# Eureka moments in Syracuse

In a charming Sicilian museum you can test the theories of antiquity’s greatest mathematician

On a quiet street in the old Sicilian city of Syracuse, hidden away from prominent landmarks such as the cathedral, the Fountain of Diana, and the Temple of Apollo, sits a remarkable little museum. Nestling in a string of buildings on the Via Vincenzo Mirabella is a large old door giving access to a desk presided over by a charming Italian-American lady. For a small fee she allows you to enter a couple of rooms crammed with interactive exhibits, mainly made of wood and cloth, celebrating the mechanical inventions and discoveries of Leonardo da Vinci and Archimedes.

Leonardo, that great Renaissance polymath and artist, came from Florence, so has no particular connection to Syracuse, but Archimedes is a different matter. A resident of the city during the third century BC, he died in 212 BC after helping to defend it during a two-year siege by the Romans. The Roman general Marcus Claudius Marcellus had commanded that no one was to harm Archimedes, one of the great intellects of the ancient world and the genius behind many of the city’s remarkable defences, such as mighty catapults that could rain fire and fury on the Roman ships, and grappling hooks connected to pulley systems that could lift ships partially out of the water, causing them to sink. Yet despite his commander’s injunction an unnamed Roman soldier killed this great scholar, then in his mid-seventies.

Why? Plutarch gives two stories. One is that Marcellus had commanded the soldier to bring Archimedes to him, and the venerable geometer declined, being engaged on some geometrical problem and saying, “Do not disturb my circles.”

An alternative story is that Archimedes was carrying mathematical instruments, which aroused the soldier’s suspicion. It’s one of those historical mysteries that may never be fully answered, but what is in no doubt is that Archimedes was one of the great, perhaps the greatest, mathematician and engineer of antiquity.

He is known in popular imagination for his Eureka story. King Hieron II of Syracuse had supplied pure gold to make a votive crown, and Archimedes was asked to determine if the goldsmith had substituted silver for some of the gold. A striking idea for testing this came to him suddenly in his bath—a feature common to many thinkers who wrestle with a problem for a long time and see the solution in a flash of insight while doing something quite different. Leaping from the bath he shouted “*Eureka!*” (Greek for “I have found it”), though quite how is not recorded and the story does not appear in Archimedes’s known works. Obviously, if you dumped the crown in a basin of water and measured how much water it displaced you would know the volume of the crown, and by weighing you could then determine its

density, but this would require very accurate measurements and would hardly inspire a eureka moment. More likely, he used a principle explained in his book *On Floating Bodies*, where a body experiences a buoyant force equal to the weight of the fluid it displaces. This suggests the following experiment. Balance the crown on a pair of scales against pure gold of the same weight, and submerge the whole thing in water. If the balance tilts, then the crown and the gold displace different amounts of water, so they have different volumes, hence different densities, and the crown is not pure gold. *Eureka!*

An eminently practical solution, but as Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life,” and in mathematics Archimedes’s methods and insights were brilliant. Following the standard works on geometry embodied by Euclid’s *Elements* he extended our knowledge to curves and curved surfaces. Using what we would now think of as the techniques of Integral Calculus, he proved that the area of a circle was exactly equal to its radius times half the circumference. When Cicero visited his tomb, he saw it was surmounted by a sphere and cylinder, celebrating Archimedes’s much-loved proof that the surface area of a sphere is the same as the curved surface of a cylinder having the same diameter and height, and moreover that the volume of the sphere is two-thirds the volume of the cylinder.

Theoretically beautiful facts, but some of his results on curves could also have highly practical uses. For example, Archimedes showed that when parallel lines meet a parabola they are reflected so that they meet in a common point. A parabola will therefore reflect light from the sun to a single focal point, and Archimedes is said to have used flat mirrors acting collectively in a parabolic arc to focus the sun’s light on a single enemy ship and cause a fire.

The parabolic mirror is one of the exhibits in the charming little museum in Syracuse, as is the principle of water displacement mentioned in connection with Hieron’s golden crown, along with the system of pulleys used so effectively in attacking the Roman ships. The principle of the lever, which Archimedes explained in his work *On The Equilibrium of Planes,* is well represented by a wonderful see-saw where you sit in a chair balanced by a heavy weight at the other end. You can then raise or lower yourself by using your feet to change the distance from chair to fulcrum. These exhibits are robustly built and interactive, unlike some museums where you cannot even lean on a glass case without being told not to touch, and my favourite is a long, flared, open box with an axle holding a heavy wheel placed at the higher and wider end. You would expect the wheel to roll down to the lower end, but on the contrary if you place it at the lower end it rolls uphill.

The reason is that the heavy wheel, shaped like two squat cones stuck together at their bases, has a radius that diminishes as you move outwards along the axle. At the wide end of the box the axle rests on the sides, but at the narrow end the sides of the wheel rest on the sides of the box, holding the axle higher. So although the sides of the box slant down towards the narrow end, the wheel rolls towards the wider end. As gravity draws the centre of mass downwards, the axle moves uphill. Magic, almost, but a great illustration of an important principle, like everything else in this Aladdin’s cave.