The MAA has a tradition in the publication of books like this one. I think it began in1969 with the first of four volumes of papers selected from its various journals (going back to 1890). The four volumes consisted of papers on Pre-calculus, Calculus, Algebra and Geometry respectively. All such articles were expositional, some were historical, and many were inspirational.
Much later, in 2004, there appeared an MAA publication that was compiled in a similar vein — except that its emphasis was historical throughout. It was given the title Sherlock Holmes in Babylon. In it, the articles appear in the chronological order of mathematical developments from ancient times up to the work of Euler in the 18th century. Fortunately, there is now a sequel to that book, which is the subject of this review. It contains forty-one papers pertaining to the history of mathematics from the early 19th century to the late 20th century.
The articles are grouped thematically into three sections: ‘Analysis’, ‘Geometry, Topology and Foundations’ and ‘Algebra and Number Theory’, and all but five of them were originally published in MAA journals in the thirty-year period 1972 to 2004. Moreover, each section begins with a Foreword providing an overview of the related topics, and each concludes with an Afterword that outlines subsequent advances in historical knowledge since the articles first appeared.
Authors of the five papers that were written before 1972 (in fact, before 1939) include G. H. Hardy, Hermann Weyl, B. van der Waerden and J. L. Coolidge. Some of the other papers have earned their authors MAA writing awards, and the subject matter and presentation of all forty articles makes them compulsive (perhaps compulsory!) reading.
To my mind, the expositional nature of this book is encapsulated by an opening comment in the very last of its main articles, written by Fernando Gouvêa in 1994. That article examines Andrew Wiles’ proof of Fermat’s last theorem, and its stated goal was ‘to give mathematicians who are not specialists in the subject access to a general outline of the strategy proposed by Wiles’.
The high quality of the individual articles is one thing that recommends the book, but they have been chosen and organized in such a way that the book in fact provides an excellent historical overview of the growth of modern mathematics since 1800. There is also a very good balance between the papers that are purely biographical, those that are semi-biographical and the articles that are more mathematical. Some of the latter can be read and appreciated by good high school students, whilst others would be more accessible to senior undergraduates of postgraduate students.
The title of this book is derived from that of its very first article called ‘Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus’, written by Judith Grabiner. This is followed by a paper, by Israel Kleiner, on the evolution of function concept. It begins with a discussion of Euler’s notion of a function and leads to examination of ideas due to Fourier, Dirichlet and Baire. This paper by Kleiner culminates with a brief account of L2 functions, generalized functions and category theory.
Consequently, within the space of twenty pages, the first two articles in themselves provide substantial understanding of the history of calculus and analysis from 1800 to the present day — and there are similar achievements in the other two main sections of the book. For example, the section called ‘Geometry, Topology and Foundations’ has papers on the history of non-Euclidean geometry, the rise and fall of synthetic projective geometry and the history of topology. Among these is Peter Hilton’s paper on 20th century developments in homotopy and homology, and there are papers on the origin of modern axiomatics, logic and the foundations of the number systems.
The third, and largest, section of the book contains three historically complementary articles on group theory. It has a paper by Fearnley-Sander that convinces us that the basis of modern linear algebra was Grassmann’s inscrutable work of 1862 (Ausdehnungslehre). Israel Kleiner is again in action with his account of commutative algebra as the love-child of algebraic number theory; and there are papers on other topics, such as the history of prime numbers, Waring’s problem and Eisenstein’s proof of the quadratic reciprocity theorem. Biographically, there is Hardy’s paper on Ramanujan and Clark Kimberling’s highly accessible description of the life of Emmy Noether.
The whole book is attractively presented, with many portraits and other illustrations, and it is both entertaining and illuminating. For example, I found the account of the working relationship between Mary Cartwright and J. E. Littlewood to be highly amusing.
In conclusion, if this book turns out to be another MAA bestseller, that will be largely due to is due the fine judgement shown by its three editors in the selection and presentation of the forty-one articles.
Peter Ruane is retired from university teaching, where his interests lay predominantly within the field of mathematics education.