All Thinkers

Archimedes

Archimedes was a Greek mathematician, scientist, and engineer. He was born around 287 BCE in Syracuse, a Greek city on the island of Sicily. We know little about his early life. His father was an astronomer named Phidias. He may have studied for a time at the great Library of Alexandria in Egypt, though this is not certain. Most of his life, however, was spent in Syracuse. In his time, Syracuse was an independent Greek-speaking city. The Roman Republic was growing stronger and would soon swallow most of the Mediterranean world. Archimedes worked closely with the king of Syracuse, Hiero II, and later with Hiero's grandson Hieronymus. He served the city as both a thinker and an inventor. In the year 212 BCE, Roman forces attacked Syracuse. Archimedes was about 75 years old. He had designed weapons to defend the city, including powerful catapults and machines that lifted enemy ships out of the water. The Romans took the city after a long siege of about two years. The traditional story is that a Roman soldier killed Archimedes during the chaos, even though the Roman general Marcellus had ordered that he be spared. According to later writers, Archimedes was working on a mathematical problem when the soldier arrived. He asked not to have his diagrams disturbed. The soldier killed him anyway. His tomb in Syracuse was lost for centuries. The Roman writer Cicero claimed to have rediscovered it nearly 140 years after his death.

Origin
Syracuse, Sicily (Hellenistic Greek world)
Lifespan
c. 287 BCE - c. 212 BCE
Era
Ancient / Hellenistic Greece
Subjects
Mathematics Physics Engineering Ancient Greek Thought Geometry
Skills
Scientific Thinking Problem Solving Critical Thinking Creative Expression Research Skills
Why They Matter

Archimedes matters for three reasons. First, he was one of the greatest mathematicians of the ancient world. He worked out methods for finding the area and volume of curved shapes. He found a very accurate value for pi, the ratio of a circle's edge to its diameter. He developed early ideas that would only be properly formalised 1,800 years later in modern calculus.

Second, he was a brilliant practical inventor. He designed war machines, water pumps, levers, and pulleys. The Archimedes screw, a long screw inside a tube used to lift water, is still in use today in some parts of the world. He showed that pure mathematics and practical engineering could be the same activity.

Third, he discovered the principle that explains why things float. The principle is named after him. It says that an object placed in a fluid pushes up on the fluid with a force equal to the weight of the fluid it pushes aside. This is now a foundation of physics. Ships, submarines, and balloons all depend on it. Archimedes set a standard of precise scientific reasoning that influenced Galileo, Newton, and modern physics. He is sometimes called the greatest scientist of the ancient world.

Key Ideas
1
The Eureka Story
2
The Power of the Lever
3
Defending Syracuse
Key Quotations
"Eureka! I have found it!"
— Reported by Vitruvius, c. 1st century BCE
This is the most famous shout in the history of science. According to the Roman architect Vitruvius, writing about 200 years after Archimedes, the great mathematician shouted this when he stepped into his bath and realised how to measure the volume of an irregular object. He is said to have run naked through the streets of Syracuse in his excitement. The story may be partly a legend. We have no record of Archimedes himself describing the event. But the principle behind it, that the volume of an object equals the volume of fluid it pushes aside, is real and important. For students, 'eureka' has become a word in many languages for the moment of discovery. The story behind it captures something true about how scientific insights often come, after long thought and in unexpected places.
"Give me a place to stand and I will move the Earth."
— Attributed by Pappus of Alexandria, c. 4th century CE
This famous boast captures Archimedes' confidence in the lever. He had worked out exactly how levers multiply force. With a long enough bar and a fixed point to push against, you could in principle lift any weight. The Earth itself was not too much. The line is reported by Pappus, a Greek mathematician writing around 600 years after Archimedes. We cannot be sure he ever said it. But it fits him. He had proved his lever rules carefully. He knew their power. For students, the line is a useful introduction to the lever and to the spirit of mathematical confidence. If you understand the rule, you can apply it in principle to any situation, however large.
Using This Thinker in the Classroom
Scientific Thinking When introducing students to scientific discovery
How to introduce
Tell students the eureka story. Archimedes had been thinking hard about how to measure the volume of a crown. The answer came in his bath. The story shows something real about how science works. You think about a problem carefully, sometimes for a long time. Then the answer comes, often in a quiet moment when you are doing something else. Discuss with students whether this matches their own experience of solving difficult problems. Many students will recognise the pattern. Effort first, then insight, often in an unexpected place.
Problem-Solving When teaching students about simple machines
How to introduce
Bring in or describe everyday levers. A door handle. A bottle opener. A see-saw. A crowbar. A pair of scissors. They all use the same principle. Archimedes worked out the exact rule over 2,200 years ago. The longer your end of the lever, the more weight you can move at the other end. Have students identify levers around them. Have them think about why long-handled tools are usually more powerful than short-handled ones. The principle is everywhere once you start looking. Knowing it helps students understand why the world is built the way it is.
Creative Expression When teaching students about thinkers who also made things
How to introduce
Show students the range of what Archimedes did. He proved abstract mathematical theorems. He also designed war machines, water pumps, and devices for moving heavy objects. The same person did both. Discuss with students how unusual this seems today. We often separate thinkers from makers. Archimedes did not. The problem of measuring a king's crown was both a mathematical and a practical problem. The lever was both a tool for builders and a topic for proof. For students, this is a useful example of how creativity can cross boundaries that look fixed.
Further Reading

For a first introduction, Sherman Stein's Archimedes: What Did He Do Besides Cry Eureka? (1999) is a clear and lively account aimed at general readers. The MacTutor History of Mathematics Archive online has excellent free articles on Archimedes. Reviel Netz's The Archimedes Codex (with William Noel, 2007) tells the dramatic story of the rediscovery of his lost works, accessible to general readers.

Key Ideas
1
Calculating Pi
2
The Floating Principle
3
The Method of Exhaustion
Key Quotations
"Do not disturb my circles."
— Reported as Archimedes' last words, attributed by various Roman writers
According to Roman writers, Archimedes was drawing geometric diagrams in the sand when a Roman soldier approached during the sack of Syracuse. Archimedes asked the soldier not to disturb his diagrams. The soldier, perhaps angered or perhaps not knowing who Archimedes was, killed him. The story has become legendary. The exact words, in a society where Greek mathematicians did not write things down quickly, may not be his. But the image is striking. A thinker so focused on his problem that even a sword does not break his concentration. For students, the story raises a serious question. Is this admirable focus or foolish detachment? Archimedes lived for mathematics. He also died for it. Both readings have merit.
"There are things which seem incredible to most men who have not studied mathematics."
— From Archimedes' work The Sand Reckoner, c. 3rd century BCE
Archimedes wrote The Sand Reckoner to show how to count extremely large numbers. He worked out a system for counting numbers as large as the grains of sand needed to fill the universe. He wanted to show that no number is too big to be expressed clearly with the right system. The line above appears as he sets up the work. People who have not studied mathematics, he says, may find such ideas unbelievable. But mathematics can handle them. For students, this is a wonderful early statement of mathematical ambition. Mathematics is not about small everyday numbers only. It can describe quantities far larger than anything in physical experience. Archimedes was making this case 2,200 years ago. The same case is still relevant today, when modern science routinely deals with numbers vast beyond ordinary imagination.
Using This Thinker in the Classroom
Scientific Thinking When teaching students about Archimedes' principle and floating
How to introduce
Demonstrate or describe a simple experiment. Place a heavy steel object in water. It sinks. Place a steel ship-shape (like a thin steel bowl) in water. It floats. The mass of steel is the same. What differs is the volume of water pushed aside. Walk students through Archimedes' rule: an object floats if it pushes aside water that weighs more than itself. Steel ships work because their shape lets them push aside huge amounts of water. The rule explains hot air balloons, submarines, and why we float more easily in salt water than fresh. One simple principle, an enormous range of consequences. Archimedes showed it 2,200 years ago.
Problem-Solving When teaching students about approximation
How to introduce
Walk students through how Archimedes worked out pi. He drew shapes inside and outside a circle. The circle's edge had to be between the two shapes' edges. With six-sided shapes the answer is rough. With many-sided shapes the answer is close. With 96-sided shapes the answer is very close. The technique is called the method of exhaustion. It is a way of squeezing in on a difficult answer using easier shapes. For students, this is a powerful general lesson in problem-solving. When you cannot solve a problem exactly, look for ways to bound it from above and below. The answer is somewhere between the two bounds. Make the bounds tighter. The answer becomes clearer.
Further Reading

For deeper reading, T.L. Heath's The Works of Archimedes (1897, Dover reprint 2002) is the classic English translation of his surviving works. E.J. Dijksterhuis's Archimedes (1956, Princeton reprint 1987) is a fine scholarly account of his mathematical achievements. For the historical context of Hellenistic Sicily, Lionel Casson's Libraries in the Ancient World (2001) and other works on Hellenistic culture provide good background. The Archimedes Palimpsest Project website has high-quality images and scholarly articles.

Key Ideas
1
The Lost and Found Palimpsest
2
Archimedes and the Birth of Mathematical Physics
3
Was He Killed for Caring Too Much About Mathematics?
Key Quotations
"Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty."
— Widely attributed to Archimedes, exact source not in surviving works
This line is often quoted as Archimedes' but is not found in any of his surviving writings. It may be a later summary of his attitude rather than his own words. Archimedes did seem to love mathematics for its own sake. According to Plutarch, he considered his discoveries about spheres and cylinders to be his finest achievement, and asked that a sphere inside a cylinder be carved on his tombstone. These were pure geometric results, not the practical machines he had built for Syracuse. For advanced students, this is a useful example to discuss. Famous figures often have lines attributed to them that capture their spirit but are not actually their words. Reading carefully means asking where a quotation really comes from. Even when the answer is 'we are not sure', that is honest scholarship.
"Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced."
— On Floating Bodies, Book 1, Proposition 5
This is Archimedes' careful statement of part of his floating principle. The language is precise. A solid lighter than the fluid will sink into the fluid until enough fluid is pushed aside to weigh the same as the whole solid. At that point, the upward push of the displaced fluid balances the weight of the object. The object floats. This single rule explains an enormous range of phenomena, from the floating of ships to the rising of hot air balloons. For advanced students, the wording shows what mathematical physics looks like when it is being created. Archimedes is not stating a rough observation. He is stating a precise theorem. The same level of mathematical care that Euclid gave to geometry, Archimedes gives to fluids. This is the foundation on which all later mathematical physics is built.
Using This Thinker in the Classroom
Research Skills When teaching students about how ancient knowledge survives
How to introduce
Tell students about the Archimedes Palimpsest. A Greek prayer book hid the only copy of one of Archimedes' most important works under its later writing. The hidden writing was not noticed until 1906. The book then disappeared, was rediscovered at auction in 1998, and is now being read with modern imaging technology. The story is dramatic. It is also typical of how ancient knowledge has come down to us. Always partly. Always with luck involved. Always relying on people centuries later to do the work of preservation and recovery. Discuss with students what other ancient works might be hidden in similar ways, waiting for modern techniques to reveal them.
Critical Thinking When teaching students about how legends form around real people
How to introduce
Walk students through the famous Archimedes stories. The eureka shout in the bath. Running naked through the streets. The mirrors that set ships on fire. The last words 'do not disturb my circles'. Discuss which of these are well attested and which are probably later legends. The mirrors story is almost certainly fiction. The eureka story comes from a writer 200 years later. The last words come from various Roman sources. Real Archimedes was a remarkable mathematician and engineer. Legendary Archimedes is a romantic figure built up over centuries. For advanced students, this is a useful exercise. Famous figures often come down to us through layers of legend. Sorting fact from story is real historical work.
Common Misconceptions
Common misconception

Archimedes used mirrors to set Roman ships on fire.

→
What to teach instead

He almost certainly did not. The story appears in writers many centuries after his death. Modern attempts to test the idea have shown it is extremely difficult to set wooden ships on fire with focused sunlight. You would need many large mirrors held still for a long time, on ships that are not moving in the water. None of his earliest biographers mention the mirrors. The story probably grew up later as part of his legend. He really did design effective war machines, including catapults and ship-grabbing claws. The mirrors are a romantic addition. This is a useful example of how famous figures attract dramatic stories that may not be true.

Common misconception

Archimedes was just a mathematician who happened to make some inventions.

→
What to teach instead

He was both a mathematician and a working engineer, and he treated the two activities as connected. He designed weapons for Syracuse over many years. He invented the screw pump, water-lifting devices, levers, and pulleys. He built a planetarium model that showed the motion of the heavens. He did not see his practical work as a side activity. He saw mathematics as the source of practical power and practical problems as a source of mathematical questions. The split between pure thinker and practical maker is partly a modern way of seeing things. Archimedes did not divide his life that way.

Common misconception

We have all of Archimedes' works.

→
What to teach instead

We do not. We have around a dozen of his works in some form. We know from ancient references that others have been lost. The Method, one of his most important works, was completely unknown until 1906, when it was discovered hidden under later writing in a medieval prayer book. There may be other lost works of Archimedes that we have not yet found. Modern imaging techniques continue to recover hidden text from the Archimedes Palimpsest. The picture we have of him is partial. New discoveries could still change it.

Common misconception

Archimedes invented calculus.

→
What to teach instead

He did not, but he came remarkably close. His method of exhaustion, used to find areas and volumes of curved shapes, captures the basic idea behind integral calculus. He used reasoning very close to mathematical limits. However, he did not develop the general algebraic techniques and the deep connection between integration and differentiation that make modern calculus so powerful. That work was done by Newton and Leibniz in the 17th century, almost 1,900 years later. They both knew Archimedes' work and were inspired by it. So while it is fair to say Archimedes anticipated parts of calculus, calling him its inventor goes too far.

Intellectual Connections
Develops
Euclid
Euclid worked at Alexandria a generation before Archimedes. He had set down the axiomatic method in geometry: start with clear definitions and a few simple postulates, then prove every result step by step. Archimedes took this method and applied it to physics. He proved his results about levers and floating bodies the same way Euclid proved his results about triangles and circles. Reading them together gives students a clear sense of how Greek mathematical thinking grew. Euclid built the method. Archimedes extended it from pure geometry into the physical world.
Anticipates
Isaac Newton
Newton, working in the 17th century, brought mathematical physics to its first great climax. He built directly on the path Archimedes had opened almost 1,900 years earlier. Newton studied Archimedes' work carefully. The method of exhaustion fed directly into Newton's development of calculus. Archimedes' approach to fluids fed into Newton's mechanics. Reading them together shows how a single thread runs from ancient Syracuse to early modern England. The scientific revolution did not come from nowhere. It built on the work Archimedes had begun.
Complements
Al-Jazari
Al-Jazari was a 12th-century Arab engineer working in what is now Turkey. Like Archimedes, he combined careful mathematical thinking with practical machine design. His Book of Knowledge of Ingenious Mechanical Devices is full of pumps, clocks, and water-raising machines. The Archimedes screw appears in his designs. Reading them together gives students a sense of how engineering knowledge crossed cultures. The ancient Greek tradition of mathematical engineering was preserved and extended in the medieval Islamic world before returning to Europe.
Complements
Al-Khwarizmi
Al-Khwarizmi was the great Persian mathematician of the 9th century who developed early algebra. Archimedes used geometric methods. Al-Khwarizmi developed algebraic methods. The two approaches together would later be combined in modern mathematics. Arab scholars carefully preserved Archimedes' works in their libraries and translated them. Without the Islamic intellectual world, Archimedes might have been largely lost. Reading them together gives students a sense of how mathematical knowledge moved across cultures and traditions.
Anticipates
Albert Einstein
Einstein admired Archimedes greatly. The spirit of using mathematics to describe the physical world precisely is something Einstein inherited from a long tradition that started with Archimedes. Einstein once said that Western science is built on two pillars: Greek geometry and Renaissance experimentation. Archimedes stands at the foundation of the first pillar and points towards the second. Reading them together shows the long arc of mathematical physics, from a man drawing circles in the sand of Syracuse to one writing equations about the curvature of space and time.
In Dialogue With
Aristotle
Aristotle's natural philosophy was the dominant view of the physical world in Archimedes' time. Aristotle relied mainly on careful observation and reasoning in everyday language. Archimedes, by contrast, used precise mathematics. The difference is important. Aristotle's physics was wonderfully thorough but sometimes wrong about quantitative details. Archimedes' physics was narrower but mathematically exact. Reading them together gives students a sense of two different scientific styles. The mathematical style would eventually win, but only after almost 2,000 years. Aristotle dominated medieval thought. Archimedes, though, was the model the scientific revolution would follow.
Further Reading

For research-level engagement, Reviel Netz's translation and study The Works of Archimedes: Translation and Commentary (multiple volumes, Cambridge University Press) is the current standard scholarly edition. Netz's own scholarship on Hellenistic mathematics is essential. Wilbur Knorr's The Ancient Tradition of Geometric Problems (1986) places Archimedes carefully in context. The journal Archive for History of Exact Sciences regularly publishes Archimedes scholarship. For the wider Hellenistic scientific world, G.E.R. Lloyd's many works are valuable.