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Publisher:

Princeton University Press

Publication Date:

1966

Number of Pages:

208

Format:

Paperback

Price:

42.00

ISBN:

978-0-691-02351-9

Category:

General

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Allen Stenger

06/22/2010

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 MathNerds.com, a math help site that fosters inquiry learning.

Preface | ||||||||

Introduction | ||||||||

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 |

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