31 October, 2024

A little under two millennia ago¹ there lived in the Egyptian city of Alexandria a Greek mathematician and inventor named Hero², and he was a total badass who invented things that you probably thought came way later, and come up with mathematical tricks that we still use to this day³.

Inventions

Illustration of an aeolipile, showing a heated sphere or cylinder of water spraying steam in two different directions, causing it to rotate. — If you know of Hero’s inventions because of his aeolipile⁴,known as “Hero’s engine”, I’ve got bad news: it was probably actually invented by his predecessor Vitruvius. But Hero *did* come up with a way to use the technique to make a pneumatic temple door that automatically opened when you lit the fires alongside it.

Hero is variably credited with inventing, in some cases way earlier than you’re expecting:

automatic doors (powered either by pressure plates or by lit fires),
vending machines, which used the weight of a dispensed coin to open a valve and dispense holy water,
windmills (by which I mean wind-powered stationary machines capable of performing useful work),
the force pump – this is the kind of mechanism found in traditional freestanding village water pumps – for use in a fire engine,
float-valve and water-pressure based equilibrium pumps, like those found in many toilet cisterns, and
a programmable robot: this one’s a personal favourite of mine because it’s particularly unexpected – Hero’s cart was a three-wheeled contraption whose wheels were turned by a falling weight pulling on a rope, but the rope could be knotted and looped back over itself (here’s a modern reimplementation using Lego) to form a programmed path for the cart

Illustration showing mechanism of action of a fire-powered automatic door: heated air expands to displace water into a vessel whose now-increased weight pulls ropes wrapped around two wheels which pull open a pair of temple doors. — It’s just headcanon, but I choose to believe that the reason Hero *needed* to invent the fire extinguisher might have involved the number of “attempting to make fire do work” inventions that he came up with.

Mathematics

If you know of Hero because of his mathematical work, it’s probably thanks to his pre-trigonometric work on calculating the area of a triangle based only on the lengths of its sides.

But I’ve always been more-impressed by the iterative⁵ mechanism he come up with by which to derive square roots. Here’s how it works:

Let n be the number for which you want to determine the square root.
Let g₁ be a guess as to the square root. You can pick any number; it can be 1.
Derive a better guess g₂ using g₂ = ( g₁ + n / g₁ ) / 2.
Repeat until g_N ≈ g_N-1, for a level of precision acceptable to you. The algorithm will be accurate to within S significant figures if the derivation of each guess is rounded to S + 1 significant figures.

That’s a bit of a dry way to tell you about it, though. Wouldn’t it be better if I showed you?

Put any number from 1 to 999 into the box below and see a series of gradually-improving guesses as to its square root⁶.

Interactive Widget

(There should be an interactive widget here. Maybe you’ve got Javascript disabled, or maybe you’re reading this post in your RSS reader?)

Maths is just one of the reasons Hero is my hero. And now perhaps he can be your hero too.

Footnotes

¹ We’re not certain when he was born or died, but he wrote about witnessing a solar eclipse that we know to have occurred in 62 CE, which narrows it down a lot.

² Or Heron. It’s not entirely certain how his name was pronounced, but I think “Hero” sounds cooler so I’m going with that.

³ Why am I blogging about this? Well: it turns out that every time I speak on some eccentric subject, like my favourite magic trick, I come off stage with like three other ideas for presentations, which leads to an exponential growth about “things I’d like to talk about”. Indeed, my OGN talk on the history of Oxford’s telephone area code was one of three options I offered to the crowd to vote on at the end of my previous OGN talk! In any case, I’ve decided that the only way I can get all of this superfluity of ideas out of my head might be to blog about them, instead; so here’s such a post!

⁴ If the diagram’s not clear, here’s the essence of the aeolipile: it’s a basic steam reaction-engine, in which steam forces its way out of a container in two different directions, causing the container to spin on its axis like a catherine wheel.

⁵ You can also conceive of it as a recursive algorithm if that’s your poison, for example if you’re one of those functional purists who always seem somehow happier about their lives than I am with mine. What’s that about, anyway? I tried to teach myself functional programming in the hope of reaching their Zen-like level of peace and contentment, but while I got reasonably good at the paradigm, I didn’t find enlightenment. Nowadays I’m of the opinion that it’s not that functional programming leads to self-actualisation so much as people capable of finding a level of joy in simplicity are drawn to functional programming. Or something. Anyway: what was I talking about? Oh, yeah: Hero of Alexandria’s derivation of square roots.

⁶ Why yes, of course I open-sourced this code.