The antikythera mechanism is one of the most mind boggling of ancient artefacts ever to have been recovered. Found in a shipwreck off the island from which it takes its name, it dates from around 100 BC and is apparently of Greek origin. Packed full with a highly sophisticated set of bronze gears and covered in fine inscriptions, it computes the positions of heavenly bodies in the sky and the phases of the moon. In essence, it is a calibrated and working model of the solar system and amounts to the world's first known desktop computer. Its predictive accuracy is remarkable, especially considering its antiquity.
As such it is astonishing enough and has been declared by one archaeologist as more precious than the Mona Lisa. But I want to explain why its significance runs far deeper than this. It is a window into the minds of the ancient Greeks themselves and it helps us to understand more about who they were and where they were coming from.
Greek mathematicians were fascinated by prime numbers, numbers which cannot be subdivided, and ascribed to them mystical significance. Not all numbers are whole numbers, but the Greeks believed that any awkward number such as π (pi) or the square root of two could be expressed as a fraction or ratio, of one whole number divided by another. You just had to search long enough and hard enough for prime numbers that were big enough. What we call irrational numbers were believed to be trivial or not to exist and some cults even declared it a heresy punishable by death to suggest that π or the square root of two were irrational. It does all sound rather primitive and superstitious, doesn't it.
Meanwhile Greek astronomers were equally fascinated by the movements of the heavenly bodies across the sky. In order to explain and predict these movements they devised and refined what we have come to call the Ptolemaic universe, at the centre of which sits the Earth and around which revolve the sun, moon and planets in a complicated rigmarole of cycles and epicycles - wheels within wheels. The Moon running on cosmic wheels? More superstitious nonsense, surely!
Now let us take a deeper look at the antikythera mechanism. Its task is to model the movements of the sun, moon and probably the known planets as precisely as it could. It achieves this using toothed gears. Like any gear wheel, they must have whole numbers of teeth. The relative rates of rotation of two meshing gears, what we call the gear ratio, is obtained by dividing one gear's number of teeth by the other's: it is a rational number. In the search for such tooth counts, prime numbers are of great importance because they help us to find the smallest number of teeth that we need to make. Many prime numbers appear in the antikythera gearing. One gear has 227 teeth - a prime number that must have taken the most extraordinary amount of time and effort to determine.
The achievement is made even more awesome by the fact that the Greeks didn't have modern numerals, nor even Roman, they used letters of the alphabet to represent numbers when absolutely necessary and did the rest with geometry. It must have been a gargantuan intellectual challenge, maintained and refined generation after generation, on a scientific and cultural par with modern physics. How do you even stumble across the number 227, never mind show that it is prime and that it is the number you want and then hand-cut a wheel with that precise number of teeth? It must have been like trying to hand-calculate accurate particle-particle scattering amplitudes before twistor theory came along. Physicists will know what I mean by that, the rest of us can be duly frightetened and impressed by the sound of it.
These gears do not all have fixed axles, some are epicyclic - mounted on other gears, wheels within wheels. The whole ensemble achieves an impressive level of accuracy - in some cases as accurate in fact as the Greek astronomers could measure, and boy had they gone to town on their measurements, even constructing whole buildings as astronomical instruments. That's why the designer was in one case satisfied with nothing less than a ratio involving 227.
Now think of our own model of fundamental physics, the Standard Model. It is complicated and a bit messy. We plug into it over a hundred arbitrary numbers and we get out of it an impressive level of accuracy - in many cases as accurate as we can measure, and boy have we gone to town on our measurements, even constructing whole buildings such as the Large Hadron Collider as measuring instruments. We use this Standard Model to describe the universe around us and we destroy the career of anybody who questions its fundamental tenets. Sounds familiar?
We can now appreciate what the antikythera mechanism really is, it is the ancient Greek Standard Model of the universe, the culmination of decades if not centuries of sustained scientific effort. They plugged into it a great many arbitrary numbers found by endless experimentation and they got out of it as much accuracy as they could measure. Those Greeks were not superstitious primitives, they were top-notch scientists who knew to use the same scientific method of reasoned theorising combined with suck it-and-see empiricism that we do to test and refine their model of reality, and when they said everything was down to prime numbers and wheels within wheels, that was because it was part of their Standard Model. They had conducted thousands of methodical and expensive measurements to the greatest possible accuracy, they had devised a rational model of what lay behind those measurements. They had searched long and hard enough until they found the numberst that made the model work. They built and tested a computational model. And the result was undeniable because it worked. Bits of it didn't come out quite right and they worked away at those, with later theoretical embellishments coming out ever more accurate. They grew their Standard Model just the way we did.
Over a thousand years later, Copernicus was still using it for calculating navigational tables even after he had discovered that the Sun lay at the centre, because it still made more accurate predictions than his own theory. Like ours, it was so successful, so accurate - and so darn useful in doing things like calculating alignments and dates for the Olympic games and, later, maritime navigation tables that anybody who disagreed with its fundamental tenets was clearly barking mad and not safe to have around. It is no more possible today to carve a career in modern physics if you deny quantum mechanics than it was then if you denied ratios.
One can begin to understand too such odddities as why numerology cast such a pervasive spell across the ancient and medieval worlds, why so many perfectly sane men spent their lives scouring through every kind of arcane subject looking for patterns of numbers that might reveal their inner secrets. The triumph of the greatest scientific model ever to be validated by experiment, the Ptolemaic cosmology, was founded on that very discipline of numerology in its hunt for the right ratios and the prime numbers that formed them. Everybody was just taking the proven discoveries and techniques of contemporary science and applying them to their own particular interests. It would have made a lot of sense at the time.
These aspects of Greek thought - the mechanical predictability of the cosmos and the character and importance of numbers - must have infused and informed almost every subject that any educated Mediterranean scholar studied from then on. When we stumble on references in their literature or artefacts we need to stop saying "how quaint" and to start looking again with fresh eyes at the intellect behind the ideas.
Significant cracks in the theory did not appear for well over a thousand years, until a raft of new technologies and techniques opened up new levels of accuracy. Proofs that root two was irrational had appeared. Arabic numerals and algebra allowed Copernicus to discover that putting the sun in the middle made astronomical calculations a lot simpler, though when he discarded those ancient epicycles he actually lost some accuracy. The telescope allowed Galileo to see the moons of Jupiter and the Danish astronomer Tycho Brahe to make measurements to a new level of accuracy - so accurate that they exposed flaws not only in the Ptolemaic model but even in the brave efforts of Johannes Kepler to update the inaccurate Copernican model. In desperation, Kepler then used Brahe's data to discover that the planets moved not in epicycles but ellipses. An ellipse cannot be reproduced using the old wheels-within-wheels. But it neatly fixed the flaws in the Copernican model, making it the most accurate available for the next few hundred years. The Ptolemaic universe at last lay as shattered as Kepler's dreams of cosmological harmony, and it was not long before Isaac Newton discovered that his new laws of motion and gravity could explain those ellipses. The modern cosmological model began to take shape and the profoundly rational intellect of the Greeks was forgotten.
Even today we call their acceptable numbers rational and their heretical numbers irrational. In two thousand years we haven't changed a bit. Only our Standard Model has. Even the technology we use to build our personal computers didn't change until the mechanical desktop calculators of my childhood were replaced by electronic ones just over forty years ago and some seventy years after the antikythera mechanism was rediscovered. Not bad for a piece of corroded junk from the bottom of the sea.
I leave you with one last thought. Will our Standard Model be as successful? Will it too last for a thousand years? Or will our own four-dimensional spacetime, our electrons, quarks and photons and our four fundamental forces of nature in due course all seem as quaint and superstitious as a clockwork moon?
Updated 2 Nov 2015