Archimedes: the law of the lever and the principle of buoyancy

The long handle of a jack when changing a car wheel. The swing pole of a well. A ship weighing thousands of tonnes anchored at Karachi port. Behind all of them are two simple principles, proved about 2,200 years ago by a mathematician of Syracuse in Sicily. His name was Archimedes. One principle tells how a small force can lift a heavy load. The other tells why water pushes every submerged object upwards.

The scholar of Syracuse

According to the MacTutor Archive at the University of St Andrews, Archimedes was born in Syracuse in 287 BC and died there in 212 BC. His father, Phidias, was an astronomer, a fact known from one of Archimedes’ own books. He probably studied in Alexandria under the successors of Euclid in his youth, as he kept sending his results to mathematicians there.

According to the ancient historian Plutarch, he was a friend and relative of Hieron II, King of Syracuse. He was famous even in his own time, mostly for the machines he built. They were used to defend the city during the Roman siege of 212 BC.

The law of the lever

The lever, a straight bar placed on a fulcrum or pivot, was in use before Archimedes. According to Chris Rorres, a mathematician at New York University, a book attributed to followers of Aristotle also stated that the farther from the pivot the force is applied, the easier it is to lift a load.

Archimedes’ achievement was to give it the form of a law in the language of geometry. In the first part of his book On the Equilibrium of Planes, he wrote that two weights balance when their distances from the pivot are in inverse proportion to their weights. In simple words: if one weight is ten times heavier than the other, the two can be balanced by placing the lighter weight ten times farther from the pivot. In the same book he also found the centres of gravity of shapes such as triangles and parallelograms.

Rorres also says experts have debated what exactly Archimedes proved and which assumptions were hidden in his proof.

Archimedes’ famous remark is linked to this principle. It was recorded about five centuries later by Pappus of Alexandria: “Give me a place to stand and I will move the earth.” Plutarch also tells a story that Archimedes, using a system of pulleys, single-handedly drew onto land a ship loaded with goods and passengers.

Buoyancy

The second major discovery is in Archimedes’ book On Floating Bodies. According to MacTutor, this book laid the foundation of hydrostatics, the study of fluids at rest. According to the English translation by the British historian of mathematics T. L. Heath, it proves that when a solid lighter than a fluid is forcibly immersed, the fluid pushes it upwards with a force. A body heavier than the fluid sinks to the bottom, but its weight in the fluid is reduced by the weight of the fluid it displaces.

This is Archimedes’ principle. It explains why an iron ship floats: its shape displaces water equal in weight to the ship itself. Archimedes also worked on the balance of floating bodies of various shapes.

The golden crown and “Eureka”

The story linked to this discovery is famous across the world, but it is not in Archimedes’ own writing. It is found about two centuries later in the book of the Roman architect Vitruvius. According to it, King Hieron suspected that the goldsmith had mixed silver into the gold of a crown. Archimedes stepped into a tub to bathe and saw that water spilled out in proportion to the part of his body that sank. Joyfully shouting “Eureka, Eureka”, meaning “I have found it”, he ran towards home. He then proved the adulteration by measuring the water displaced by lumps of gold and silver equal in weight to the crown.

According to Rorres, the method described in the story would have required measuring very small differences with the instruments of the time. Many experts therefore think Archimedes used a balance and the principle of buoyancy. In Heath’s view, Proposition 7 of On Floating Bodies gives the real solution to this question.

A new discovery from erased writing

Several of Archimedes’ books remained lost for centuries. According to the Archimedes Palimpsest Project, beneath the pages of a 13th-century prayer book lie erased texts centuries older. They include two books of Archimedes found nowhere else: The Method and the Stomachion. The manuscript was sold at auction on 29 October 1998 and then placed in the Walters Art Museum in the United States, where its text was read using modern imaging.

In life today

The principle of the lever works in every pair of scissors, the claw of a hammer, a bicycle pedal and the handle of a hand pump. The principle of buoyancy is the basis of the design of ships, boats and submarines, and every boat in our coastal life, from Gwadar to Karachi, floats on it. Archimedes showed that behind everyday machines are laws that can be proved by mathematics. This thinking became the foundation of all later physics.