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Ioan James introduces and profiles sixty mathematicians from an era which saw mathematics freed from its classical origins to develop into its modern form. The characters, all born between 1700 and 1910, come from a wide range of countries, and all made an important contribution to mathematics, through their ideas, their teaching, their influence, and so on. The 2003 book is organised chronologically into ten chapters, each of which contains potted life stories of six mathematicians. The players James has chosen to portray are sufficiently representative that their stories, when read in sequence, convey in human terms something of the way in which mathematics developed.

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• Preface(p. xi)
• Prologue(p. xiii)
• 1 From Euler to Legendre(p. 1)
• Leonhard Euler (1707-1783)(p. 1)
• Jean-le-Rond d'Alembert (1717-1783)(p. 8)
• Joseph-Louis Lagrange (1736-1813)(p. 14)
• Gaspard Monge (1746-1818)(p. 20)
• Pierre-Simon Laplace (1749-1827)(p. 28)
• Adrien-Marie Legendre (1752-1833)(p. 36)
• 2 From Fourier to Cauchy(p. 42)
• Joseph Fourier (1768-1830)(p. 42)
• Sophie Germain (1776-1831)(p. 47)
• Carl Friedrich Gauss (1777-1855)(p. 58)
• Simeon-Denis Poisson (1781-1840)(p. 69)
• Jean Victor Poncelet (1788-1867)(p. 76)
• Augustin Cauchy (1789-1857)(p. 81)
• 3 From Abel to Grassmann(p. 91)
• Niels Abel (1802-1829)(p. 91)
• Carl Jacobi (1804-1851)(p. 97)
• Lejeune Dirichlet (1805-1859)(p. 103)
• William Rowan Hamilton (1805-1865)(p. 109)
• Joseph Liouville (1809-1882)(p. 116)
• Hermann Grassmann (1809-1877)(p. 123)
• 4 From Kummer to Cayley(p. 129)
• Ernst Kummer (1810-1893)(p. 129)
• Evariste Galois (1811-1832)(p. 134)
• J.J. Sylvester (1814-1897)(p. 141)
• Karl Weierstrass (1815-1897)(p. 150)
• P.L. Chebyshev (1821-1894)(p. 159)
• Arthur Cayley (1821-1895)(p. 165)
• 5 From Hermite to Sophus Lie(p. 173)
• Charles Hermite (1822-1901)(p. 173)
• Leopold Kronecker (1823-1891)(p. 177)
• Bernhard Riemann (1826-1866)(p. 182)
• Henry Smith (1826-1883)(p. 189)
• Richard Dedekind (1831-1916)(p. 195)
• Sophus Lie (1842-1899)(p. 200)
• 6 From Cantor to Hilbert(p. 208)
• Georg Cantor (1845-1918)(p. 208)
• Gosta Mittag-Leffler (1846-1927)(p. 215)
• Felix Klein (1849-1925)(p. 219)
• Sonya Kovalevskaya (1850-1891)(p. 230)
• Henri Poincare (1854-1912)(p. 237)
• David Hilbert (1862-1943)(p. 246)
• 7 From E.H. Moore to Takagi(p. 258)
• E.H. Moore (1862-1932)(p. 258)
• Jacques Hadamard (1865-1963)(p. 263)
• Felix Hausdorff (1868-1942)(p. 271)
• Elie Cartan (1869-1951)(p. 279)
• Emile Borel (1871-1956)(p. 283)
• Teiji Takagi (1875-1960)(p. 292)
• 8 From Hardy to Lefschetz(p. 299)
• G.H. Hardy (1877-1947)(p. 299)
• Oswald Veblen (1880-1960)(p. 306)
• L.E.J. Brouwer (1881-1966)(p. 313)
• Emmy Noether (1882-1935)(p. 321)
• R.L. Moore (1882-1974)(p. 326)
• Solomon Lefschetz (1884-1972)(p. 333)
• 9 From Birkhoff to Alexander(p. 340)
• George Birkhoff (1884-1944)(p. 340)
• Hermann Weyl (1885-1955)(p. 345)
• George Polya (1887-1985)(p. 350)
• Srinivasa Iyengar Ramanujan (1887-1920)(p. 357)
• Richard Courant (1888-1972)(p. 362)
• J.W. Alexander (1888-1971)(p. 372)
• 10 From Banach to von Neumann(p. 380)
• Stefan Banach (1892-1945)(p. 380)
• Norbert Wiener (1894-1964)(p. 386)
• P.S. Aleksandrov (1896-1982)(p. 392)
• Oscar Zariski (1899-1986)(p. 401)
• A.N. Kolmogorov (1903-1987)(p. 407)
• John von Neumann (1903-1957)(p. 412)
• Epilogue(p. 417)
• Further Reading(p. 423)
• Collections(p. 426)
• Acknowledgements(p. 429)

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CHOICE Review

James (Univ. of Oxford) offers a fantastic opportunity--introductions to the 60 most influential mathematicians born between 1700 and 1910, a period when mathematics was undergoing major shifts in form, technique, and emphasis. The biographies are arranged chronologically by birth date, starting with Leonhard Euler and ending with John von Neumann. The majority of the mathematicians are well known--Laplace, Gauss, Cauchy, Cayley, Riemann, Cantor, Ramanujan, for example--while some are less well known yet important. For each mathematician, James's emphasis is on telling their life story within the context of the changing mathematical scene, while paying little (if any) attention to details of their specific mathematical achievements. Readers interested in pursuing the mathematical details can use the supplementary lists of additional references and collected works (for all but 16 of the mathematicians). The epilogue is the author's attempt to reach some broad conclusions across these 60 mathematicians, addressing items such as the lack of female mathematicians (only three), lack of geographical spread, diverse family backgrounds, varied roles as teachers, and evidence of precocious or gifted talents in mathematics. A portrait of each mathematician is included with their biography. Overall, an interesting and valuable work. ^BSumming Up: Highly recommended. General readers; lower-division undergraduates through faculty. J. Johnson Western Washington University