Legend has it that there was once a man, an old Greek to be exact, who stopped the most powerful army in the world for three years: that of Rome itself. They say that he mounted some curved mirrors on the walls of his city. When the Roman ships besieging the city approached, the ancient Greek concentrated the sun’s rays on his candles and made them burn. It is also said that when the Romans saw ropes and logs hoisted over the city walls, they weighed anchor and sailed away under full sail. The name was Archimedes and he was from Sycarusa, a Greek city on the island of Sicily.

Unlike all the Greek scientists who had preceded him, he was an eminently practical man. **He devoted himself to applying science to everyday life** . For Pythagoras, Thales or Euclid, mathematics was conceived as an abstract entity, something that served to study the ultimate order of the universe , without connection to daily life.

Archimedes had an imagination like no other. For example, his contemporaries thought that the number of grains of sand in the sea was too great to count. **Archimedes not only contradicted them but invented a way to do it** . And not only to count the grains on the beaches, but also the number of grains that would be needed to cover the Earth and to fill the universe. According to Archimedes, the number of grains of sand needed to fill a sphere the size the Greeks believed the universe to be is 10 ^{80} , that is, a one followed by 80 zeros. And, look what a coincidence, this is the number of protons and neutrons that according to cosmologists exist in the universe.

In the year 213 BC Rome laid siege to the city of Siragusa. The old scientist who is said to have held off the Roman army was meditating on the beach when the city was finally taken in 211 BC The soldiers had been ordered not to kill the old man. During the sack of the city Archimedes was in the sand on the beach trying to solve a geometry problem. While Archimedes was doing his calculations in the sand, a soldier ordered him to go meet Marcellus, the Roman general who had taken the city, but he refused, saying that he had to finish working on the problem : “noli turbare circos meos ” (do not spoil my circles), replied . In response, the soldier killed him.

It is said that Archimedes had arranged that a sphere inscribed in his cylinder should be placed in his tomb, since he was very proud of his calculation of the relationship between the volumes of both solids: the volume of the sphere is two thirds that of its cylinder. Today this result is easy to calculate using integrals but Archimedes obtained it with a lot of effort.

Over time, the location of Archimedes’ tomb was lost, but centuries later, Cicero, in his *Tusculan Disputations* (an attempt to popularize Greek philosophy among the Romans), relates how he helped the Syracusans find the lost tomb: “While I was quaestor, I succeeded in discovering his tomb, unknown to the Syracusans, and whose existence they denied, that it **was completely surrounded and covered with brambles and bushes** . I kept in my memory some brief senarios [system in which numbers are represented using only 0 to 5], […] which indicated that a sphere with a cylinder had been placed on top of the tomb. While I was looking around the area, I noticed a small column that barely rose above the bushes, in which there was the figure of a sphere and a cylinder.

Despite Cicero’s best efforts, the tomb was lost again, but not its memory. Something that is celebrated every four years, when the Fields Medal is awarded, the highest award of the international mathematical community: in it is his effigy and the cylinder and sphere.

But what Archimedes is truly famous for is the principle that bears his name. It goes like this: every body immersed in a fluid experiences a vertical and upward thrust equal to the weight of the volume of liquid displaced. Many of us learned it by heart, but what does it mean?

The story of Archimedes’ principle begins when King Hieron asked a goldsmith to make him a crown. **Hieron gave him the gold, but when the goldsmith handed him the crown, the king was suspicious** . He believed that he had deceived him and that he had replaced some of the gold with copper or silver. Hieron, intrigued, asked Archimedes to find out if the crown was pure gold… without damaging it, of course.

Archimedes was intrigued. He knew that copper and silver were lighter than gold. That meant that a kilo of silver occupies more volume than one of gold. Since Archimedes knew the amount of gold Hiero had delivered to his goldsmith, the solution was obvious. If the crown were made of pure gold, it would occupy a certain volume. If the goldsmith was clever and had replaced some of the gold with an equivalent weight of silver or copper, then the volume of the crown would be greater. **The problem was that there was no way to find out the volume of the crown without melting it down** .

Archimedes was mulling over the problem without success. But one day, while he was in the public toilets, he found the solution. Getting into a bathtub, he saw how the water overflowed. All of us have experienced the same thing, but it is in these small details that the brain of a genius is seen to work. Archimedes jumped as if propelled by a spring. I had just solved the problem. So excited was he that he ran naked into the street shouting “Eureka! Eureka!”

What had Archimedes realized? From something very simple: his body had displaced the water out of the bathtub. How much? Here was the crux of the matter. Exactly the volume of his body! This is what happens: our body has a certain volume and when we get into the bathtub we occupy that same volume. And now comes the magisterial point of Archimedean reasoning. **To occupy that space in the bathtub we have had to displace a volume of water equal to ours** . And if the bathtub is filled to the top, the water spills out.

Archimedes, now calmer, filled a container with water and placed the crown in it. Then he measured the volume of water that had overflowed: that was its volume. Then he took a piece of gold with the same weight as that given by King Hieron to the goldsmith and put it in the water. If that were the amount of gold used in the crown, it would displace the same volume of water. Bad business. Archimedes discovered that the displaced volume was smaller: the goldsmith had wanted to cheat the king. Archimedes was rewarded by Hiero and the goldsmith as well. The wise man received congratulations and the craftsman lost his head.

**References:**

Torija, R. (2007) Archimedes. Around the circle, *Editorial Nivola*