Decoding the Heavens
Page 12
First, the easy part. Price confirmed that the small crown wheel drove the mechanism by engaging with the big four-spoked wheel, which he grandly called the ‘Main Drive Wheel’, because it then drove the rest of the machine’s gear trains. The shaft of the crown wheel extended out through the side of the case. Price was undecided how the crown wheel itself would have been turned – by hand via a handle on the side of the case, he concluded, or even by a spectacular water clock like the Tower of the Winds.
The drive wheel was positioned directly behind the zodiac dial on the front of the case and turned around the same centre. It takes the Sun one year to go all the way through the zodiac, so Price deduced that this big wheel would have driven a pointer that showed the position of the Sun in the sky. Five turns of the side handle would have moved the wheel and pointer through roughly one turn – a time period of one year.
From there, things got a bit more complicated. The rotation passed through three connected pairs of meshing gears, ending up at a gearwheel that turned around the same centre as the main drive wheel, but with a narrower axle that passed back up to the front of the mechanism, through the middle of the drive wheel’s hollow shaft. This presumably drove a second pointer around the front dial.
What did the second pointer show? To say for sure, Price needed to know the speed at which it moved relative to the Sun pointer. And by counting the teeth on the gears he could calculate what happened to the speed of rotation at each step. For example, as described in Chapter 2, if a wheel with 20 teeth drives a wheel with 10 teeth, then every turn of the first wheel will result in two turns of the second. This can be written mathematically.
20
* * *
10 = 2
And in a similar pair of meshing gear wheels, say with 90 teeth and 30 teeth, every turn of the first wheel will result in three turns of the second.
90
* * *
30 = 3
The two pairs can be connected together by a common axle, running through the second wheel of the first pair and the first wheel of the second pair. Because they share an axle, these wheels have to turn at the same rate, meaning that the output of the first pair becomes the input of the second pair. This can be written as follows:
20
* * *
10 × 90
* * *
30 = 6
In other words, for every full turn of the first wheel, the final wheel of the four will turn six times. Of course it would be easy to achieve such a simple result with a single pair of gearwheels – with 60 teeth and 10 teeth, say – but combining two, three or more pairs of gears allows for more complicated ratios than can be easily achieved with a single pair (smaller wheels are simpler to cut than big ones, for example, and it’s harder to get the spacing of the teeth right with prime numbers).
Price sifted through the tooth counts that Emily and Charalambos Karakalos had provided for the six wheels in the first gear train, trying to work out what the overall ratio had been. What was the ancient maker trying to compute? The obvious answer, of course, was that the second pointer had shown the motion of the Moon. But he needed to prove it.
Calculating the position of the Moon from that of the Sun isn’t straightforward. Although the calendar we use today divides years into exactly twelve months, the Moon doesn’t go round the Earth exactly twelve times for every time that the Earth goes round the Sun. So a simple calendar can either show the timing of the Sun and seasons, or the motions of the Moon. But it can’t do both – the two soon get out of sync. Our modern system of counting the days is driven by the Sun, and our calendar matches the seasons in a yearly cycle. This means that on a particular date each year the Sun will be in roughly the same position relative to the Earth. January is always winter (in the northern hemisphere at least) and July is always summer. The summer solstice – the longest day of the year, when the northern hemisphere is tipped maximally towards the Sun – falls unfailingly on 20 or 21 June.
The price we pay for following the Sun is that our calendar has lost touch with the Moon. The date of the full moon varies from month to month, and every year the pattern is different (this is why Easter, which is calculated according to the first full moon after 21 March, moves around in the calendar). The days of our months conveniently follow the same pattern every year – we know that March will always have 31 days, and April will always have 30 – but they no longer reflect the lunar phases.
Nowadays it doesn’t matter that much. For most of us the Moon’s phase is of little importance in daily life. But for the Greeks, as for most ancient peoples, following the Moon’s motion was critical for everything from the timing of frequent religious festivals and civic duties to whether you’d be able to see that night.
The Moon orbits the Earth – from our point of view circling through the sky with respect to the background stars – about every 27.3 days. This is called the sidereal month (from sidus, which is Latin for ‘star’). The time period from full moon to full moon, called the synodic month, is a little longer, on average 29.5 days. The Greeks knew that although the Moon’s movement doesn’t fit neatly into a year, it does come back to almost exactly the same position with respect to the Sun and the Earth every 19 years. Within each 19-year cycle, there are 235 synodic months (give or take a couple of hours) and the Moon circles through the sky 254 times.
So the Greeks combined the movements of the Sun and Moon in a 19-year repeating calendar called the Metonic cycle, after an astronomer called Meton who lived in Athens in the fifth century BC. He was the first Greek we know of to use it, although he almost certainly got the idea from the Babylonians. Their priest-astronomers had been observing the heavens for centuries before that, and were well acquainted with the relationship.
According to this cycle, the number of sidereal months in one year is 254/19. So Price realised that once you have a wheel that turns with the Sun through the sky, you can multiply its rotation by this ratio to calculate the speed of the Moon. The counts that Emily and Charalambos had given him for the six wheels in this train were 65 (although they couldn’t rule out 64 or 66), 38, 48, 24, 128 and 32. That gives the following gear train:
65
* * *
38 × 48
* * *
24 × 128
* * *
32 = 260
* * *
19
It is tantalisingly close to the 19-year cycle. Price played around with the numbers, hoping that the ratio he needed would appear. Changing the first wheel to 64 teeth – at the lower end of the range that the Karakaloses had given – brought the top number to 256. Then all it took was to tweak the 128-tooth wheel to 127, surely within the range of possible error. The train now read:
64
* * *
38 × 48
* * *
24 × 127
* * *
32 = 254
* * *
19
Price sat back in his chair and lit his pipe, sucking on the end as he watched the particles of smoke dance in the light of his desk lamp. The mechanism was giving up its secrets to him at last! And they were beautiful. The results of centuries of astronomical observations had been converted first into mathematics and then made real again, carved quite precisely into six wheels of shining bronze. The gear train reminded him of a computer programme: put in the Sun, and get out the Moon. As the owner turned the handle on the side of the box – driving the main wheel and the Sun pointer round once for every year – the second pointer on the front dial would have shown the Moon’s position in the sky, whirling around the zodiac scale just over twelve times faster than the stately Sun.
But there was a snag. Each time one gearwheel meshes with another, the direction of rotation is reversed. So the gear train Price had just worked out, with its three pairs, would have sent the Moon spinning in the opposite direction to the Sun. That couldn’t be right. But Price soon came up with an ingenious solution. Rather than the main drive wheel ca
rrying the Sun pointer, he calculated that there must have once been a second wheel of the same size just in front of it, now lost, driven by the other side of the crown wheel. This would have turned at the same speed as the main drive wheel, but in the opposite direction, so this second wheel must have carried the Sun pointer, around the same way as the Moon.
This wasn’t the end of the gearing. It seemed to Price that the two speeds of rotation achieved so far – corresponding to the movements of the Sun and the Moon in the sky – each fed further back into the machine, into a cluster of gearwheels that were mounted on a bigger turntable. Price was stumped . . . until he had a crazy idea.
When he reconstructed the missing clock in the Tower of the Winds, Price had succeeded where others failed by looking at things from the point of view of the ancient craftsman he was trying to second-guess. The stars above the lit streets of Connecticut aren’t as bright as they would have been in ancient Greece, but now he looked up anyway, watching the ghostly silver crescent of the Moon wax and then wane as the stars completed their graceful arcs behind it. Each fresh moon was like a new life, its cycle by far the most dramatic thing in the night sky. The maker of the device would surely have wanted to capture it.
Calculating the phase of the Moon is basically the same as working out the number of synodic months that have passed. If you start at full Moon, for example, then after each whole number of synodic months that passes, the Moon will be full again. For every half number of synodic months, the Moon will be new, and so on. The number of synodic months in any time period is intimately connected with the number of sidereal months and years, because the phase of the Moon depends not just on its position with respect to Earth, but the position of the Sun.
Imagine the Earth as the tip of the hour hand on a giant clock face in space, with the Sun at the centre. The Earth inches its way around the clock face as the Moon circles around the Earth in turn. At full Moon, all three fall into a straight line with the Earth in the middle, the Sun’s rays shining past us to illuminate the Moon head on as we look at it. If the Moon then completes exactly one rotation around the Earth it will come back into the same position with respect to the background stars – say, from one o’clock to one o’clock. But because the Earth is itself sweeping around the Sun, one orbit of the Moon isn’t enough to bring the three bodies back into line. The Earth is now at two o’clock with respect to the Sun. So the next full moon doesn’t occur until the Moon has travelled the extra twelfth of a circle. In one year, these extra twelfths add up to one extra sidereal month. The relationship holds in general – the number of sidereal months in a particular time period is equal to the number of synodic months that have passed, plus the number of years. In one 19-year period, for example, 235 + 19 = 254.
The Greeks didn’t necessarily think of it in such heliocentric terms, but thanks to the Babylonians and their 19-year cycle, they knew the numerical relationships involved. And just as you can add the number of years and synodic months that pass to get the number of sidereal months, so you can subtract years from sidereal months to get synodic months (for example, 254 - 19 = 235).
Price was looking at a gear train in which two speeds of rotation – representing the speed of the Moon, and in the reverse direction the speed of the Sun – were fed into a cluster of linked gearwheels mounted on a turntable, such that their relative motion had driven the turntable around. Two inputs, one output.
And so the answer came to him. It must be a differential gear – a set-up already familiar to him from the astronomical clocks of Renaissance Europe. Whereas the parallel gear arrangements he had noted so far could multiply and divide rates of rotation according to the ratios of their numbers of teeth, a differential gear could add and subtract.
Differential gears are complicated, a whole new level of gearwork, and to find one in such an old device made the Antikythera mechanism more astounding than ever. If Price was right there was no question of viewing the mechanism as an early stumbling attempt at mathematical gearing, the beginning of a technological line that might have died out as quickly as it began. To come up with a differential gear takes a virtuoso talent, both in mathematics and in practical execution, and it must have been the culmination of generations of experience.
There was already a hint that the differential gear was known in ancient times: a legend that around 2600 BC, the Chinese Yellow Emperor Huang Di had a chariot topped with a wooden figure that always pointed south. A differential gear could in theory achieve this by subtracting the revolutions of one wheel from the other, thus keeping track of any changes in direction. But it’s probably just a story. No surviving text describes a working model until the third century AD, and there’s no description of how it might have worked until the eleventh century.
The first differential gear known in the West – and the first used anywhere for a mathematical purpose – was in the eighteenth century. Its origins aren’t clear, but it was possibly invented by the British watchmaker Joseph Williamson, who wrote in 1720 that he had constructed one for use in a clock, so that as well as showing the time, the mechanism could calculate the varying speed of the Sun through the sky.
The differential gear is an impressive invention because using one to make a calculation involves working out that the way in which its various parts move relative to each other is governed by a precise mathematical relationship. Two wheels driven independently of each other are both connected to a third wheel in such a way that it moves round with a speed that is half the sum of the speeds of the two input wheels.
In the Antikythera fragments Price saw the remains of a triangle of three little wheels, all mounted on a bigger turntable. He had worked out that one of these wheels turned with the speed of the Sun, driven directly from the main shaft, and one of them turned in the opposite direction, with the speed of the Moon. The third wheel in the triangle was a pinion that connected the other two in such a way that as they turned relative to each other they pushed around the turntable on which they were both sitting. In effect, the motion of the Sun through the sky was being subtracted from its lunar equivalent. Multiply the resulting movement of the turntable by two and the machine was calculating the phase of the Moon.
Price followed the gear train on down and concluded that this rate of rotation was then transmitted to the rings of the lower back dial to show the 235 synodic months of the 19-year cycle, with the position of the pointer within each segment corresponding to the Moon’s changing phase. The subsidiary dial would have showed the twelve synodic months of the lunar year.
That just left the upper back dial. He could see that it was a series of concentric rings with a subsidiary dial divided into four, but only part of the gear train leading to it survived. After playing around with the numbers he had, Price guessed that it had shown the months of a four-year cycle, presumably so that the user could keep track of the 365-day calendar as it shifted against the seasons. He didn’t know what all the different rings were for. It didn’t really matter. He had decoded the stunning differential gear, and at last he understood the essence of the Antikythera mechanism. It was, he announced, a ‘calendar computer’. It calculated the movements of the Sun and Moon as seen from Earth, in order to track the days and months of the year and, through the parapegma text, to predict the corresponding positions of the stars.
Price had also uncovered some hints as to where the information encoded within the mechanism came from. It was intriguing, for example, that the 19-year cycle it used had come originally from the Babylonians. They used centuries of observations to come up with equations that could predict the positions of celestial bodies, which they saw as messages that communicated the activities and intentions of the gods. They compiled astronomical tables on clay tablets that contained endless progressions of numbers, each line recording the incremental changes in the position of the Moon, say, and using simple algorithms to extrapolate those movements into the future – like lines of computer code carved in stone.
For
all their precision, the Babylonians showed little interest in how the solar system was actually arranged; to them the night sky was basically a light show. The Greeks, on the other hand, were obsessed with coming up with geometrical models of the heavens. They wanted to explain the celestial movements – what orbited what, and how – not simply to predict them. In fact, they weren’t too bothered about detailed observations at all. The arrangement of the heavens was a philosophical matter and proposed models were judged more on beauty than precise correspondence to reality.
The accurate dials and pointers of the Antikythera mechanism, although certainly Greek, showed a reliance on numerical relations that seemed much closer to the arithmetical spirit of the Babylonians. Whoever had invented the device seemed to have combined the two traditions. They surely had links to the East.
Price wrote up his findings into a 70-page opus called Gears from the Greeks, which he published in June 1974. The Antikythera fragments already represented the oldest geared mechanism – or mechanism of any kind – in existence, and by far the most sophisticated device that has survived from antiquity. But the discovery of a differential gear was breathtaking. It combined astronomical knowledge, abstract mathematical understanding and mechanical skill in a way that would not be matched again until the Renaissance. Yet the Antikythera mechanism was made with an assurance and skill that made it all look easy.
Price remained convinced that this technology had not died out. When Greco-Roman civilisation collapsed in the early centuries AD, much learning, including that of mathematics and astronomy, was transferred first to the Islamic world, and in later centuries back to Europe. Price argued in his paper that a knowledge of the gearing of the Antikythera mechanism had been carried through to safety each time.