I've been spending the last couple of days going over proofs for "Heron's Formula," which is a way to find the area of any triangle if you know or can figure out the lengths of all three sides. If s stands for the "semi-perimeter" of the triangle, that is, half the sum of the lengths of all three sides, then Heron's formula for area is the square root of: s multiplied by s minus the length of side a, s minus the length of side b, and s minus the length of side c. That's (square root) (s(s-a)(s-b)(s-c).).

Heron of Alexandria was a mathematician, physicist, and engineer back in the late Hellenistic world of the first century CE. (That's "A.D." for the older types.) He inherited a Greek and Near Eastern mathematical tradition which stretches back not only to Archimedes and Pythagoras but to ancient Babylon where the Pythagorean theorem and algebra may already have been practiced. Reading this biography of Heron, I find it amazing how technologically and mathematically sophisticated the ancients were. And they didn't even have slide rules, let alone pocket calculators.

If I were asked to prove Heron's formula by myself, I'd be completely baffled. I can do, on my own, only the simplest algebraic and trigonometric proofs; I have yet to really get into the technique of doing more complex ones. But fortunately the helpful Schaum's text has a detailed proof of the formula, and there's another, equally good proof at Jim Wilson's math course site at the University of Georgia.

The proof flies through all sorts of mathematical twists and turns, often relying on trigonometric and algebraic identities and substitution. It multiplies things out, and re-arranges them. It uses old algebra standbys like solving for a single variable, factoring out quantities, finding the difference of squares, and canceling out common factors in a fraction. It also uses trigonometric standards like the cosine formula, which in turn depends on the majestic Pythagorean theorem, which was ancient even when Heron mathematized two thousand years ago.

I could just memorize the formula, which I have indeed done, but the proof of this formula carefully followed is a lesson in proof technique which I will find valuable in my later mathematical work. It demonstrates the use of identity and transformation, how to make one thing into another thing which is equivalent but fits with yet another thing. The proof of Heron's formula, to use another metaphor, meshes, turns, and clicks like antique gears, like one of Heron's own late Hellenistic inventions. It's the technique that I need to remember. Take the identity equivalent of something, and plug it into another expression where you find it, and multiply it out. See if what you get fits anywhere else or looks like something else that could be relevant. I wonder if this is where mathematical creativity starts.

In my usual way of learning math, I wrote the entire proof down on a sheet of notebook paper, turned horizontally. Even with all that space, I had to cram a lot onto the page, and it looks rather like an oriental carpet pattern. But I can read it, and it will go into my archives as a reference.

By the way, Heron's name has nothing to do with long-necked waterfowl. The name "Heron" is probably the English version of the ancient Greek name "Hieron," which comes from the Greek word hieros, which means "holy" or "divine."

Posted at 2:56 am | link