Bernhard Riemann (1826-1866) is a central figure in the history of mathematics. Laugwitz's 1996 book in German is an excellent place to begin a study of his life, work, and influence. The 1999 translation by Abe Shenitzer into English is especially welcome, because previous works by Riemann or directly related to Riemann are mainly in German.

Laugwitz's book is divided into five parts with titles and approximate page lengths as follows

0. Introduction (60)

1. Complex Analysis (120)

2. Real Analysis (40)

3. Geometry; Physics; Philosophy (70)

4 Turning points in the conception of mathematics (50).

Throughout the book, a single theme is hit hard. This theme, captured in the book's subtitle and repeated as the title of Chapter 4, is that Riemann is the person most responsible for turning mathematics from a principally algorithmic science to a principally conceptual science. This theme is introduced dramatically on page *ix* by means of a very enticing quote from a 1959 letter from Carl Ludwig Siegel to André Weil:

The degeneration of mathematics began with the ideas of Riemann, Dedekind, and Cantor, which progressively repressed the reliable genius of Euler, Lagrange, and Gauss.

The theme is developed throughout the book in a more nuanced way. For example, Siegel is revealed as less of an extremist in pages 177-178.

There are two other works in English that I also recommend for beginning a study of Riemann. They are much shorter, and focused on different themes. Raghavan Narasimhan's introduction to Riemann's collected works focuses on the "quality of permanence" in Riemann's mathematics. The first part of Riemann, Topology, and Physics by Michael Monastyrsky focuses on the important role played by physics in Riemann's work.

There is a tone of seriousness throughout Laugwitz's book. One indication of this seriousness is that we are advised on page xiii to have "ready access" to Riemann's collected works. Another indication is that sections are numbered a.b.c, as in 0.4.6, not very common for a biography! Similarly there is excellent referencing throughout, by page number when appropriate. I welcome such a scholarly approach. I especially liked the fact that quotes are given almost always both in English and in the original German. Many readers will feel the original German brings them a bit closer to Riemann and his contemporaries.

There is similarly an admirable concern for accuracy throughout the book. For example, Riemann's first biographer, his slightly younger colleague Richard Dedekind, describes Riemann's death as peaceful and almost beautiful. I have seen reworded versions of Dedekind's description repeated uncritically in several places. But Laugwitz is dubious, thinking that Dedekind may have just made it up to please Riemann's widow (page 29). Laugwitz's style generally is to keep conclusions explicitly tentative in the many cases when evidence is not sufficient, writing "We must be aware that every interpretation is risky and that the available writings admit different interpretations" (page 278).

Since Laugwitz's theme is that Riemann's work changed mathematics on a fundamental level, he is obliged to discuss not only Riemann but also mathematics before and after Riemann. The translator comments "The book is an intellectual panorama of mathematics from Leibniz to Bourbaki" (page *xvii*). Indeed the names of Euclid, Descartes, Newton, Euler, Fourier, Gauss, Abel, Cauchy, Dirichlet, Dedekind, Cantor, Klein, Hilbert, Einstein, and Weyl all make it into the table of contents. Laugwitz's expertise on historical matters is most impressive, but I think some readers will be frustrated by the shifting focus.

Chapter 0 concerns mainly Riemann's life. Its first sentence is "The external circumstances of Riemann's short life can be quickly set down." Throughout, I get a sense that the life of Riemann the person is low on the list of Laugwitz's interests. While I wouldn't want drama coming from invented events, I am disappointed that Laugwitz doesn't try harder to capture the considerable drama that is really there. Reading Chapter 0 does not give one the feeling of reliving Riemann's life. For example, Laugwitz describes Riemann's declining 1860s before his glorious 1850s. With respect to catching the real drama of Riemann's life, I prefer Monastyrsky's chronological narrative.

As Laugwitz explains on page *x*, Chapters 1-3 are organized to echo Riemann's three "qualifying papers". Riemann's 1851 thesis is on the foundations of one-variable complex analysis, through what we now call the Riemann mapping theorem. It forms the starting point of Chapter 1. His 1854 habilitation paper on trigonometric series includes a careful discussion of what we now call the Riemann integral. It forms the mathematical core of Chapter 2. Finally his 1854 habilitation lecture outlines what we now call Riemannian geometry, with a heavy emphasis on physics and philosophy. It plays the central role in Chapter 3. I think many readers would have appreciated an appendix with a simple list of Riemann's papers, as the chronology is quite involved. For example, the two 1854 works just mentioned are "early works" but were published only posthumously. I found myself making repeated use of the list in Riemann's collected works, itself spread out annoyingly over pages *V-VI*, 30-32, 597-598.

Chapter 4 compares the algorithmic-to-conceptual turning point in mathematics to other events called revolutions by consensus. An oft-cited example is the discovery around 1830 of non-Euclidean geometries. Laugwitz argues that the algorithmic-to-conceptual turning point is greater than the "non-Euclidean revolution". He argues it is less recognized by historians only because it involves higher mathematics. He describes Dedekind's and then Hilbert's role in carrying forward the algorithmic-to-conceptual change. Readers inclined toward philosophy will probably particularly enjoy this chapter; it complements Narasimhan's purely mathematical assessment of Riemann's work.

The translator begins his remarks on page *xvii* with "I don't always agree with the author, but I find him stimulating and enlightening." I share the translator's viewpoint, and feel thanks are due to both author and translator for making it much easier to enter into the literature on Riemann.

**References**

Gesammelte Mathematische Werke, Wissenschaftlicher Nachlass und Nachträge , by Bernhard Riemann, edited by Raghavan Narasimhan . Springer Verlag, 1990. Hardcover, vi+911pp, $175.00. ISBN 3540500332.

Riemann, Topology, and Physics, by Michael Monastyrsky. Springer Verlag, 1999. Hardcover, 210pp, 2nd edition. ISBN 0817637893

For an English translation of the 1854 habilitation lecture "On the hypotheses which lie at the foundation of geometry" and also an extract from a 1861 paper which introduced the Riemann curvature tensor see: A Comprehensive Introduction to Differential Geometry (Volume 2) , by Michael Spivak. Publish or Perish, 2nd Edition, 1990. Hardcover, $25.00. ISBN 0914098853. For an English translation of the 1859 paper "On the number of primes less than a given magnitude" see Riemann's Zeta Function , by H.M. Edwards. Academic Press, 1974. Hardcover, xiii + 313pp. ISBN 0122327500. These two books also contain very readable expert commentary on the papers translated. Riemann's collected works also contains, besides Narasimhan's overview, two other articles offering expert analysis in English, one analyzing a paper by Riemann on shock waves, the other a paper by Riemann on rotating fluid ellipsoids.

A considerable amount of Riemann material is on the web at http://www.maths.tcd.ie/pub/HistMath/People/Riemann/. This site includes a transcription of a large portion of Riemann's collected works (in the original languages, i.e., mostly in German), biographical information, and English translations of the same two papers mentioned above. There are plans to add further translations and commentary.

Riemann's collected works, listed above, has a 42-page bibliography, but I could not find any more English translations of his original papers. However a large number of these papers appeared in a French translation around 1900, reprinted in 1968: *Oeuvres mathématiques de Riemann*, by Bernhard Riemann, translated by L. Laugel. Librairie scientifiques et technique Albert Blanchard, Paris. Hardcover, xxxvi+455pp.

David P.Roberts is assistant professor of mathematics at University of Minnesota, Morris.