We all know the story of Archimedes flooding his bathroom, leaping out of his bath, and dashing stark naked through the streets shouting “Eureka!” in excitement at his discovery of the Archimedes Principle of water displacement.

But exactly which city’s streets did the world’s first flasher actually dash through?

For years I thought it was somewhere in Greece, since my maths teacher at school told me he was Greek. Actually, it was the citizens of Syracuse in Sicily who were privileged to to see the bare wedding tackle of the world’s cleverest man.

And do you know he was messing about in the bath in order to solve a crime? Forget Inspector Montalbano; Archimedes was Sicily’s first famous detective.

Archimedes was born in 287 B.C. in the coastal city of Syracuse. The ancient Romans and Greeks considered Archimedes the greatest mathematician who had ever lived. He also made great innovations in astronomy, geometry, mechanics, hydrostatics and optics and he made far more inventions – certainly far more which actually worked – than Leonardo Da Vinci.

Archimedes’ genius for engineering lay in his profound understanding of the six simple machines which are, to this day, the components of every functioning piece of machinery in existence. These machines are the lever, the wheel and axle, rotating gears, the rope and pulley, the screw, and the ramp or wedge. He figured out mathematical formulae to calculate how big and strong each one needs to be in order to do the required job. It was the lack of this mathematical knowledge that meant so many of Leonardo Da Vinci’s inventions, which look brilliant on paper, don’t actually work in real life.

Archimedes invented a system of ropes and pulleys with multiple gears, and used them to tow a fully manned and loaded galley up the beach all by himself, to show the king of Syracuse how gears can drastically multiply the strength of one person.

He also worked out the mathematical relationship between the power needed to operate a lever, its length, and the positioning of the fulcrum. As soon as a North African contemporary of Archimedes, named Eratosthenes, had worked out the size of the earth, Archimedes calculated the size of lever needed to move it, and told the king: “All I need is somewhere to stand, and I can move the earth.”

Archimedes’ best known use of the six simple machines is the modification of the screw which came to be named after him, the Archimedes Screw. It is simply a spiral tube turned with a lever, which can force water to flow upwards against gravity. He originally invented this to pump water out of a large ship, and 2,000 years later one was used to pump water out of the world’s first ocean-going steamship in 1839, the SS Archimedes. Archimedes screws are still used in Sicily in the salt works, to pump sea water from one pool to another as it progressively evaporates until it leaves nothing but pure edible salt crystals.

Archimedes created the mathematical formulae for calculating the volume of many solids, including those with curved surfaces. To do this he invented integral calculus. Newton and Leibniz argued bitterly over who had invented calculus first in the 17th century, but the reading by X-ray of the Archimedes palimpsest in the 1990s proves Archimedes beat them both by over 2,000 years.

In case you were not really listening in maths lessons at school, you can take it from me that it is impossible to design anything from a road to a car to a house or anything else without relying on this branch of mathematics. Archimedes designed a crane which was used to lift enemy ships out of the sea and hoist them up into the air – obviously all the enemy sailors would drop out and land in the sea – then the ship was dropped back into the water at right angle to make it sink. This piece of apparatus has been reconstructed in modern times and it worked terrifyingly well.

Archimedes also studied the movements of the planets and created a working planetarium which showed their relative positions and movements around the sun. Greek astronomer Aristarchus of Samos (310BC – 230BC) first showed that the earth moves around the sun, and this was well know throughout Sicily in Archimedes’ time. The Catholic church did all it could to quash this fact, and it was eventually “forgotten” and revived by Copernicus about 1,800 years later.

Archimedes’ genius for engineering and general problem-solving made him the king’s “Go-To” man for all problems. The best known story about Archimedes in modern times is the tale of how he solved a crime scientifically, CSI-style. King Hiero had given a large amount of gold to a goldsmith, to make a crown for a temple statue. When the crown came back it was very beautiful, but looked a bit of a funny colour. King Hiero suspected the goldsmith had kept some of the the gold for himself, and replaced it with the same weight of silver.

Could Archimedes prove whether he had really done this, King Hiero wanted to know? The crown was very nice, he emphasised (even if it did look a bit silvery) so Archimedes had to figure it out without melting it down.

Silver and gold have different densities, so Archimedes needed to know both the weight and the volume of the crown. The weight was easy, but how to figure out the volume of something so twiddly? He realised that if he filled a container to the brim with water and then put the crown in it, the exact volume of the crown would be displaced and could be measured.

Legend has it that he tested this idea by filling his bath to the brim and getting into it, and was so excited he ran outside stark naked shouting “Eureka! I’ve got it!” In reality he would not have shouted Eureka, as the ancient Greek for for “I’ve got it” was “Heureka”. Since the letter H was a tiny weeny letter in the Greek alphabet, just the size of an apostrophe, people always seem to forget it.

The test proved that the goldsmith had indeed mixed silver into the crown, and he was duly punished.

This primary-school version of events, known as the Archimedes Principle, is really a very simplified explanation of what Archimedes actually did. In Archimedes’ own book called “On Floating Bodies”, and in descriptions of the story written soon after his death, the real method of calculation relied on complex hydrostatics. A body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Using this principle, Archimedes compared the density of the golden crown with that of pure gold by balancing the crown on a scale with a gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly because, although they were the same weight, they were different volumes and therefore displacing different amounts of water. This technique, whilst less fun than flooding the bathroom and than dashing about naked shrieking in Ancient Greek, is far more accurate.

Whilst this is our favourite story about Archimedes, the ancient Greeks loved a different one: the tale of how he made the attacking Roman ships burst into flames by shining “burning lenses” at them. Mirrors angled inwards, like the curve of a satellite dish, were already known in the ancient world as a means of concentrating sunlight and starting fires. The Romans had one in the Temple of Vesta, the Goddes of fire, to help make sure the sacred fire never went out accidentally. The sacred fire which the Ancient Greeks kept burning at the Olympic Games had to be pure and sacred, which meant it must be ignited by the sun alone and not by wooden kindling, so the Greeks used “burning lenses” to light it, which were essentially very powerful lenses made from glass bottles of water. The Greeks also used these burning lenses to cauterise wounds.

Archimedes thought on a completely different scale. He set up a fancy array of “burning lenses” which made the approaching Roman ships burst into flames. It worked so well that practically the whole fleet was incinerated and the technique was supposedly used again by Proclus defending Constantinople in the 6^th century. Whilst the American TV show “MythBusters” failed to recreate Archimedes’ feat of optics, it has been sucessfully replicated by the Comte de Buffon in 1740, Greek scientist Ioannis Sakkas in 1973, and students at MIT in 2005. The MythBusters team used metal mirrors and a plain plywood ship, even though the ancient sources are quite clear that Archimedes used “burning lenses” which would have been glass bottles of water, and ancient Roman ships were always waterproofed with tar which made them far more flammable than plain modern plywood (which is normally treated with fire retardant before it leaves the factory). It is hard to imagine that, among all Archimedes’ achievements, the one his contemporaries most admired never actually happened.

Despite his success in burning the Roman ships, the Romans still managed to land and sack Syracuse. Archimedes was apparently sorting out some of his most precious scientific models and instruments at the time, and told the Roman soldier who tried to arrest him “Don’t disturb my circles.” The soldier lost patience and killed Archimedes, to the great anger of his commander Marcellus.

Archimedes’ tomb was adorned with a metal sphere set inside a cylinder, touching its sides and the top and bottom, representing his personal favourite among all his discoveries. He had proven mathematically that the volume of the sphere will always be two-thirds that of the cylinder, and had asked his friends to commemorate this on his tomb.

When Cicero, the Roman orator and politician, was governor of the Roman Colony of Sicily, he went hunting for Archimedes’ tomb. Archimedes was still regarded as the greatest scientist of all time yet Cicero found his tomb abandoned and overgrown with weeds. He cleaned it up so it was again a suitably respected memorial to a Sicilian who remains, over two thousand years later, one of the greatest scientists of all time.