*A Mathematician’s Journeys: Otto Neugebauer and Modern Transformations of Ancient Science *is a collection of some of the papers presented at a 2010 conference concerning the life and work of Otto Neugebauer (1889–1990) that marked the twentieth anniversary of his death. As the editors note, there are three principal focuses: Neugebauer’s career in the 1920s and 1930s in which he was in turn a mathematician, a historian of mathematics, and a historian of the exact sciences; the historiography of ancient Egyptian and Mesopotamian mathematics; and the historiography of Babylonian astronomy. Taken as a whole, the papers provide a very useful introduction to these topics and the various mathematicians and historians who helped develop the study of the exact sciences in antiquity.

The first section consists of three essays that provide biographical details of Neugebauer’s life, of his development as a scholar, and his approach to scholarship. David E. Rowe sketches in some detail the intellectual development of Neugebauer and of his interests from physics to mathematics to the history of the exact sciences as he moved from Graz to Munich to Göttingen. Rowe discusses the culture of Göttingen and Richard Courant’s activities as founding director of its Mathematical Institute, for his thesis is that Neugebauer was greatly influenced throughout his life by the mathematical culture of Göttingen and his close personal relationship with Courant.

Reinhard Siegmund-Schultze’s essay develops the thesis that throughout his life, but highlighted by the era of the Nazi rise to power, Neugebauer sought to “maintain and expand the rationalistic and internationalist ideals long associated with Göttingen science and mathematics, and that this commitment was central both in his organizational work and his approach to history.” [p. 63]

Lis Brack-Bernsen’s essay focuses on Neugebauer’s visits to Denmark: first, in 1924-25, when he wrote both his only paper in pure mathematics and a review of an edition of the Rhind Mathematical Papyrus, and second, during the years 1934–1939, when the political situation in Germany had made his position at Göttingen and with *Zentralblatt* intolerable. Both visits were facilitated by the mathematician Harald Bohr.

The second section of five papers begins with an essay by Jim Ritter describing Neugebauer’s work in ancient Egyptian mathematics. This begins with a survey of mathematics and Egyptology at Göttingen and the historiography of Egyptian mathematics in the early 1920s. This is followed by a summary of Neugebauer’s work in Egyptian mathematics beginning in 1923, through his 1926 dissertation and ending in 1931, when he published his last substantive research article on the topic. Neugebauer’s interest in Egyptian astronomy culminated in his publication, with R. A. Parker, of the three volumes of *Egyptian Astronomical Texts* in the 1960s.

The remaining papers in this section discuss Neugebauer’s contributions to the study of Mesopotamian mathematics. Jens Høyrup’s essay discusses the historiography of Mesopotamian mathematics through the early works of Neugebauer in the 1930s. Included are comparative translations of cuneiform texts by Ernst Weidner and Neugebauer and by Carl Frank and Neugebauer; among other things, we see the development of an understanding of the features of the Babylonian mathematics and its sexagesimal system of numeration. Høyrup also introduces the rivalry between approaches to the texts of the mathematician Neugebauer and the philologist François Thureau-Dangin.

Béatrice André-Salvini’s short essay on Thureau-Dangin gives additional context for their rivalry, while Christine Proust’s paper discusses correspondence between Neugebauer and other Assyriologists on the decipherment and translation of texts. She includes in the appendices copies of a number of these letters as well as illustrations depicting the difficulty in transcribing the texts — many of the scholars worked from photographs of the tablets that were of varying quality.

Duncan Melville’s paper, which concludes the material on Mesopotamian mathematics, provides a valuable summary of recent research and research questions. His conclusion is quite apt: “All historians of Mesopotamian mathematics owe Neugebauer an enormous debt. His work is still continually consulted and referenced. But the discipline has moved on, asking new questions and re-evaluating old evidence. It continues to evolve.” [p. 260]

The final section of this collection concerns Babylonian astronomy. Teije de Jong provides the context for Neugebauer’s work by discussing developments in Babylonian mathematical astronomy from 1880–1955, when Neugebauer’s *Astronomical Cuneiform Texts* (*ACT*) appeared. He also briefly mentions the contemporary scholar Bartel Leendert van der Waerden and Neugebauer’s younger colleague Abraham J. Sachs.

John M. Steele discusses the reception of Neugebauer’s *ACT *by analyzing the published reviews of this work, insightfully divided into those that appeared in journals from four different areas: history of science, Assyriology, oriental studies, and science. He also provides a history of the compilation and publication of this work, compares it with Sach’s *Late Babylonian Astronomical and Related Texts*, and summarizes the impact of *ACT* on Assyriology and the history of science.

In the final short essay, Mathieu Ossendrijver describes Neugebauer’s approach to analyzing the Babylonian mathematics astronomy texts: his emphasis was on discovering algorithms that would reproduce the Babylonian results. Ossendrvijver points out that there has now been a change in focus from understanding Babylonian astronomy in our own terms to interpreting its algorithms in Babylonian terms.

It was a pleasure to learn more about Neugebauer’s life and work, for I knew him primarily through his *Mathematical Cuneiform Texts* and *The Exact Sciences in Antiquity.* It was fascinating to read of his work with *Zentralblatt* and the founding of *Mathematical Reviews*. The politics before the Second World War come alive in the specific example of his struggles to continue his work during the 1930s and 1940s. I was pleased to read that his interest in ancient mathematics may have been piqued by being asked to write a review of an edition of the Rhind Mathematical Papyrus. Finally, the histories of the history of Egyptian mathematics, of Mesopotamian mathematics, and of Babylonian astronomy will add depth to the classes I teach.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.