Catapults and cube roots
Since the dawn of time, people have been looking to technology for help in their military endeavors. We all know that many inventions that nowadays are used in “civilian” settings were born out of need in war times.
Catapults are perhaps the oldest large weapons ever invented. Previously the efficacy of a weapon was limited by the strength and ability required to use it: swords and clubs need to be swung around, heavy bows need to be drawn. And if you think that the bow is a generally “easier” weapon to handle (mostly because you are supposed to distance yourself from the battle a bit to even use it), well, remember the story of Odysseus and the bow only he could draw?
So ancient people wanted mechanical bows that could shoot bigger arrows, or hurl bigger weights toward their enemies. This is where the designs of the first catapults and mechanical bows enter the scene. Ancient Greece had both wars and several scientific minds willing to put their genius behind the design of new weapons (provided they were adequately funded by the current elite!).
The story of the invention of the catapult is a convoluted one: most texts we have from the time refer to different scientists or earlier models as sources of inspirations. Heron of Alexandria points to the gastraphetes as the forerunner to the first catapults and stone throwers. The gastraphetes, or “belly-bow”, was a large mechanical bow, with a mechanism of slides for drawing the string. It is unclear who invented it, or where it was first deployed.
But a couple of centuries before Heron we meet Philo of Byzantium, an engineer and writer who authored the Mechanike syntaxis, a nine-book treatise covering mathematics, geometry, military engineering, and many other fields. One of the few books to survive until today is the Belopoeica, or “On artillery”. In this book, Philo recounts that after powerful kings devolved many resources to increase their military prowess, the fundamental principle of building a “stone thrower” was reduced to a single element, the diameter of the circle through which the torsional spring (the one “drawing” the weight or arrow back) is passed. More precisely, the diameter of the hole holding the spring, and therefore the diameter of the spring itself, should be proportional to the cube root of the weight of the projectile we want to throw.
How did they get this result? I think empirical tests plus a ruler. That must have been quite surprising! It’s not common for a cube root to show up in practical applications.
The same principle is recounted later, again by Hero of Alexandria, and by Vitruvius, a Roman engineer.
But now our ingenious minds had an even larger problem: to design new catapults, they needed to be able to compute cube roots of arbitrary numbers, to some degree of precision.
The first important intuition comes from the name of the operation itself: a cube. Given a cube of side L, its volume is L³. If we manage to somehow build a second cube whose volume is double the volume of the first one, i.e. 2L³, and we measure its side, that will be the cube root of 2 multiplied by L. Divide by L and we find the cube root of 2. This trick was called duplication of the cube. Of course it can be done with any other multiplication factor, not just 2, in order to find cube roots of other integers.
But building such a cube in reality is quite hard. Even if I have one cube to start from, how do I know whether the volume of the new cube is the correct one?
The second interesting intuition comes from arithmetic, from something called (I think) proportional mediums.
Let’s take the following proportions:
2: y = y : x = x: 1
Given the basic definition of proportionality it’s easy to see that x² = y and y² = 2x. Let’s get rid of y and solve by x.
Now we combine together the terms with x, switching to fractional exponents to represent roots.
Now elevating both sides to the power of 2/3 gives me:
Here it is! x is equal to the cube root of 2, i.e. the leftmost term of the proportion we started with. The rightmost term is nothing else than L, the side of the original cube, which I set to 1 for simplicity.
This is quite cool, but equations alone are still not helping our poor Greek engineer who needs actual numbers to build a catapult. If he were able to represent these proportions (or equivalent ones) with actual lines, he could measure the length of x and get the actual value. At least two mathematicians invented devices that did just that: one is the mesolabe, invented by Erathostenes, and the other is Philo’s line, invented by (yes) Philo of Byzantinium. Their ingenious ideas and mechanical results are going to be covered in the next posts.
