André-Louis Cholesky (1875-1918) was a French military officer who served as a topographer in the army. He studied at the *École Polytechnique* and was sent to Crete, Tunisia and Algeria for measurements. He was also participating in correspondence courses at the *École Spéciale des Travaux Publics* founded by *Léon Eyrolles* in 1891. During the war he was assigned to the artillery but in the period 1916-1918 he was also director of the Geographical Service in Romania. After his return to France he took part in the second Battle of Picardy where he was fatally wounded. He died in Bagneux near Soissons at the end of WW I. Cholesky is famous among topographers, certainly in France, while most mathematicians will be familiar with the Cholesky method for the factorization of symmetric matrices. Cholesky never published the method himself, but Ernest Benoît, a military colleague, published the method in 1924. Benoît had also written a short biography before (an English translation of 1975 is included in this book).

Claude Brezinski is professor emeritus of numerical analysis, but he has been keen on the history of sciences and has published papers and books in both domains. Already in 1996 he wrote about Cholesky. That was when he got access to the documents at the *École Polytechnique* that, according to the French law, became public 120 years after Cholesky's birth. His research got an enormous momentum when in 2003 he was contacted to help classify Cholesky's archives that the Cholesky family wanted to donate to the *École Polytechnique*. The result is reflected in this book.

The logical start is a detailed biography of Cholesky. It has many illustrations, mostly pictures of Cholesky, and (translated) citations from several documents.

Raymond Nuvet, the vice-mayor of Montguyon, Cholesky's birth place, is largely responsible for a second chapter that sketches the family history from Cholesky's great great grandfather till his grandchildren. The roots lay most probably somewhere in Poland, but the precise origin is uncertain.

Chapter 3 explains some elements from topography. That involves triangulation: one has to measure all the angles of the connected triangles, but only the length of one side of a starting triangle. All unknown lengths, and hence the coordinates of the vertices, can be computed by solving a system of equations. To deal with measurement errors, a least squares solution is computed. In this context the system is usually underdetermined. Another aspect of topography is leveling. Because not all measurement points will be in the same horizontal plane, their relative elevation has to be measured and taken into account. Cholesky developed a method of double-run leveling in 1910, which is still used today.

The following chapter is more extensive and deals with Cholesky's method to solve (symmetric) linear systems. The history starts with the least squares technique attributed to Gauss, the method of Gaussian elimination, variants by Doolittle and others, and of course a discussion (and a translation) of Cholesky's unpublished notes (the original French version appears as an appendix). An analysis of the notes shows the skills of Cholesky. He discussed the computational complexity, the convergence of the square root computation, and gives a rounding error analysis. Brezinski writes *"If this paper was submitted today to a numerical analysis journal, it would be recommended for publication without any hesitation".* The chapter continues by explaining how the method was re-invented, and how it gradually was spread among the computing community and what current research is dealing with (iterative methods, preconditioning, etc.).

Other work of Cholesky (military and topographical) is surveyed in a short chapter, and another chapter sketches a biography of Léon Eyrolles, especially the early evolution of the *École Spéciale des Travaux Publics* where Cholesky was a professor. Eyrolles is also the founder of the publishing company *Éditions Eyrolles*. In the archives of Cholesky, also an unpublished book was found about graphical computation. It is typeset in the original French version in an appendix and it is placed in its historical context and extensively discussed in a separate chapter by Dominique Tournès who is professor of mathematics and the history of mathematics.

The next chapter is devoted to Ernest Benoît. The authors had a hard time to find the correct information. In all reports he was mentioned as *Commandant Benoît* (or with another military title as appropriate). Not even an initial for his first name was known and the name Benoît is a common name in France. Nevertheless his history is tracked down and a translation of his eulogy of Cholesky is included in English translation.

The last chapter is an inventory of other documents from the Cholesky archive. It contains translations of military reports about Cholesky and of diary booklets written by Cholesky during his field work in France in 1905.

This book is the only book about Cholesky that is currently available. After the little bits that were available in the few publications before, the wealth of facts that is brought here about Cholesky's life and work is overwhelming and will be almost impossible to surpass. Whatever the reader did not find here can be found in the archives in the *Fonds Cholesky* at the *Éole Polytechnique* to which the authors refer if needed. The style is rather factual and there are many illustrations. All the historical and mathematical context that a reader can wish for is provided. Sometimes pushed to the extreme. For example, every name of a person occurring in the text is followed by place and date (day, month, and year!) of birth and death (if known of course). This is clearly very informative, but you can imagine that it sometimes hinders readability a bit when several names occur in the same sentence.