I am not a physicist.

My high school physics class was taught by a man who was probably brilliant. I say “probably” because he was so bad at explaining things that it was impossible to tell. His lectures went like this: he would state a problem and draw some sort of diagram on the board. He would then proceed to mumble incomprehensible things while drawing (apparently random) arrows and equations on the diagram. Then he would write down the correct answer. Our “final exam” consisted of the multiple-choice portion of an old AP physics exam; I got a score of 17.

Out of 200.

It was the highest score in the class.

In college, I occasionally thought about taking a physics course, but it never quite fit my schedule. Besides, I had this vague sense, left over from high school, that physics was messy, inelegant, and full of equations that didn’t yield any intuitive insight into anything.

I was pleasantly surprised, then, to enjoy Mark Levi’s new book, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems (Princeton University Press, 2009). [Full disclosure: PUP kindly sent me a free review copy. Does that make me biased? Well, probably.]

Here’s the idea: you have a thorny mathematical problem to solve. For example, let’s say you want to prove the Pythagorean Theorem (which is, in fact, the subject of Chapter 2). You then (this is the hard part) come up with some sort of appropriate idealized physical system, such as a freely rotating triangular tank full of water, or a frictionless ring sliding on a semicircle and attached to two springs, and so on. You then make some sort of observation that qualifies you as being at least three years old, such as, “The tank full of water doesn’t move. Because a tank full of water doesn’t just move by itself.” And then you write down some equations which represent the fact that the water doesn’t move, and hey presto! The Pythagorean Theorem!

The book is chock-full of these seemingly magical physical thought experiments involving bicycle wheels, pistons, springs, soap films, pendulums, and electric circuits, with applications to geometry, maximization and minimization problems, inequalities, optics, integrals, and complex functions, obviously collected over an (intellectually adventurous) lifetime. I doubt it will actually help me solve problems anytime soon, but in some sense that isn’t really the point; it gave me new intuition and insight into both physics and mathematics, and a new appreciation—missed in high school—for the elegance and beauty found in the study of a physical world that can be described (perhaps unreasonably so) by mathematical equations.

The prose is pithy, and I had no trouble following the gist of the arguments. My one complaint is that the details are often hard to follow for a non-physicist like me: although the book claims to be self-contained, with an appendix covering all the necessary background, the appendix is so concise as to be not all that useful unless you already know some physics—in which case you probably don’t need the appendix in the first place! But in some sense, this is more a complaint about my lack of education than about the book. It isn’t meant to be (nor should it be) a physics tutorial, so it’s appropriate for it to assume some background; I’m pretty sure that anyone with a (decent!) high school course in physics could follow most of it well enough.

Overall, I highly recommend it to anyone who is (even slightly) interested in physics, and appreciates mathematical elegance and cleverness. It would make a great gift for almost anyone, whether a high school student or university professor, armchair physicist or professional mathematician.