For two years the Roman army sat outside Syracuse, and for most of that time it lost. Not to soldiers. To one old man’s machines.

The historians who wrote it down describe cranes that swung over the sea walls and dropped stones heavy enough to sink a ship, and an iron hand on a chain that hooked the bow of a galley, lifted it out of the water and let it fall stern-first. Catapults were sized for every range, so that wherever a Roman ship stood, something was already aimed at it. The Roman commander, Marcellus, reportedly told his own engineers that they were at war with a geometrical giant who used their ships as cups to ladle the sea.

The old man was Archimedes, about seventy-five years old by then, and by every account he considered all of this beneath him. Machines were what you built when a king asked. What he loved was geometry.

Plutarch, who collected the stories three centuries later, says Archimedes forgot to eat when he was working, that he drew figures in the ashes of the fireplace, and that after a bath, while being rubbed with oil, he would trace diagrams with his finger on his own oiled skin. It is the earliest recorded case of a touchscreen, and it was not a good one.

Armies that lose to engineering do what armies always do: they wait. In 212 BC the Romans got over a weakly guarded stretch of wall during a festival, and the city fell. Marcellus gave orders that Archimedes was to be taken alive.

Archimedes, meanwhile, was drawing in the sand.

“Noli turbare circulos meos.”
Do not disturb my circles.— attributed to Archimedes, to the soldier who had just walked into his diagram

A soldier came into the courtyard and told him to come to Marcellus. Archimedes would not get up until he had finished his problem. The soldier killed him. Marcellus is said to have been deeply grieved, to have turned away from the man who did it, and to have looked after Archimedes’ relatives. That did not bring back the problem.

I have liked this story for as long as I have been able to read things. So, apparently, has every engineer who ever had a drawing torn off their desk five minutes before it was finished. But this column has one rule — the truth before the comfortable version — so here is the first hard truth.

He probably never said it

Plutarch, our best source, does not give one account of the death. He gives three. In the first, the soldier orders Archimedes to come and is refused. In the second, the soldier comes to kill him and Archimedes begs for time to finish. In the third, Archimedes is carrying his instruments to Marcellus — sundials, spheres, devices for measuring angles — and soldiers who think the box is full of gold kill him for it. None of the three has him say anything about circles.

The line first appears in a Roman collection of moral anecdotes written more than two centuries after the event, and even there it is not the famous one: noli, obsecro, istum disturbare — “please, I beg you, don’t disturb this”, the “this” being the dust he was drawing in. A Byzantine writer, more than thirteen centuries later, has him say “stand away from my diagram.” The neat Latin sentence we quote today is a later polish of that.

In other words, the most famous sentence in the history of mathematics has been through more revisions than any drawing set I have produced, and nobody drew a single revision cloud.

What was he drawing?

Nobody knows. But we know what he cared about, so let me show you one strong candidate — and you can draw it yourself.

Archimedes wanted to know how long the edge of a circle is compared to its width — the number we now call π. He had no decimal notation, no algebra, no calculator. What he had was an idea: trap the circle between two polygons, one drawn inside it and one drawn around it. The circle’s perimeter must lie between theirs. Then double the number of sides, and double again. The trap closes.

At 96 sides Archimedes wrote down his result: π is greater than 3 10/71 and less than 3 1/7. That is 3.1408 < π < 3.1429, about two and a half thousand years ago, with a stick.

Now the part that every structural engineer should frame and hang above their desk. If you compute the 96-sided polygons exactly, the true bounds are 3.14103 and 3.14271. Archimedes’ bounds are slightly wider than that. Why? Because every time he approximated a square root, he rounded it in the safe direction: the lower bound always a little down, the upper bound always a little up. He gave away a little precision so that the answer could never be wrong.

That is a safety factor. It is the same idea as taking characteristic strength a little below the average and design load a little above it. Two thousand two hundred years before the first steel code, the man in the sand already knew the rule: you may be imprecise, but you may not be unsafe.

The first book on statics

If π does not move you, try this. Archimedes wrote a short treatise called On the Equilibrium of Planes. It starts from a handful of postulates — equal weights at equal distances balance; if you add weight to one side, that side goes down — and proves from them the law of the lever and where the centre of gravity of a triangle, a parallelogram and a parabolic segment lies.

That is statics. Not a rule of thumb from a workshop, but a chain of proofs. Every calculation report I produce has a page early on that checks one thing: do the support reactions add up to the applied loads? It is the most boring page in the report and the one I would never remove. Archimedes would recognise it instantly. He would probably ask why it took us so long to automate it.

There is also the line everybody knows: “Give me a place to stand and I will move the Earth.” It comes to us through a mathematician who lived five hundred years later, so treat it the way you now treat the circles. But notice what it really is. It is not a boast. It is a correct statement about lever arms, with an engineer’s honest footnote attached: the support is the hard part. Every foundation designer I have worked with agrees.

The soldier is not always the villain

Engineers love this story because we all cast ourselves as Archimedes. And we all have a Roman soldier. The client who needs it by Friday. The contractor who moved a column 400 mm “because it was in the way, it’s fine.” The phone that rings in the middle of a stability check. In my case the soldier usually arrives as a revised model file, at 17:55, while the analysis from the previous revision is still running.

But read the story once more, slowly. The walls had been breached. There was fighting in the streets. The city he had defended for two years was being looted around him. And the greatest mind of the age was looking at the ground.

Focus is a virtue right up to the moment it becomes a failure to notice that the city has fallen.

The skill is not “never look up.” The skill is telling the two kinds of interruption apart. A request to change the font in the title block is a soldier: let him wait. A revision that moves a column line is the city falling: your beautiful circles are now about a building that no longer exists, and finishing them is not dedication, it is waste. Most of the engineering disasters I have read about did not happen because someone was interrupted. They happened because someone was not.

So here are three rules I actually work by. None of them is ancient, but all of them are in the story.

1. Draw in something that survives foot traffic. The diagram died in the courtyard. The mathematics did not, because Archimedes wrote it out and sent it to colleagues in letters. A calculation that lives only in your head dies with your concentration. Write it down so that someone else can check it.
2. Make the work interruptible. I can be stopped in the middle of an analysis. What makes that survivable is that every step writes its result to a file before the next one starts. If a soldier steps on step four, I lose step four — not the whole siege.
3. Look up on a schedule. Before you start a long calculation, and again before you sign it, ask one question: is the input still the building that will be built? It takes thirty seconds. It is the difference between Archimedes and a well-informed Archimedes.

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The grave, the scribe and the forger

About 137 years after the siege, a Roman official named Cicero was serving in Sicily and went looking for Archimedes’ tomb. The people of Syracuse told him there was no such thing. Cicero searched anyway, and outside one of the city gates, buried in brambles, he found a small column carrying the figure of a sphere inside a cylinder. Archimedes had asked for that figure on his grave: he had proved that the sphere has exactly two-thirds of the cylinder’s volume, and two-thirds of its surface, and he considered it the best thing he ever did.

Cicero had the brambles cut back and noted, with visible satisfaction, that the most famous city of the Greek world would have forgotten its greatest citizen if a man from a small Italian town had not shown them where he was buried. So, to summarise the project lifecycle: one Roman killed him, another Roman found him. Nobody can say the Romans did not do their own maintenance.

The circles had more to survive. In 1229 a scribe needed parchment, took an old copy of Archimedes’ works, scraped the text off, cut the pages, and wrote a book of prayers over them. In the twentieth century somebody painted fake medieval pictures over a few of the pages, apparently to raise the price. In 1998 the book was sold at auction for two million dollars, and over the following decade scientists used X-rays and light in many wavelengths to read what was underneath.

What they recovered included a text that exists nowhere else: The Method, a letter in which Archimedes explains how he found his results before he proved them. He imagined the shapes sliced into thin strips and hung on a balance, and weighed them. The sphere-in-cylinder on his tomb was discovered with a lever. The greatest mathematician of antiquity was, when nobody was looking, thinking like an engineer.

Stepped on by a soldier, scraped off by a scribe, painted over by a forger — and still not erased.

So draw your circles. Write them down. Look up now and then, and learn to tell a soldier from a falling city.

Do not disturb my circles. But if you must — I keep a copy.