Thousands of years ago, before the Babylonians, someone noticed that if you have ropes that are 3, 4, and 5 units long, and you stretch them into a triangle, the angle between the 3 and 4 length ropes will be 90 degrees – or a right angle. Today, the far descendants of that realization cavort in n-dimensional spaces and non-Euclidean geometries. And, in *Hidden Harmonies: The Lives and* *Times of the Pythagorean Theorem*, Robert and Ellen Kaplan take you on that journey of at least 5,000 years.

The Pythagorean Theorem, which is the focus of *Hidden Harmonies: The Lives and* Times of the Pythagorean Theorem, is named after the famous Greek mathematician and philosopher Pythagoras, although he is probably not the one who discovered it. It states that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the famous example above, 32 + 42 = 52, or 9 + 16 = 25. Although the Babylonians knew many examples of Pythagorean triples, including some with very large numbers, it was the genius of the Greeks that allowed them to go from what is essentially very clever number play to math – to say that this is true, not in this case and that case, but in ALL cases, and then to prove that they were right.

Bob and Ellen Kaplan are a husband and wife team who love math. They teach math in the Math Circle, which I have written about (see resources), and they write about math. People who write popular math books have to decide just who they are writing for. There seem to me to be three broad choices: 1) Non-math people. These books try to convince someone who doesn’t like math that they really should 2) Math amateurs. These books assume you know and like math, and present you with some interesting material. This audience may have majored in math in college (or may not) but they are familiar with some basic math ideas, at least at the high school level, or perhaps the intro college level 3) Math and science professionals. These are books written for people who know a lot of math, certainly studied it in college, and are now exploring a different part of the mathematics universe. *Hidden Harmonies: The Lives and* *Times of the Pythagorean Theorem* is firmly in the second class.

The Kaplans assume you know some math. They assume you are willing to work a bit at the book, because if you are a math amateur you know that the working at a math book is part of the joy of it. More than this,. the Kaplans assume that you are an intelligent person. This is not a “for dummies” book.

*Hidden Harmonies: The Lives and* *Times of the Pythagorean Theorem* starts with some fundamentals about the PT, then gives a great many proofs of it, then shows extensions of it to all sorts of domains the Greeks never dreamed of. Along the way they introduce you to some fascinating people, and discuss some fascinating ideas. It’s a fun trip.

What are these hidden harmonies of the title? Well, one part of the miracle of math is that they are everywhere, and you’ll see many in the book. Here is one:

sin(x) = x – x3/3! + x5/5! – x7/7! …….

where sin() is the sine function from trigonometry (the opposite side divided by the hypotenuse) and ! is the factorial, and x is an angle, expressed in radians.

What? How does a ratio of the sides of a triangle relate to powers and factorials????

It’s a hidden harmony. If it strikes you as cool, then *Hidden Harmonies: The Lives and* *Times of the Pythagorean Theorem* is for you.