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The Enjoyment of Math

Hans Rademacher and Otto Toeplitz
Princeton University Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a serious math book that has minimal prerequisites: geometry and college algebra, but no trig or calculus. It contains 28 largely independent chapters that solve a variety of famous and difficult math problems, mostly in the areas of plane geometry and number theory. The problems include: Fermat’s last theorem for exponent 4, unique factorization in number fields, a number of geometrical maximization problems including several versions of the isoperimetric problem, some transfinite numbers, the 5-color map coloring theorem, and the arithmetic mean - geometric mean inequality. There’s no analysis per se in the book, but several topics depend on the analytic ideas of continuity and variation.

This book was first published in German in 1930 and in English in 1957 as The Enjoyment of Mathematics, and is still in print today in both languages. This implies that there is still an audience for it, but it is hard to imagine exactly what this audience is. The book was developed out of a series of public lectures and was intended as a “popular math” book. While it is very clear and well-written, the reasoning in all the chapters is very intricate (especially in the geometric problems), and the book is much more difficult than anything that appears in popular math books being written today. It’s also too difficult for a math appreciation text. The modern (2000) Preface to the German edition suggests that the book is suited for bright high-school students who are hungry for learning, and maybe this is its real audience today. Eager college students would instead get enrichment activities related to their course work, and professional mathematicians, although they would admire the book, would probably look up the material in a more specialized text and not here.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  1. The Sequence of Prime Numbers
  2. Traversing Nets of Curves
  3. Some Maximum Problems
  4. Incommensurable Segments and Irrational Numbers
  5. A Minimum Property of the Pedal Triangle
  6. A Second Proof of the Same Minimum Property
  7. The Theory of Sets
  8. Some Combinatorial Problems
  9. On Waring's Problem
  10. On Closed Self-Intersecting Curves
  11. Is the Factorization of a Number into Prime Factors Unique?
  12. The Four-Color Problem
  13. The Regular Polyhedrons
  14. Pythagorean Numbers and Fermat's Theorem
  15. The Theorem of the Arithmetic and Geometric Means
  16. The Spanning Circle of a Finite Set of Points
  17. Approximating Irrrational Numbers by Means of Rational Numbers
  18. Producing Rectilinear Motion by Means of Linkages
  19. Perfect Numbers
  20. Euler's Proof of the Infinitude of the Prime Numbers
  21. Fundamental Principles of Maximum Problems
  22. The Figure of Greatest Area with a Given Perimeter
  23. Periodic Decimal Fractions
  24. A Characteristic Property of the Circle
  25. Curves of Constant Breadth
  26. The Indispensability of the Compass for the Constructions of Elementary Geometry
  27. A Property of the Number 30
  28. An Improved Inequality
  Notes and Remarks