Cover -- copyright page -- Title page -- Introduction -- Contents -- Analysis -- Foreword -- Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus JUDITH V. GRABINER -- The practice of analysis: fromNewton to Euler -- The algebra of inequalities -- Changing attitudes toward rigor -- The concepts of the calculus -- Conclusion -- References -- Evolution of the Function Concept: A Brief Survey ISRAEL KLEINER -- Introduction -- Precalculus developments -- Euler's Introductio in Analysin Infinitorum -- The vibrating-string controversy -- Fourier and Fourier series -- Dirichlet's concept of function -- Pathological functions -- Baire's classification scheme -- Debates about the nature of mathematical objects -- Recent developments -- References -- S. Kovalevsky: A Mathematical Lesson KAREN D. RAPPAPORT -- Highlights in the History of Spectral Theory L. A. STEEN -- 1 Principal axes theorem -- 2 Infinite systems of linear equations -- 3 Integral equations -- 4 David Hilbert -- 5 Hilbert-Schmidt spectral theory -- 6 The Lebesgue integral -- 7 Quantum mechanics -- 8 John von Neumann -- 9 Von Neumann-Stone spectral theory -- 10 Gelfand-Naimark theorem -- 11 Unfinished business -- References -- Alan Turing and the Central Limit Theorem S. L. ZABELL -- 1 Introduction -- 2 The central limit theorem -- 3 Turing's fellowship dissertation -- 4 Discussion -- 5 Epilogue: Bletchley Park -- References -- Why did George Green Write his Essay of 1828 on Electricity and Magnetism? I. GRATTAN-GUINNESS -- 1 Honor to Green -- 2 Three strands in eighteenth-century mechanics -- 3 Poisson and the appearance of divergence theorems -- 4 Green and the place of surface integrals -- 5 Sources and influences -- 6 Options for publication -- 7 On Green's second period -- 8 Recognition -- References.

Connectivity and Smoke-Rings: Green's Second Identity in its First Fifty Years THOMAS ARCHIBALD -- Introduction -- Green's 1828 essay -- Thomson rediscovers Green -- Riemann and multiply-connected regions -- Helmholtz and vortices -- Tait, Thomson, smoke-rings and atoms -- Tait's quaternion version of Green's theorem -- Conclusion -- References -- The History of Stokes's Theorem VICTOR J. KATZ -- Origins of the theorems -- Vector forms of the theorem -- Generalization and unification -- Appearance in texts -- References -- The Mathematical Collaboration of M. L. Cartwright and J. E. Littlewood SHAWNEE L. MCMURRAN AND JAMES J. TATTERSALL -- Dr. David Harold Blackwell, African-American Pioneer NKECHI AGWU, LUELLA SMITH, AND AISSATOU BARRY -- The education of David Blackwell -- Professional career: scholar, teacher, and administrator -- World recognition: the leader and civic scientist -- Family life and personal tidbits -- An example to remember -- References -- Afterword -- Geometry, Topology and Foundations -- Foreword -- Gauss and the Non-Euclidean Geometry GEORGE BRUCE HALSTED -- A Review of Gauss's Werke, Vol. VII -- References -- History of the Parallel Postulate FLORENCE P. LEWIS -- The Rise and Fall of Projective Geometry J. L. COOLIDGE -- 1 The early period -- 2 The great period -- 3 The gradual decline -- Notes on the History of Geometrical Ideas DAN PEDOE -- 1 Homogeneous coordinates -- 2 The principle of duality -- References -- A note on the history of the Cantor Set and Cantor function JULIAN F. FLERON -- Evolution of the Topological Concept of "Connected" R. L. WILDER -- Introduction -- The evolution -- Bolzano's contribution -- Cantor's contribution -- C. Jordan's contribution -- Schoenflies' contribution -- The work of W. H. and G. C. Young -- The Lennes and Riesz definitions -- Concluding remarks.

A Brief, Subjective History of Homology and Homotopy Theory in this Century PETER HILTON -- Question-and-answer session -- The Origins of Modern Axiomatics: Pasch to Peano H. C. KENNEDY -- C. S. Peirce's Philosophy of Infinite Sets JOSEPH W. DAUBEN -- Mathematics in America in the nineteenth century -- Georg Cantor (1845-1918) -- Charles Sanders Peirce (1839-1914) -- Conclusion -- References -- On the Development of Logics between the two World Wars I. GRATTAN-GUINNESS -- 1 Introduction -- 2 Logicism and its critics -- 3 Formalism and its fate -- 4 Recursion and computability: American logic -- 5 The emergence of intuitionism -- 6 Beyond first-order and finitary logics -- 7 The rise of Polish logic -- 8 Conclusions and comparisons -- References -- Dedekind's Theorem:√2 ×√3 =√6 DAVID FOWLER -- 1 Dedekind's theorem -- 2 The continued fraction representation -- 3 The decimal representation -- 4 The unit fraction representation -- 5 The Eudoxian representation -- 6 A geometrical description -- 7 Mathematics and the real numbers -- References -- Afterword -- Algebra and Number Theory -- Foreword -- Hamilton's Discovery of Quaternions B. L. VAN DER WAERDEN -- Introduction -- A brief history of complex numbers -- "Papa, can you multiply triples?" -- The leap into the fourth dimension -- Octonions -- Product formulas for the sums of squares -- References -- Hamilton, Rodrigues, and the Quaternion Scandal SIMON L. ALTMANN -- The men involved: Hamilton and Rodrigues -- The discovery of quaternions -- In praise of Hamilton: the algebra of quaternions -- The trouble starts -- Quaternions and rotations: the first steps -- An optical illusion: the rectangular rotation -- The comical transformation (this heading contains a misprint) -- The Rodrigues programme -- The resolution of the paradoxes -- The decline -- Epilogue -- References.

Building an International Reputation: The Case of J. J. Sylvester (1814-1897) KAREN HUNGER PARSHALL AND EUGENE SENETA -- The Foundation Period in the History of Group Theory JOSEPHINE E. BURNS -- Introduction -- Lagrange, Ruffini, Galois -- Cauchy, the founder of group theory -- References -- The Evolution of Group Theory: A Brief Survey ISRAEL KLEINER -- 1 Sources of group theory -- 2 Development of "specialized" theories of groups -- 3 Emergence of abstraction in group theory -- 4 Consolidation of the abstract group concept -- dawn of abstract group theory -- 5 Divergence of developments in group theory -- References -- The Search for Finite Simple Groups JOSEPH A. GALLIAN -- 1 The alternating groups and the classical linear groups -- 2 Range problem 1-660 -- 3 PSL(m, pn) -- 4 Range problem to 1092 -- 5 Permutation representations and character theory -- 6 Odd order problem -- 7 Dickson's simple groups -- 8 The Mathieu groups -- 9 Range problem to 6232 -- 10 Burnside's p^a q^b theorem -- 11 The Chevalley groups -- 12 Groups of Lie type -- 13 Sporadic simple groups -- 14 Thompson's N-paper -- 15 The p^a q^b r^c problem -- 16 The range problem to1,000,000 -- References -- A Simple Song -- Genius and Biographers: The Fictionalization of Evariste Galois TONY ROTHMAN -- 1 Introduction -- 2 Sources -- 3 Early life and Louis-le-Grand -- 4 L'Ecole Normale -- 5 Arrest and prison -- 6 The duel and theories surrounding it -- 7 The last night -- 8 Harsher words -- References -- Hermann Grassmann and the Creation of Linear Algebra DESMOND FEARNLEY-SANDER -- 1 Introduction -- 2 Grassmann's life -- 3 The invention of linear algebra -- 4 Products -- 5 Inner products -- 6 Linear transformations -- 7 Geometry -- 8 Contemporary and later developments -- 9 Conclusion -- References -- The Roots of Commutative Algebra in Algebraic Number Theory ISRAEL KLEINER.

Introduction -- Algebraic number theory and unique factorization -- Kummer and ideal numbers -- Dedekind and ideals -- Dedekind's legacy -- Postscript -- References -- Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem REINHARD C. LAUBENBACHER AND DAVID J. PENGELLEY -- Eisenstein's proof -- Eisenstein versus Gauss -- References -- Waring's Problem CHARLES SMALL -- A History of the Prime Number Theorem L. J. GOLDSTEIN -- APPENDIX A: Samples from Gauss's Tables -- APPENDIX B: Gauss's Letter to Encke -- APPENDIX C: Corrections to Gauss's Tables -- References -- A Hundred Years of Prime Numbers PAUL T. BATEMAN AND HAROLD G. DIAMOND -- Early work on primes -- First proofs of the prime numbertheorem -- Later developments -- Sources -- References -- The Indian Mathematician Ramanujan G. H. HARDY -- Emmy Noether CLARK H. KIMBERLING -- 1 Her passing -- 2 Early years -- 3 Excerpts from Weyl's address -- 4 Her contribution to physics -- 5 World War I years -- 6 Excerpts from Alexandroff's address -- 7 In America -- 8 Missing letters recovered -- "A Marvelous Proof" FERNANDO Q. GOUVEA -- 1 Preliminaries -- 2 The actors -- 3 The play -- References -- Afterword -- Surveys -- Foreword -- The International Congress of Mathematicians GEORGE BRUCE HALSTED -- A Popular Account of Some New Fields of Thought in Mathematics G. A. MILLER -- A Half-Century of Mathematics HERMANN WEYL -- Introduction. Axiomatics -- Part I. Algebra, Number Theory, Groups -- Rings, Fields, Ideals -- Some achievements of algebra and number theory -- Groups, vector spaces and algebras -- Finale -- Part II. Analysis, Topology, Geometry, Foundations -- Linear operators and their spectral decomposition. Hilbert space -- Lebesgue's integral, Measure theory, Ergodic hypothesis -- Topology and harmonic integrals.

Conformal mapping, meromorphicfunctions, Calculus of variations in the large.