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Publisher:

American Mathematical Society and London Mathematical Society

Publication Date:

2005

Number of Pages:

168

Format:

Hardcover

Series:

History of Mathematics 28

Price:

49.00

ISBN:

0-8218-3860-1

Category:

Anthology

[Reviewed by , on ]

David M. Bressoud

02/21/2006

The title of this book is misleading. The subtitle is more accurate. Most accurate would have been *Olinde Rodrigues, His Times and His Mathematics*. Rodrigues is a fascinating and little known mathematician. One of the first Jewish mathematicians of the modern era, he earned his doctorate at the Université de Paris in 1815, the same year that the monarchy was restored. Opportunities for an academic career were slim to non-existent, and so he turned first to actuarial science — helping to create the actuarial tables that would be used in France throughout the 19^{th} century — and then to banking. Though a successful banker, his life was shaped by two passions, social justice and mathematics.

In 1823, Rodrigues became the financial supporter, companion, and collaborator to Claude-Henri de Rouvrey, comte de Saint-Simon. He helped Saint-Simon to complete his final work, *Nouveau Christianisme*, and after Saint-Simon’s death in 1825, Rodrigues became part of the leading triumvirate of *Saint-Simonisme*, a lay religion espousing a visionary social philosophy with mystical undertones. From within this movement, an early precursor of French socialism, Rodrigues argued for universal suffrage, the abolition of slavery, and workers’ holidays. He worked to educate the bourgeoisie of the plight of workers, and in 1841 he published a 500-page edition of workers’ poetry.

Rodrigues wrote 17 mathematics papers in his lifetime. The first nine, which appeared 1813-16, involve equations of motion and lines of curvature. Here is where we find the one result that today carries his name, the representation of the n^{th} Legendre polynomial in terms of the n^{th} derivative of (x^{2} – 1)^{n}. Except for his actuarial work, his mathematics was quiescent until 1838–45 when he published the remaining eight papers. Here the topics were more diverse. He began with questions in combinatorics, studying Catalan numbers and discovering the generating function for permutations weighted by inversions. In 1840, he published his study of the group of rigid motions in **R**^{3}, probably the most influential of his mathematical work, though it would never receive proper credit.

This volume is a collection of essays on Rodrigues and his work. The first, jointly written by Simon Altmann, David Siminovitch, and Barrie M. Ratcliffe, is a superb overview of his life and work. Two essays explore his personal life and role in the Saint-Simon movement in greater detail. Five essays explore his mathematics: Ivor Grattan-Guinness on the early work, Richard Askey connecting the work in combinatorics with that in orthogonal polynomials and setting it into its historical context, Ulrich Tamm who explains the work on Catalan numbers and also references the role of Rodrigues’s formula in Zeilberger’s proof of the alternating sign matrix conjecture, Jeremy Gray who explains the 1840 paper on rigid motion and its significance, and Edouardo L. Ortiz who connects Rodrigues’s work to that of Hamilton on quaternions. Ortiz also includes an intriguing family tree that reveals that Rodrigues’s sister’s husband’s sister’s husband’s brother was Joseph Bertrand. Bertrand’s brother-in-law was Charles Hermite. Two of Bertrand’s nephews by marriage were Paul Appell and Émile Picard. And Paul Appell’s son-in-law was Émile Borel. That in itself is a glimpse into the mathematical community of 19^{th} century France.

David M. Bressoud is DeWitt Wallace Professor Mathematics at Macalester College in St. Paul, Minnesota.

- S. Altmann and E. L. Ortiz -- Introduction
- S. Altmann, D. Siminovitch, and B. M. Ratcliffe -- Olinde Rodrigues and his times
- B. M. Ratcliffe -- Towards a better understanding of Olinde Rodrigues and his circle: Family and faith in his life and career
- P. Ferruta -- Euphrasie and Olinde Rodrigues: The 'woman question' within Saint-Simonism
- I. Grattan-Guinness -- Rodrigues's early work in mathematics, 1813-1816
- R. Askey -- The 1839 paper on permutations: Its relation to the Rodrigues formula and further developments
- U. Tamm -- Olinde Rodrigues and combinatorics
- J. Gray -- Olinde Rodrigues's paper of 1840 on a group of transformations
- E. L. Ortiz -- After Rodrigues: From rotations to quaternions
- Index

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