What Would You Do
If You Were Archimedes?

You run naked through the streets of Syracuse shouting Eureka — and what burns into history is not the nakedness or the shout, but the proof: every submerged object experiences an upward force equal to the weight of the fluid it displaces, universally, without exception. Archimedes' Principle states: a body submerged in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. This explains why ships float, why helium balloons rise, why submarines can submerge and surface, why anything floats or sinks. It is one of the fundamental principles of fluid mechanics and was not improved upon until the 19th century. The story of the bath is recorded by Vitruvius, writing about 200 years after Archimedes' death; the Eureka legend may be embellished. But the principle itself is in Archimedes' treatise On Floating Bodies, which survives. Whether or not he ran naked through the streets, he did derive the mathematics of floating and sinking with complete rigor — it was not matched until Archimedes' works were rediscovered and understood in the Renaissance.

You spend years building machines that hold off Rome — and the man himself considered all of it beneath mention. His surviving writings contain no treatise on any machine he built. Plutarch, writing about Archimedes, records that Archimedes considered his mechanical inventions beneath him — the work of a craftsman, not a philosopher — and that he wrote no treatise about them, considering them diversions produced at the request of King Hiero. Archimedes' surviving letters emphasize the mathematical results: the calculation of pi, the relationship between sphere and cylinder volumes (his favorite discovery, which he asked to be carved on his tomb), the method of exhaustion. Whether Plutarch was accurately describing Archimedes' attitudes or projecting Greek philosophical snobbery about practical work is debated. But his surviving mathematical works are clearly the product of someone who cared most about pure mathematics — the engineering works survive mainly in historical descriptions, not in Archimedes' own treatises.

Monks scraped off the greatest mathematics of antiquity to reuse the parchment for prayers — and for a thousand years, no one knew what they had erased. Modern X-ray imaging found it anyway, in a prayer book sold at auction for $2 million. In 1906, a medieval prayer book was discovered in a monastery. The monks had scraped off earlier text to reuse the parchment. Modern imaging technology (X-ray fluorescence, ultraviolet light) revealed the scraped-off text: a work by Archimedes, the Method of Mechanical Theorems, previously lost. The Method reveals something extraordinary: Archimedes used infinitesimal reasoning — thinking of areas as sums of infinitely many infinitely thin slices — to discover results, then proved them rigorously using the method of exhaustion. He essentially used integral calculus to find results, then translated the results into rigorous Euclidean proofs. The infinitesimal method is calculus. Newton and Leibniz rediscovered it 1,900 years later. The Archimedes Palimpsest (as the prayer book is now called) is at the Walters Art Museum in Baltimore and was sold at auction in 1998 for $2 million.

Rome sends 60 warships and land forces to take a single city — and one man makes them flee from ropes and shadows. Roman soldiers who had conquered the Mediterranean turned and ran when they saw a rope thrown over the wall. The Siege of Syracuse lasted from 214 to 212 BC. Marcellus arrived with 60 quinqueremes (large warships, each rowed by 300 men) plus land forces. The standard assumption was that a naval assault on a coastal city walls would succeed quickly. Archimedes' machines disrupted every assault. Plutarch writes that Roman soldiers, seeing a rope thrown over the wall or a piece of timber appearing, would immediately flee — convinced that Archimedes was about to unleash something terrible. The Roman general Marcellus reportedly began making jokes about it to cover his frustration: "Archimedes uses our ships to ladle sea-water into his wine-cups" and called Archimedes "this geometrical Briareus." The city finally fell not to the Roman assault but to treachery — Syracusans who opened the gates. Marcellus had given orders that Archimedes was not to be harmed.

You designed it 2,500 years ago, and it is still pumping water in sewage treatment plants today. Not as a museum piece — as working infrastructure on every inhabited continent. The Archimedes Screw remains in active use in the 21st century. It appears in: irrigation systems across the developing world, where it can be human-powered; sewage treatment plants, where it lifts wastewater from one stage to another; hydroelectric generation, where modern "Archimedean turbines" generate electricity from flowing water (the screw used in reverse); fish-friendly water pumping systems, since fish can pass through the screw undamaged (unlike conventional pumps); grain handling equipment on farms. The design has been refined over 2,500 years but the principle is unchanged. The Museum of the History of Science in Florence has a Roman example from around the 1st century BC. Models of the original are sold as educational toys. The most durable technology human civilization has produced is often its oldest, because the oldest technology has had the longest time to prove that it works.

The greatest mathematician of antiquity asks for one thing on his tombstone — a sphere inside a cylinder — and 137 years later a Roman politician has to rescue the tomb from weeds because the Greeks let it fall into ruin. The tomb has been lost again ever since. Cicero, visiting Sicily as quaestor in 75 BC, writes that he found Archimedes' tomb — neglected and overgrown — near one of the city gates of Syracuse. He identified it by the sphere and cylinder carved on the tombstone, as Archimedes had requested. This was 137 years after Archimedes' death. Cicero cleaned it up and had the inscription read aloud. He writes that it was evidence that Greeks were better at mathematics than Romans, since Romans had let the tomb fall into ruin while Cicero, a Roman, had to rescue it. The tomb's location was lost again after Cicero's time and has never been found. The sphere-cylinder ratio (2:3 for both volume and surface area) is now proved in any geometry textbook, typically in a paragraph. Archimedes required a sophisticated multi-step argument to derive it for the first time.

MIT students, a Greek naval team, and the US Navy have all tried to burn a ship with mirrors. Two of them got the wood to smolder. The debate started in 250 BC and is still technically open. The Archimedes heat ray has been tested multiple times in modern settings. The 2009 MIT experiment used 127 polished bronze mirrors (approximately what ancient craftsmen could produce) focused on a ship at 100 feet distance on a sunny day; the wood caught fire after about 10 minutes. However, the MythBusters TV show attempted the same experiment under more realistic conditions (moving ship, cloudy sky, time pressure) and rated it "busted." A Greek team successfully burned a wooden boat using a similar array of mirrors in 1973. The consensus is: the physics works, but the tactical conditions required (stationary target, bright sun, enough time) would be unusual in an active naval battle. The most plausible version is that Archimedes used mirrors to blind and confuse Roman sailors — effective as a weapon even if fire was difficult to achieve. Plutarch mentions only the catapults and cranes, not fire.

He was so far ahead that no mathematician matched his work for 1,900 years — not because no one was trying, but because the level of his integration and approximation simply wasn't reached again until the 17th century. The circles he was drawing when the soldier found him are gone. Marcellus was reportedly devastated by Archimedes' death and had him given a proper burial. He later met Archimedes' relatives, according to Plutarch, and showed them respect. The mathematical achievements Archimedes made were so far ahead of his time that they sat largely unmatched for 1,900 years — not because no one was working on mathematics, but because the level of his work on integration, spherical geometry, and approximation wasn't reached again until the 17th century. Specifically: his method of exhaustion used limits and infinitesimals in the same way Newton's calculus would. His proof of the area of a parabola is essentially integration. His bounds on π (between 223/71 and 22/7) weren't improved until 500 AD by Chinese mathematician Zu Chongzhi, who reached 7 decimal places. The circles he drew in the sand — whatever problem he was working on when the soldier found him — will never be known. That's the part the legend can't recover.