NB I was sent this book as a review copy.

Four impossible puzzles, all described in detail during the height of classical Greek Mathematics. All simple to define and yet so tempting that it has taken not only the brain power of many, many thousands of mathematicians (amateur and professional alike), but also two millennia to show that however hard you may try, these puzzles are just not possible. The puzzles are:

• Squaring the circle: With only a compass and a straight edge, draw a square with the same area as that of a given circle.
• Doubling the cube: With only a compass and a straight edge, draw the edge of a cube with volume twice that of a cube whose edge is given.
• Constructing regular polygons: Given a compass and a straight edge, construct a regular n-gon in a given circle for $n\ge 3$.
• Trisecting an angle: Given a compass and a straight edge, and a given angle, construct an angle that is one third of the original.

So simple to define. You might not imagine that one could write a 400 page book on these four problems, and yet, when you dig into the ancient history, the mathematics which has been directly or indirectly influenced by these problems, the extensions to these problems, the connection with modern mathematics, and their final resolution, you discover that actually such a book is as fascinating as it is enlightening.

The book runs largely chronologically, with tangents in between each chapter giving some interesting tale, or puzzle, or historical side-note which adds greatly to the flow of the book.

The level is largely that which an advanced high school student could follow, with the occasional idea which may be a little too advanced, but this does not detract from the flow of the book itself.  The last chapters of the book get on to complex numbers and transcendental numbers (all relevant to the questions posed above), and these are also very nicely laid out historical explanations of how these ideas came about.

As somebody working within a mathematics department, while I had heard of these problems, I did not quite understand how important they had been, and how much time had been diverted to the often fool-hardy search for a solution, even after it had been shown that they were indeed impossible. A quick internet search will show you that there are still people out there trying to crack these impossible problems.

The writing style is very easy to read with diagrams throughout and just about the right level of detail so that you can really keep up with every argument, even though not all proofs are laid out in detail.

Overall this book was a pleasure to read and I would recommend it for anybody who wants a lovely overview of many areas of the history of mathematics, with a focus on some very easy to understand problems.

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