A mathematical excursion.

I went for a walk a week or two back around an Iron Age hill fort here in Devon. It was overgrown with large beech trees, but its leaf-littered ramparts and ditches remained imposing, 2000 years after they were built. From the bottom of the ditches to the tops of the ramparts they were 6 metres high in places, perhaps more.

It set me thinking about how much work it must have taken to build them. They’re often regarded as prodigious feats of engineering. Could I make an educated guess at how much work was required?

Firstly, how big was the workforce? The hill forts very often have the traces of numerous roundhouses within their ramparts. A typical roundhouse would have been 30 feet or 9 metres in diameter. That’s an area of 63 square metres. Packed together in a hexagonal grid pattern, each roundhouse would have occupied a hexagon 70 square metres in area. The area of one of the largest hill forts in England, Maiden Castle in Dorset, is 18 hectares, or 180,000 square metres. So you could have fitted over 2500 roundhouses into Maiden Castle. If each roundhouse contained a man and his wife and their children, that’s 5000 adults. So that’s a ballpark figure for the available workforce.

How much work does this workforce need to do? Maiden Castle isn’t circular, but a great many hill forts are almost perfectly circular. The radius of a circle enclosing 18 hectares is about 240 metres. And so the circumference of that circle is 2 times pi times 240, or about 1500 metres.

So 5000 adult men and women would have to build 1500 metres of ramparts and ditches. That’s 0.3 metres of it each. And 0.3 metres is almost exactly one foot.

How do you build a 6 metre high rampart and ditch? Well, you dig out earth from where you want the ditch to be, and you pile it up next to it where you want the rampart to go. Since you are building a 0.3 metre thick slice of the rampart, you have to dig a 0.3 metre thick triangular ditch, 3 metres deep. Assuming that the slope of the rampart is about 40 degrees, the volume of that triangular slice is 3.2 cubic metres. And assuming a soil density of 2000 kg/cubic metre, that’s a mass of 6400 kg. And the centre of mass of this triangle is one third of the way up from its base – 1 metre. So this entire triangular slice has to be lifted 2 metres.

The physical work required to raise a mass m against gravitational acceleration g by a height h is m times g times h, or m.g.h. So the work done in raising 6400 kg up by 2 metres against a gravitational acceleration of 9.81 m/s^2 is 126,000 units of work. These particular units of work, using kilograms and metres and seconds, are called Joules. So that’s 126,000 Joules.

Horses can perform work at the rate of one horsepower, or 745 Joules per second, or 745 watts. Humans can work at about 0.1 hp, or 75 watts. So to perform 126,000 J of work at 75 J/s would take 1680 seconds, or 28 minutes.

So, * Maiden Castle was built in half an hour!* And that’s without the children helping out.

Is that plausible? If you think of a kilo as being a 1 kg bag of sugar, how long do you think it would take you to lift one of those from the floor and put it on a shelf a bit over your head? You could easily pick up one in each hand, and it would take a couple of seconds to do that, no? Even on that estimate, it only takes 6400 seconds, or an hour and a half, to lift 6400 kilos.

Well, it would clearly have taken rather longer than this to build a rampart and ditch. Earth doesn’t come nicely packaged in handy-size paper bags. Earth has to be sliced up with spades. And then it has to be packed into bags or baskets. Hauling a basket filled with earth up 2 metres is just one the tasks that has to be performed. And then the basket has to be emptied. And the soil tamped down.

But even if it’s supposed that it takes ten times that amount of work, digging, filling baskets, emptying baskets, compacting soil, that’s still only 5 hours. If Maiden Castle wasn’t built in half an hour, it may very well have been built in a day. And the next day they added another rampart and ditch. And the day after, another one.

And it may well have been necessary to build them this quickly. The hill forts in Britain are very often within sight of each other. And this would have allowed warning signals to be transmitted very rapidly by signal fires from one fort to the next. You’d get a few days notice of the impending arrival of an invading army. Quick everybody, run up the hill and start digging!

Another odd conclusion that comes out of this analysis is that the bigger the hill fort, and the more people you could house inside it, and who you could call upon to work in its construction, the quicker it was built. It took less time to build big hill forts than little ones. It would have taken one man many weeks to build himself a one-man hill fort with a single roundhouse and a 6 metre deep rampart and ditch around it. And just a day or two to build Maiden Castle.

In fact, we can work it out. A hill fort with a single 9 metre diameter roundhouse in it would need a 3 metre high rampart around it, and a 3 metre deep ditch outside that. A single man would have to dig out a circular ditch with a radius of about 15 metres. That would be about 850 cubic metres, and 1,700,000 kilograms of earth. That’s 265 times as much work as a single individual had to do to build Maiden Castle with 5000 helpers. So if they took just one day to build Maiden Castle, he’d have taken 265 days, or 37 weeks, to build his one man hill fort.

So, by the time the invading army showed up a week after the warning fires had been seen, the people in Maiden Castle would be safely behind several ramparts and ditches, and our single man would have only dug out a couple of metres of his perimeter ditch. He would have been captured and killed or enslaved. Perhaps that explains why there aren’t many mini hill forts in Britain.

So, was it really such a prodigious feat to have built huge earthworks like Maiden Castle? It wasn’t a physically prodigious feat. We’ve just found out that it could have been built in a day or two. The prodigious feat was to organise people to work supremely hard together in concert, in their assigned places all round the perimeter. They must all have been trained beforehand what they had to do, where they had to go, and what tools to bring with them.

I think you may have overlooked a point when calculating the work required to move things from low places to high places, particularly in the ‘bags of sugar’ example. To move these bags of sugar you also have to move yourself, an average of perhaps half the height you are moving the load. For a person of mass 50kg, over 90% of the work consists of moving yourself up and down.

I think you may have overlooked a point when calculating the work required to move things from low places to high places, particularly in the ‘bags of sugar’ example. To move these bags of sugar you also have to move yourself, an average of perhaps half the height you are moving the load. For a person of mass 50kg, over 90% of the work consists of moving yourself up and down.

Quite so. And it hadn’t entirely escaped my thinking.

Bending down to pick something off the floor entails lowering one’s entire head and torso, and then raising it again.

It would perhaps be much more energy efficient to crouch on the floor, and use just one’s arms to toss the sugar bags up to the required height. People can become extraordinarily adept at doing this. I once saw a piece of film on which somebody was carrying bricks piled on top of his head, and was tossing one brick after another up to form a growing pile of them.

In a digging a rampart, it would also be inefficient to dig out a basket full of soil from the bottom of the ditch, and then carry it bodily up to the top of the rampart to empty there – because you’d be raising your whole body several metres each time.

It would probably be far more efficient to have a pulley system that carried the baskets up the ramp. This pulley system would largely require just the use of the arms to pull on a rope, or to turn a wheel.

Of course there’s also the weight of baskets or bags to take into account.

A further thing to take into account is that the human frame acts to some great extent like a spring, absorbing energy and then re-releasing it. e.g. when walking, each time you put your foot down, the leg flexes and absorbs energy, which it re-releases when you use that leg to propel yourself upwards and forwards.

I think it’s perfectly possible that 90% of the work could entail raising and lowering the body, but I suspect that it would be a great deal less using the most efficient method.

Frank

Quite so. And it hadn’t entirely escaped my thinking.

Bending down to pick something off the floor entails lowering one’s entire head and torso, and then raising it again.

It would perhaps be much more energy efficient to crouch on the floor, and use just one’s arms to toss the sugar bags up to the required height. People can become extraordinarily adept at doing this. I once saw a piece of film on which somebody was carrying bricks piled on top of his head, and was tossing one brick after another up to form a growing pile of them.

In a digging a rampart, it would also be inefficient to dig out a basket full of soil from the bottom of the ditch, and then carry it bodily up to the top of the rampart to empty there – because you’d be raising your whole body several metres each time.

It would probably be far more efficient to have a pulley system that carried the baskets up the ramp. This pulley system would largely require just the use of the arms to pull on a rope, or to turn a wheel.

Of course there’s also the weight of baskets or bags to take into account.

A further thing to take into account is that the human frame acts to some great extent like a spring, absorbing energy and then re-releasing it. e.g. when walking, each time you put your foot down, the leg flexes and absorbs energy, which it re-releases when you use that leg to propel yourself upwards and forwards.

I think it’s perfectly possible that 90% of the work could entail raising and lowering the body, but I suspect that it would be a great deal less using the most efficient method.

Frank

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