Questions to ask Archimedes
8 min read
Good questions for Archimedes of Syracuse (about 287 to 212 BC) ask how he measured pi, why he wanted a sphere and cylinder on his tomb, how the lever and floating bodies work, how he found results by imagining shapes on a balance, and how his machines defended Syracuse against Rome. The catch: famous stories like “Eureka” were written down centuries after his death. This page gives 22 questions and marks which answers come from his own works.
What we actually know about Archimedes
Very little of Archimedes' life is documented, and almost nothing in his own words is personal. He lived in Syracuse, a Greek city in Sicily, under King Hiero II. In The Sand Reckoner he names his father, Phidias, as an astronomer. His treatises are written as letters to mathematicians in Alexandria, such as Conon, Dositheus and Eratosthenes, which suggests he spent time there. He died in 212 BC, killed by a Roman soldier when Syracuse fell after a siege of about two years in the Second Punic War.
Everything else comes from later writers: the historian Polybius (2nd century BC), Livy and Vitruvius (1st century BC), Cicero, and Plutarch's Life of Marcellus (about AD 100). That gap matters for your interview. Answers about mathematics can come straight from his works. Answers about his bath, his death or his mirrors rest on stories told 150 to 1,000 years later.
Pi Day, and how to use these questions
Archimedes is a favourite for Pi Day on 14 March, because his Measurement of a Circle gave the first rigorous bounds for pi. For a class interview, choose questions from his own works and from the legends, and in your write up say which is which. Teachers notice that.
To practise, you can talk with an AI recreation of Archimedes on Yuga, by voice or text. It is built from his treatises and the ancient accounts, but it is an AI and can be wrong. Cite the treatises (in T. L. Heath's translation, for example) and the ancient authors, not the app.
Questions answered by his own mathematics
Source: his surviving treatises. In Measurement of a Circle he fitted polygons of up to 96 sides inside and outside a circle to prove that pi lies between 3 10/71 and 3 1/7. On the Sphere and Cylinder proves that a sphere has two thirds the volume and surface of the cylinder that just contains it. The Sand Reckoner builds a number system big enough to count the grains of sand that would fill the universe, and mentions Aristarchus' idea that the Earth goes round the Sun. In the preface to On Spirals he admits he once sent friends results without proofs, slipping in two false ones to catch anyone who claimed them as their own.
- How did you trap pi between two numbers using polygons with 96 sides?
- Why did you stop at 96 sides?
- How did you prove that a sphere is two thirds of its cylinder?
- Why did you want a sphere inside a cylinder carved on your tomb?
- Why did you try to count the grains of sand in the universe?
- What did you think of Aristarchus' idea that the Earth moves around the Sun?
- How do you work with huge numbers when Greek numerals have no zero?
- How did you find the exact area of a slice of a parabola?
- Why did you include two false theorems among the results you sent to Alexandria?
Questions about levers, balance and floating
Source: On the Equilibrium of Planes and On Floating Bodies. He proved the law of the lever from a few simple postulates and found the centres of gravity of flat shapes. In On Floating Bodies he showed that a body in a fluid is pushed up by a force equal to the weight of the fluid it displaces. His letter to Eratosthenes, known as The Method, explains that he often found a result first by imagining thin slices of a shape balanced on a lever, then proved it by strict geometry.
- What is the law of the lever, in your own words?
- Why do you start from postulates instead of experiments?
- Why does a body float or sink?
- What is the method you described to Eratosthenes, and why did you share it?
- Did you think your balancing method counted as a proof?
- Did you invent the water screw that carries your name? (The attribution comes from Diodorus Siculus, writing in the 1st century BC, and similar devices may be older.)
Questions about the siege of Syracuse
Source: Polybius, Histories Book 8, and Livy, Books 24 and 25, the earliest accounts, with Plutarch's more colourful version. They describe catapults set for different ranges, cranes that dropped weights on ships, and a “claw” that lifted Roman ships out of the water and let them fall. The Roman commander Marcellus is said to have ordered that Archimedes be spared. Plutarch also says Archimedes thought practical engineering beneath him; that is Plutarch's view and may not be his.
- How did the claw lift a Roman ship out of the sea?
- Why did the Romans give up attacking from the sea?
- Did you enjoy building machines, or did you do it only for King Hiero?
- Why do you think Marcellus wanted you taken alive?
- What were you doing when the city fell in 212 BC?
Legend or record? Six famous stories checked
| Story | Earliest source | Verdict |
|---|---|---|
| “Eureka” in the bath, testing King Hiero's crown | Vitruvius, On Architecture, Book 9 preface (about 25 BC) | Written about 200 years later; a legend, though the physics is real |
| “Give me a place to stand and I will move the Earth” | Pappus of Alexandria (4th century AD); a similar boast in Plutarch | Late report; a lever boast fits his work, but the words are not his |
| Lifting a fully loaded ship alone with pulleys | Plutarch, Life of Marcellus 14 | Told about 300 years later; possible, unprovable |
| Setting Roman ships on fire with mirrors | Writers of the 2nd century AD and later | Not in Polybius, Livy or Plutarch; most historians doubt it |
| “Do not disturb my circles” to the soldier | Later retellings of Plutarch's account | Plutarch has him absorbed in a diagram; the famous words came later |
| Sphere and cylinder on his tomb | Plutarch; Cicero, Tusculan Disputations 5 | Cicero says he found the overgrown tomb in 75 BC; fairly strong |
Two good questions follow from the table: “Did you really run through the streets shouting Eureka?” and “Could you really move the Earth with a lever?”. Ask them, then let the answer explain where the story comes from. More in did Archimedes shout Eureka.
Sample answers in Archimedes' words
These are from On the Equilibrium of Planes, Book 1, in T. L. Heath's translation. They are mathematical statements, not chat, which is exactly how he wrote.
Q: What is the first rule of balance?
“Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.”
Q: So what is the law of the lever?
“Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.”
Q: Where is the balance point of a parallelogram?
“The centre of gravity of a parallelogram is the point of intersection of its diagonals.”
Questions he could not answer
The symbol π was first used for the circle ratio in 1706, by William Jones, so ask about “the ratio of the circumference to the diameter”, as he would say. He had no algebra with letters, no decimal point and no calculus, though his slicing method comes close to ideas of integration found 1,800 years later. He could not know what happened to his books: The Method survived in a single copy, scraped off and overwritten as a prayer book in 1229, identified by the scholar Johan Ludvig Heiberg in 1906, sold at auction in 1998 and imaged with modern technology at the Walters Art Museum in Baltimore.
Where to read Archimedes
T. L. Heath's The Works of Archimedes (1897), with a 1912 supplement containing The Method, is the classic English translation and is in the public domain. Plutarch's Life of Marcellus, chapters 14 to 19, and Polybius Book 8 are short and readable. The Archimedes Palimpsest Project website explains the recovered manuscript, and Reviel Netz and William Noel's The Archimedes Codex (2007) tells the story for general readers. For his place among other mathematicians, see famous mathematicians in history and key figures of ancient Greece.
Questions people ask
- What are good questions to ask Archimedes?
- Ask how he trapped pi between 3 10/71 and 3 1/7 using 96 sided polygons, why he wanted a sphere and cylinder on his tomb, how the law of the lever works, why things float, how his machines defended Syracuse, and whether he really shouted Eureka. The first four have answers in his own treatises.
- Did Archimedes really shout Eureka?
- The story comes from the Roman architect Vitruvius, writing about 200 years after Archimedes died. It says he noticed the water rise as he got into a bath, saw how to test King Hiero's crown, and ran home shouting Eureka. No earlier source has it, so treat it as a legend.
- How did Archimedes calculate pi?
- In Measurement of a Circle he drew regular polygons inside and outside a circle, doubling the number of sides from 6 up to 96, and calculated their perimeters. Because the circle lies between them, pi must lie between 3 10/71 and 3 1/7, or about 3.1408 and 3.1429.
- How did Archimedes die?
- He was killed by a Roman soldier in 212 BC, when Syracuse fell after a siege of about two years. Plutarch gives more than one version; the best known says he was absorbed in a geometry problem. The Roman general Marcellus is said to have ordered him spared and to have regretted his death.
Sources
- T. L. Heath (trans.), The Works of Archimedes (1897; supplement with The Method, 1912)
- Plutarch, Life of Marcellus, chapters 14 to 19 (about AD 100)
- Polybius, Histories, Book 8; Livy, History of Rome, Books 24 and 25
- Vitruvius, On Architecture, Book 9, preface
- The Archimedes Palimpsest Project
- Reviel Netz and William Noel, The Archimedes Codex (2007)