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Algebra in Ancient and Modern Times

V. S. Varadarajan
American Mathematical Society and Hindustan Book Agency
Publication Date: 
Number of Pages: 
Mathematical World 12
[Reviewed by
Ed Sandifer
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The AMS does not publish very many books that are accessible to high school students. Algebra in Ancient and Modern Times is an exception. It gives an historical introduction to elementary algebra, from Euclid and the infinitude of primes to Hamilton and the quaternions.

The book is Volume 12 of the Mathematical World series, jointly published by the AMS and the Hindustan Book Agency. Other books in the series include four books about Japanese high school curriculum and five books based on Russian curricula. Varadarajan mostly follows the development of algebra in the Western tradition, but he earns inclusion in the "World" series by giving as much treatment as will fit to developments in India, Arabia and China and still keep the length of the book at just 142 pages.

The work is divided into three roughly equal sections. The first section, "Some history of early mathematics" will be mostly familiar to people who have read any standard book on the history of mathematics. At times, it reads a bit like several student projects edited together to form a single chapter, but for the most part, it is concise and accurate, with a great variety of exercises every few pages. Also, there are a few rare gems sprinkled in. Few authors mention Fibonacci's book Flos, published in 1225, overshadowed as it is by his Liber abacci and Liber quadratorum.

The second section is "Solutions for the cubic and biquadratic equations." This section is a little shorter than the other two sections, but it is the really distinctive part of the book. There are a number of reasonably good treatments of the solution to the cubic equation, but this is the only complete exposition of the solution of the biquadratic, or fourth degree equation that I have ever seen. The author gives a complete and concise treatment of the controversies surrounding the solution of the cubic equation, and also shows how it was the solution to the cubic equation, not the solution of the quadratic equation, that led to serious study of complex numbers.

The final section, "Some themes from modern algebra" is a glimpse into the Pandora's box of algebraic creatures unleashed following the development of complex numbers in the 1500's. We get to peek at quaternions and Clifford algebras, and we hear of the accomplishments of Gauss, Euler, Hilbert and Wiles.

This is a fine book on two counts. First, as mentioned above, there is the singularly excellent treatment of the solution of biquadratic equations. Second, it paints a strong picture of mathematics as a very long sequence of accomplishments, each building on the ones before, in a way that beginning mathematicians can understand and appreciate it. It paints the picture in a concise and economical style, the style that mathematicians find elegant.

I would particularly recommend Algebra in Ancient and Modern Times to strong high school students, to high school algebra teachers, to people who want a history of mathematics with a lot of mathematics in the history, and to anyone who needs to know how to find an analytic solution to a nasty fourth degree polynomial.

Ed Sandifer ( is a professor of mathematics at Western Connecticut State University, Contributed Papers Coordinator for the Northeastern Section of the MAA, and an avid runner.

Some history of early mathematics

  • Eucild-Archimedes-Diophantus
  • Pythagoras and the Pythagorean triplets
  • Aryabhata-Brahmagupta-Bhaskara
  • Irrational numbers: construction and approximation
  • Arabic mathematics
  • Beginnings of algebra in Europe
  • The cubic and biquadratic equations

Solution of the cubic and biquadratic equations

  • Solution of the cubic equation
  • Solution of the biquadratic equation

Some themes from modern algebra

  • Numbers, algebra, and the real world
  • Complex numbers
  • Fundamental theorem of algebra
  • Equations of degree greater than four
  • General number systems and the axiomatic treatment of algebra
  • References
  • Chronology
  • Index