Archimedes was a mathematician, inventor, and astronomer who was one of the most celebrated mathematicians of all time. He was famous for getting so absorbed in his studies, that he forgot about social conventions. Famously, he is said to have realised a principle of mathematics when he saw the displaced water after getting in the bath. So excited at realising the importance of this he shouted “Eureka” and rushed out into the street unclothed. We cannot be certain this entertaining anecdote is 100% true, but he was a genuinely great mathematician, who in many ways was centuries ahead of his contemporaries. His works were later rediscovered by both Arabic and Renaissance scientists who first replicated his results and then built on them.

Archimedes was born in Syracuse, a Greek city-state in Sicily. He was educated at Alexandria, Egypt – famed for its knowledge and learning. He then returned to Syracuse where he became famed as both a mathematician, inventor, astronomer and philosopher. He continued to correspond with mathematicians back in Alexandria, and even during his lifetime, he gained a strong reputation for being a mathematical genius. Archimedes was close to the ruler of Syracuse, King Hiero II who employed Archimedes in the defence of the city against the Roman invasion.

Archimedes greatest love was theoretical mathematics. He wrote several treatises and corresponded with other mathematicians of the day. He had a brilliant mind and made many important developments in the subject. This includes a development of calculus using infinitesimals. Archimedes development of calculus remained un-improved until the 15th Century. Archimedes also developed the most accurate prediction of pie to date. Using the method of approximation, he showed pi has to be greater than 223/71 and less than 22/7. His favourite proof involved proving that the volume and surface area of a sphere was 2/3 of a cylinder of the same height and diameter.

Archimedes had an inquisitive nature and was willing to challenge conventional views. For example, it was widely considered that the number of grains of sand on the earth was infinite – or at least impossible to calculate. However, in *The Sand Reckoner*, Archimedes proceeded to make a calculation using a new system of counting which made use of powered numbers based on the myriad (10,000). He proposed a myriad x myriad = 100 million. Archimedes calculated the number of grains of sands to fill the universe would be eight vigintillions or 8 x 10^{63}.

As well as mathematics, Archimedes made discoveries in the field of mechanics. Although he did not invent the lever, he made descriptions about its use, the mathematical underpinning of levers, and made practical innovations to help sailors lift heavier objects than they could on their own. Archimedes work on the lever led to one of his most famous statements

“Give me a place to stand on, and I will move the Earth.”

Archimedes was also known as an outstanding astronomer; he made observations on solstices and calculated the distance to the sun and planets through the use of Pythagorean theory.

As well as theoretical mathematics, Archimedes was asked to help deal with matters of state. For example, Archimedes was asked to prove whether a crown was made purely of gold or whether it was made up of silver. However, Archimedes wasn’t allowed to damage the crown. One account states Archimedes determined the density of the crown by measuring how much water was displaced when the crown was submerged in water and then dividing the mass of the crown by the weight of the crown. From knowing the density, Archimedes could work out whether it was pure gold. An anecdotal story of how Archimedes got the idea for this method was sitting in a bathtub of water. When he saw the water displaced, the idea came to him. Archimedes was so excited. He shouted “Eureka” – ‘I have found it’ and then took to the streets without time to put his clothes on. Unsurprisingly, this great anecdote wasn’t found in Archimedes own writings, but the legend has stuck to Archimedes.

“Any floating object displaces its own weight of fluid.”

— Archimedes of Syracuse

An alternative explanation for measuring the density of a crown may have been Archimedes own principle about the buoyancy of an object in a fluid. In *Floating Bodies* (c. 250 BC) Archimedes wrote:

“Any object, totally or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.”

The implication of this principle is that when a body is partially immersed in a fluid, it experiences an apparent loss in weight that is equal to the weight of the fluid displaced by the immersed part of the body.

Though Archimedes great love was theoretical mathematics, King Hiero II of Syracuse employed Archimedes to build inventions for the civil benefit and military defence of the city. Archimedes was involved in constructing and possibly inventing the Archimedes screw, which is able to lift liquid and solids uphill against the force of gravity. It was an important device which could lift water from low-lying lands to higher lands.

During the Second Punic War, the city was under siege from the Romans and General Marcus Claudius Marcellus. Archimedes invented many military defences. This included a system of mirrors deflecting the sunlight to the Roman ships – either to set them alight or blind the sailors heading to shore. Another weapon was the Claw of Archimedes – it involved a long metal hook suspended from a crane-like arm. It was used to drop on invading ships to lift them from the water and possibly sink it. He is also credited with inventing a more powerful and accurate catapult. Plutarch wrote in glowing terms about the impact Archimedes had on the defence of the city.

“When… the Romans assaulted the walls in two places at once, fear and consternation stupefied the Syracusans…. But when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons… that came down with incredible noise and violence… they knocked down those upon whom they fell in heaps, breaking all their ranks and files.”

The Roman General Marcellus wanted to keep alive the famous Archimedes, but Archimedes was killed by a Roman soldier shortly after the city fell (in 212 BC). One anecdote suggests Archimedes was absorbed in his mathematical studies using a compass to draw circles when a Roman soldier demanded he surrendered and follow him. Archimedes replied. “Do not disturb my circles” or according to Velerius Maximus – “I beg of you, do not disturb this.’ The Roman soldier in a fit of anger killed Archimedes on the spot.

He was buried in Syracuse with a model sculpture of his sphere and cylinder. The tomb of Archimedes was later rediscovered by Cicero in about 75BC.

### Influence of Archimedes

Archimedes was in many ways ahead of his time. Despite numerous breakthroughs in mathematics, there was not sufficient skilled and intelligent mathematicians to make use and develop Archimedes work in the classical period.

“Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again with Descartes and Newton.”

Eric Temple Bell, *The Development of Mathematics* (1940)

Many of his works were lost or fell out of general circulation. However, some works were preserved and when they were rediscovered and re-printed, they gave a major boost to a reinvigoration of mathematics in Asia and Europe. Archimedes works were translated into Arabic in the 9th Century AD. In 1544, a version of Archimedes works in both Latin and Greek were published in *Editio Princeps* in Basel. This was a very influential work. Galileo was a great admirer of Archimedes and – influenced by his writings, he invented a hydrostatic balance for weighing metals in water and air. Perhaps even more important was Archimedes influence on mathematicians Rene Descartes and Pascal Blaise. In 1906, the *Archimedes Palimpsest* was discovered – and, although overwritten with prayers in the 13th Century, underneath they discovered original writings of Archimedes that had been first written in the 10th Century AD. It includes original sources of *On Floating Bodies* and *“The Method of Mechanical Theorems.”*

Citation: Pettinger, Tejvan. “Biography of Archimedes”, Oxford, UK *www.biographyonline.net* Published 8 July 2019.

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