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# Archimedes: Early Years and Mathematics

by Ron Kurtus (revised 3 June 2009)

** Archimedes** (287-212 BC) was a great ancient Greek

*and scientist. He lived during a time of turmoil and war. Archimedes was educated in Egypt by followers of the famous mathematician, Euclid.*

**mathematician**Although he invented several important devices, he spent most of his efforts in mathematics, where he devised ways to calculate areas and volumes, defined pi and formulated integral calculus.

But it was his inventions such as a water pump and discoveries such as hydrostatics, that made him famous in his time. Some of his inventions are still used today. Finally, at age 46, there seemed to be peace in his city.

Questions you may have include:

- How was Archimedes' education important?
- What did he discover in mathematics?
- What were the influences of war in his life ?

This lesson will answer those questions.

## Education

Archimedes was born in Syracuse, Sicily in about 287 BC. Although Sicily is near Italy, at that time Syracuse was a Greek city. His father Phidias had some wealth and considered himself an astronomer.

There was much turmoil in the area during those years. In 275 BC, Hiero—a relative of Phidias—led a military takeover of Syracuse.

### Sent to Alexandria

When he was in his teens, Archimedes was sent to Alexandria, Egypt to study mathematics with teachers who had learned from Euclid. Not only was this as a way to gain his education, but his father also felt it was better to send his son away from the warfare being waged around Syracuse.

## Inventions and warfare

Besides studying mathematics, Archimedes also was interesting in creating inventions to solve common problems. But warfare also infuenced his life.

### Invented pump

While in Alexandria, he invented a device now known as Archimedes' screw. It was first used to pump water out of ships and was later used in irrigation. This type of water pump is still used in many parts of the world today.

Archimedes Screw easily pumps water to higher level

### Returned to Syracuse

Archimedes had completed his studies and returned home from Alexandria and spent the rest of his life in Syracuse. Unfortunately, there was more warfare.

When Archimedes was 22, in 265 BC, Hiero was declared king of Syracuse. Then in 264 BC, First Punic War between Carthage and Rome for control of Sicily began. Syracuse was initially allied with Carthage against Rome. A year later, after the Romans threaten Syracuse, Hiero signed peace treaty with Rome.

In the years that followed, Archimedes developed a close, friendly relationship with Hiero, the king of Syracuse, and his son Gelo. The king would often ask Archimedes to solve some difficult problem for him, and he soon considered Archimedes a "national treasure."

### Explained levers and pulleys

In 260 BC, when he was 27 years old, Archimedes explained how lever and pulleys worked. Levers are one of the basic tools and were probably used in prehistoric times, but Archimedes' explanation facilitated their use.

He later demonstrated to the king how effective levers and pulleys can be employed to move large objects.

## Mathematics and more warfare

Archimedes had became a master at mathematics, especially geometry. He spent most of his time working on solving new problems. Sometimes he became so involved in his work that he forgot to eat.

### Communicated with mathematicians

For years after he left Alexandria, Archimedes would often communicate with mathematician friends who remained in Alexandria. He would send his fellow mathematicians statements of his latest theorems, but he would not send the proofs of those theorems. The reason was that some of the mathematicians would claim the results as their own. Without being able to figure out the proof, they could not claim credit.

### Math discoveries

Some of the mathematical problems Archimedes solved concerned areas and volumes of geometric figures. He had to devise a better number system and a new way to determine the formulae for the areas and volumes of spheres, cylinders, parabolas, and other plane and solid figures.

#### Circles and spheres

Archimedes showed that the surface of a sphere is four times that of a great circle, that the volume of a sphere is two-thirds the volume of a circumscribed cylinder, and that the surface of a sphere is two-thirds the surface of a circumscribed cylinder including its bases.

#### Pi

In his measurements of circles, Archimedes showed that the exact value of pi (π) was between the numbers 3^{10}/_{71} and 3^{1}/_{7}. He found this by approximating a circle by a regular polygon having 96 sides. This was the most accurate approximation of pi at that time.

#### Integration

One of the methods he used to find the areas, volumes and surface areas of many bodies was an early form of integration. This was considered his greatest mathematical invention, leading to the field of Calculus.

To determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and added the areas together.

#### Number system

Unhappy with the unwieldy Greek number system, Archimedes proposed a number system capable of expressing extremely large numbers. Using that system, he calculated the number of grains of sand that would fill the known universe. In modern notation, that number is 8x10^{63}.

#### Center of gravity

He then applied his calculations and methods of geometry to physical objects, discovering fundamental theorems concerning the center of gravity of plane figures and solids.

### End of war

In 241 BC, the First Punic War between Carthage and Rome ended. Fortunately, Syracuse had made peace with Rome and was not involved in this 20 year battle. Rome had gained control of all of Sicily outside of a small area surrounding Syracuse. Archimedes was 46 at the time.

The next year, Hiero's son Gelo began co-ruling Syracuse with his father. There was then relative peace for about 20 years. War had been an issue for most of Archimedes' life up to this point. Although there peace during those years, problems with Rome were on the horizon.

## Summary

Archimedes was born in Syracuse, Sicily, but he was educated in Egypt by followers of the famous mathematician, Euclid. Upon his return to Syracuse, he became a favorite of the king, to whom he was a distant relative. Archimedes spent most of his early life solving mathematical problems, including devising ways to calculate areas and volumes, defining pi, and formulating integral calculus. War or the threat of war was present for the first half of his life.

### Lessons learned

Lessons learned from the life of Archimedes include:

- A good education is important
- Intelligent and wealthy parents can help your education
- Parents with good social connections are useful
- Observation and creativity can lead to greatness
- War and political forces can ruin people's lives

Learn and think

## Resources and references

### Websites

**All the Grains of Sand** - From Nova

**Mathematical Methods of Archimedes**

### Books

## Questions and comments

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archimedes.htm**

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## Where are you now?

## Archimedes: Early Years and Mathematics

## Later Years and Science