This book contains short, self-contained essays on elementary mathematics. Many describe work of outstanding mathematicians, Gauss among them, and show the incisiveness and ingenuity so characteristic of first rate mathematical work, no matter what the level. I have chosen the topics treated here because I remember how exciting and stirring I found them on first encounter. I want to share this sense of wonder with a larger audience and believe that many people, though unaware of it, have the minimal mathematical background needed to understand and enjoy the ideas described here; the basic material of a standard first high school course in algebra and geometry suffices. It is my earnest hope that some readers will come under the spell of these fascinating topics.
The book is modeled after The Enjoyment of Mathematics by Rademacher and Toeplitz, in my opinion one of the best popular mathematical expositions ever written.
The references given in each essay usually include my sources as well as appropriate material for those readers who wish to pursue the topic treated a little farther. The references in the bibliography at the end of this monograph are outstanding books in the general field of elementary mathematics and will nurture a growing interest. In this connection I want to mention specifically the periodical Scripta Mathematica which, specially during the years 1935-1957, was a veritable treasure trove of mathematical riches.
In some essays I have treated rather specific concrete problems, often neglecting or merely hinting at generalizations, while in others I have tried to generalize and abstract a bit more. The reader should be aware that the treatment of any particular topic here is just one of several possible approaches. At the end of most essays I have added related exercises whose solutions appear at the end of the book. Since the essays are independent of one another, they may be read in any order.
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