Study Guide: Archimedes
Archimedes (ΑΡΧΙΜΗΔΗΣ) was a mathematician, physicist, astronomer, engineer, inventor, and weapons-designer. He was both a person of his own time and far ahead of his time.
Some Achievements
- Invented the sciences of mechanics and hydrostatics.
- Discovered the laws of levers and pulleys, which allow us to move heavy objects using small forces.
- Discovered a fundamental concepts of physics—the center of gravity.
- Calculated PI to the most precise value known at his time. His upper limit for PI was the fraction 22⁄7. This value was commonly used around the world until the invention of computers enabled us to quickly calculate even more accurate values.
- Discovered and mathematically proved formulas for the volume and surface area of a sphere.
- Showed how exponents could be used to write bigger numbers than had ever been thought of before.
- Proved that to multiply numbers written as exponents, the exponents should be added together.
- Directly inspired Galileo Galilei and Isaac Newton to investigate the mathematics of motion. Archimedes’ surviving works (tragically, many have been lost) finally were printed in 1544. Leonardo da Vinci was lucky enough to see some of the hand-copied works of Archimedes before they were printed.
- Was the first person to apply lessons from physics—such as the law of the lever—to solve problems in pure mathematics.
- Invented war machines such as a highly accurate catapult that blocked the Roman conquest of Syracuse for many years.
The Ancient Greeks
The ancient Greeks were the first civilization to do “modern” science. Although many other cultures made scientific discoveries, these were made for individual, practical reasons, such as how to build stronger temples or predict when the heavens would be right for planting crops.
The ancient Greeks investigated the physical world for the sheer pleasure of discovering knowledge and wisdom. They studied geometry to discover its inner logic and its beauty.
For example, Democritus proposed—for no particular practical reason—that all matter was made of tiny particles called atoms and that these atoms could not be split into smaller particles, and that they were in constant motion and colliding with one another. He produced logical arguments for this idea.
Early Years
Archimedes was born into this Greek scientific culture in the city-state of Syracuse on the island of Sicily, around 287 BCE. His father, Phidias, was an astronomer. He may also have been related to Hiero II, King of Syracuse.. In his work The Sand Reckoner he wrote that his father was an astronomer.
Archimedes spent most of his life in Syracuse. But in pursuit of learning, he sailed to the Egyptian city of Alexandria, where one of the world’s greatest library had been built. The Library of Alexandria, with its meeting rooms and lecture halls, had become the focal point for scholars in the ancient world.
Some of Archimedes’ work is preserved in copies of the letters he sent from Syracuse to his friend Eratosthenes. Eratosthenes was in charge of the Library of Alexandria, and was no mean scientist himself. He was the first person to accurately calculate the size of our planet accurately.
Immersed in the scientific culture of Ancient Greece, Archimedes blossomed into one of the finest minds our world has known. He was the Einstein of his time, or perhaps we should say that Einstein was the Archimedes of his time.
The Annoying Mathematician
During the Renaissance, 2,000 years after Archimedes’ time, mathematicians looked again at his work. They knew his results were correct, but they couldn’t figure out how he found them. Archimedes was frustrating, because he gave clues, but did not reveal his full methods. It seems that he enjoyed challenging other mathematicians. He would often give the correct answer to problems, and then see if others could solve the problems for themselves.
Discovery of Ancient Texts
The mystery of Archimedes’ mathematics wasn’t solved until 1906, when Professor Johan Heiberg discovered a book in an ancient library, in the city of Constantinople. (now Istanbul, Turkey) The book was a Christian prayer book written in the thirteenth century, when Constantinople was the last outpost of the Roman Empire. The book Heiberg found is now called theArchimedes Palimpsest.
Heiberg discovered that the book’s prayers had been written on top of mathematics. The monk who wrote the prayers had tried to remove the original mathematical work; only faint traces of it remained. It turned out that the traces of mathematics were actually copies of Archimedes’ lost original works, which had been copied in the 10th century. The book contained seven treatises from Archimedes including The Method, which had been lost for many centuries. Archimedes had written The Method to reveal how he did mathematics. He sent it to Eratosthenes to be lodged in the Library of Alexandria.
In it Archimedes wrote, “I presume there will be some current as well as future generations who can use The Method to find theorems which we have not discovered.” By reading The Method, modern mathematicians learned just how far ahead of his time Archimedes was and the techniques he used to solve problems.
Archimedes and the Golden Crown
King Hiero II gave a weighted amount of gold to a craftsman to make him a crown. The crown he got back weighed the same, but King Hiero was suspicious. He thought the craftsman had stolen some gold and replaced it with silver in the crown. He couldn’t be sure, so he sent for Archimedes and explained the problem to him. For many days Archimedes struggled with the problem, and the whole city waited and wondered if he could solve the riddle.
After many days Archimedes found a solution while taking a bath, and noticing the water level rise as he got into the tub. He was so excited that he is said to have leaped up and run naked through the streets of Syracuse shouting ‘Eureka,’ (‘I’ve found it.’) It seems that even thousands of years ago, scientists had a reputation for being a little crazy!
Archimedes knew that gold is denser than silver, so a one centimeter cube of gold would weigh more than a one centimeter cube of silver. The problem was that the crown was irregularly shaped, so although its weight was known, its volume wasn’t.
Archimedes solved the riddle by sinking the crown in water. If we did this measurement using modern equipment, we would find the 1 kg of gold would raise the water level by 51.8 ml and the 1 kg of silver by 95.3 ml. If King Hiero’s crown weighed 1 kg, and it raised the water level by 52 ml or so, then the crown would be pure gold. If the water level rose more than this, then some of the gold had been replaced by silver.
Archimedes found that the crown was a mixture of gold and silver, which was bad news for King Hiero, and even worse news for the craftsman!
Death and Legacy
Archimedes died during the conquest of Syracuse in 212 BC when, while working on a math problem he was killed by an enraged Roman soldier.
He was buried in a tomb on which was carved a sphere within a cylinder. This was his wish, because he believed his greatest achievement was finding the formula for the volume of a sphere.
Many years later, Cicero, the Roman Governor of Sicily, went looking for the tomb of Archimedes. He found that it had become overgrown with weeds and bushes, which he ordered to be cleared.
Today we do not know where Archimedes’ tomb is—it has been lost, probably forever. Much of his work has also been lost forever, but what we know of it leaves us in awe of his achievements. More than 300 years after Archimedes’ death the Greek historian Plutarch said of him:
“He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”