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Two Millennia of Mathematics: From Archimedes to Gauss

George M. Phillips
Publication Date: 
Number of Pages: 
CMS Books in Mathematics
[Reviewed by
Mary Shepherd
, on

This is a wonderful book from the Canadian Mathematical Society, sixth in their series of Books in Mathematics. The title should not be misunderstood. This book makes no claim to be a thorough coverage of two thousand years of mathematics. Instead, it contains relatively few selected topics, chosen because of the author's interest in them. It is a glimpse into the topics that one can meet when simply following one's interests. It treats most of the topics from both the historical and the modern perspective. It seems to follow the musings of a person who has enjoyed mathematics and has let his joy take him to many different, but in some sense related, topics. It is gently written with many examples and clear exposition.

There are five chapters and just over 200 pages. This should tell the reader that it cannot be a comprehensive historical treatment of mathematics. The chapters generally have about 5 sections with problems at the end of each section. No answers are given, and I found some to be easy to work and some quite challenging. There is a bibliography at the end of the book that gives a flavor of the history covered and contains several non-western sources

The chapter titles give some hint about what each holds. The first three chapters have a basic theme of estimation. The first is "From Archimedes to Gauss" and deals with the various methods employed since Archimedes to estimate the value of pi. Those interested in methods of approximation will find several approaches and techniques are used to get estimates and bounds for pi.

The second chapter was one of the more interesting to me, especially from a historical perspective. It is amazing to realize what insights some mathematicians had into numbers in the 16th and 17th centuries long before calculators and computers. The title of the chapter is "Logarithms". It gives much of the history and mathematics of the computation of the earliest logarithm tables.

The third chapter is entitled "Interpolation" and discusses some methods to generate polynomials which have specified values at certain points. It is a natural extension of the second chapter. And it goes further into finite differences and other methods used historically to generate interpolating polynomials. It even includes a section on multivariate interpolation for functions of more than one variable.

I found the fourth chapter on "Continued Fractions" to be quite interesting, going from the Euclidean algorithm to Fibonacci numbers and on to other recurrence relations. This is the first chapter in which I really wanted to get into the examples presented. I am sure others will see the possibilities with undergraduates of the preceding chapters. I can see this chapter being used in undergraduate projects or for supplemental reading. If I had one complaint in the book, though, it was also in this chapter. Some might be bothered by the non-standard notation used for continued fractions, but it is not hard to become comfortable with it.

And then the final chapter was on "More Number Theory". It touched on prime numbers, congruences, quadratic residues, Diophantine equations and ends with some comments about Ramanujan. I particularly enjoyed the discussion of Pythagorean triples. Again, I wanted to get my hands dirty and do many of the computations and problems in the section. The comments about algebraic integers gave me some ideas that might be fun to explore with undergraduates.

All in all, I really enjoyed the book and wanted to get into the problems, more so as I got further into the book. There seem to be fewer errors in this book than in many that have been published recently (although I did notice three or four). I don't think it would work as a textbook, but I believe it would be great supplemental reading for undergraduates, or maybe as a place for an undergraduate to get ideas for presentations or research in a math history or capstone courses. I think that people who do mathematics just for the fun of it — amateurs — will enjoy the book, also. And, as I said, I wanted to get into some of the problems and I thought there were topics from it that I could use to enrich courses I teach, from pre-calculus to abstract algebra.

I highly recommend this book. Read it and enjoy the wonderful ramblings of a person who enjoys mathematics and lets the math take him wherever it will.

Mary Shepherd ( is Assistant Professor of Mathematics at Northwest Missouri State University. Her research interests are in differential geometry and undergraduate mathematics education. Other interests include music, running, hiking and needlework. She is married to a non-academic; they have two sons, both in college.

From Archimedes to Gauss.- Logarithms.- Interpolation.- Continued Fractions.- More Number Theory.