s Republic of China and Taiwan, and to secure the participation of Soviet mathematicians in the International Congresses. Lehto could not disclose all details of the extensive IMU files. For example, the files on the discussions on invited lectures are sealed for 60 years. But in general the account is open and critical, even with respect to the outcome of several initiatives taken by the IMU. In Lehto’s opinion the IMU’s only successful bibliographical enterprise was the “World Directory of Mathematicians”, which was largely due to K. Chandrasekharan.
Not being a professional historian, Lehto’s bibliography is somewhat erratic and does not use all relevant historical work. Some general historical judgements, esp. on the development of applied mathematics during the 19th and 20th centuries are disputable. Erroneously the author does not mention in his acknowledgements the American archives, especially the AMS Archives in Providence, which he nevertheless uses extensively, because Americans such as M. H. Stone were instrumental in the rebirth of the IMU after the war.
All in all the book is a very welcome and useful publication esp. in preparation of the coming International Congress of Mathematicians to be held in Berlin in August 1998.