The Pythagorean Theorem

A 4,000 Year History
by Eli Maor

I was disappointed by this book. I was hoping for a history of the thinking behind the Pythagorean Theorem. This is one of the most important ideas in mathematics, and it was independantly figured out by the Mesopotamians, the Greeks, the Chinese, and the Indians. What was the thinking that made it possible for each of these civilizations to figure it out? From Maor’s presentation it is apparent that the solution to the theorem was motivated by practical considerations. However, in each case the proof of the theorem was based on abstract analysis.

There is a subtle point to appreciate here. All four civilizations knew of the classic “windmill diagram” demonstrating the Pythagorean Theorem:

While this diagram illustrates the theorem, it does not prove that the theorem works with all possible right triangles. It is important to note that the Greeks were the only one of the civilizations to produce an actual proof. Everybody else was satisfied to demonstrate that it worked in some cases, and assume that it worked in all cases. Over 200 unique proofs of the theorem have been devised, but the simplest breaks the windmill diagram into a series of component triangles like so:

And then uses those triangles to prove the theorem. I’ll not spare the reader the details of the proof. What’s important is that the Greeks saw the need to prove the theorem formally, and nobody else saw that need.

The book is really a mathematician’s history of the theorem; it slogs through every proof with mathematical rigor. A rule in publishing is that every mathematical formula in a book cuts the market size in half. If this be true, and there are 7 billion people on this planet, then I would someday like to speak with the other person who read this book.