Archimedes (c. 287–212 BC) lived in Syracuse, a rich Greek city on the island of Sicily. He wrote to the mathematicians of Alexandria in Egypt, and many of his books open with a friendly letter to one of them. He was a mathematician first, but the stories people told about him for centuries were about his machines: pulleys that let one man move a ship, and engines that defended Syracuse against the Roman army.
Key idea
The problem
Builders in Archimedes' time already used levers, pulleys and screws. They knew that they worked, from long practice. Nobody had shown why, or how to predict exactly what a machine would do before building it. Archimedes wanted rules as certain as Euclid's geometry.
How the idea came
In On the Equilibrium of Planes he starts from a few simple claims anyone would grant, like "equal weights at equal distances balance." From those, step by step, he proves the law of the lever.
Two magnitudes, whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.
"Reciprocally proportional" means the heavier weight sits closer to the pivot, in the same ratio. Twice the weight, half the distance. In modern words, $W_1 d_1 = W_2 d_2$.
Worked example: Balance the beam
- Weight times distance must match: $6 \times 2 = 3 \times d$.
- $12 = 3d$, so $d = 4$ m.
Answer: 4 m: half the weight needs twice the distance.
Plutarch, writing about 300 years later, says Archimedes told King Hiero that with enough force any weight could be moved, and showed it by pulling a loaded ship ashore with a set of pulleys, by himself. A later saying gives him the words "give me a place to stand and I will move the Earth." The exact words are not in his own books, but the idea is his law of the lever.
Why things float
In On Floating Bodies he proved what happens to a solid put in water.
A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced.
Worked example: Weighed in water
- It lost $50 - 30 = 20$ N of weight.
- By Archimedes' rule, that loss is the weight of the water displaced.
Answer: 20 newtons of water, about 2 liters.
The crown: what we know and what is a story
The most famous Archimedes story is told by the Roman architect Vitruvius, about two hundred years after Archimedes died. King Hiero gave a goldsmith gold for a crown, then heard that some gold had been swapped for silver. Here is how Vitruvius tells what happened next.
The latter, while the case was still on his mind, happened to go to the bath, and on getting into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, without a moment's delay, and transported with joy, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, "Eureka, eureka."
In Vitruvius' version, Archimedes dropped the crown, a lump of pure gold of the same weight, and a lump of silver of the same weight into a brimming vessel, and measured the water that spilled. Silver is less dense than gold, so the same weight of silver takes up more room. The crown spilled more water than the gold lump, so it held some silver.
Story or history?
The bath and the run through the streets are a legend: no writing by Archimedes mentions a crown, and Vitruvius lived two centuries later. Many historians also doubt the spill method as told, because the difference in spilled water would have been tiny for a crown. Some suggest Archimedes would rather have weighed the crown against gold while both hung in water, using his own law of floating bodies. The idea, that a mixture of metals takes up a different volume than pure gold of the same weight, is sound.
Machines of war, and the end
When Roman forces under Marcellus attacked Syracuse during the Second Punic War, ancient writers say Archimedes' catapults and cranes held them off for a long time. Polybius, Livy and Plutarch all describe the machines. The story that he set ships on fire with mirrors appears only in much later writers, and modern tests have not settled whether it could work; treat it as a legend. Syracuse fell in 212 BC, and Archimedes was killed by a Roman soldier. Plutarch gives several versions of how it happened and says Marcellus was grieved by it.
| When | What happened |
|---|---|
| c. 287 BC | Born in Syracuse, Sicily |
| 3rd century BC | Writes On the Equilibrium of Planes, On Floating Bodies, Measurement of a Circle, The Sand-Reckoner and other works |
| 214–212 BC | Roman siege of Syracuse; his machines help defend the city |
| 212 BC | Syracuse is taken; Archimedes is killed |
| c. 20 BC | Vitruvius tells the story of the crown and "Eureka" |
| 75 BC | Cicero, in Sicily, finds Archimedes' overgrown tomb, marked with a sphere and a cylinder |
Worked example: Counting years across BC
- Both years are BC, so the later year is the smaller number.
- $212 - 75 = 137$ years.
Answer: 137 years.
What it led to
Archimedes' law of the lever is the start of the science of statics, which every bridge and crane designer uses. His rule for floating bodies explains why a steel ship floats: its hollow shape pushes aside a great deal of water. Galileo, eighteen centuries later, admired him above all the ancient writers and built on his methods.
Common mistake
Read the source on the Great Books Ladder
- Archimedes: The Works: read the lever and floating-body proofs in Heath's translation
Read more: Archimedes.