Archimedes and the Golden Crown
by Rohini Chowdhury
For almost a hundred years, the ancient Greek city of Syracuse had been at war with Carthage, and riven by internal strife as successive rulers fought each other for the throne. Till, in 275 BCE, the Syracusan troops, tired of the inefficiencies of their leaders, elected commanders from amongst themselves. One of these was a young general called Hiero.
Now, Hiero had a natural flair and talent for leadership and politics. He managed, through his connections, to enter the city and take over its government, but so smoothly and efficiently, that the citizens of Syracuse, who usually did not approve of soldiers choosing their own commanders, did so in this case. Then, after a great battle in 265 BCE, in which Hiero led the Syracusans to victory against their enemies, the people of Syracuse chose Hiero to be their king.
Hiero was grateful to the gods for his success and good fortune, and to show his gratitude, he decided to place in a certain temple, a golden crown in their honour. The crown was to be shaped like a laurel wreath. Hiero weighed out a precise amount of gold, and appointing a goldsmith, commanded him to fashion out of the gold a wreath worthy of the gods.
The goldsmith did as he had been ordered, and on he appointed day, he delivered to the king an exquisitely wrought crown, shaped, as the king had ordered, like a laurel wreath. The wreath seemed to weight exactly as much as the gold that the king had given the goldsmith. Hiero was pleased, and paid the goldsmith handsomely. The goldsmith, receiving his payment, went away.
Hiero made preparations for the ceremony to place the wreath in the temple that he had chosen. But a few days before the ceremony, he heard rumours that the goldsmith had cheated him, and given him a crown not of pure gold, but of gold that had silver mixed in it. The goldsmith, said the rumours, had replaced some of the gold that Hiero had given him, with an equal weight of silver.
Hiero was furious to learn that he might have been tricked. But he was a fair-minded man and wished to determine the truth before he punished the goldsmith.
If the goldsmith had indeed cheated him and mixed silver into the gold, then the goldsmith would have to be punished, and the crown could no longer be given as an offering to the gods. But if the goldsmith had been honest, then the crown remained what it had been intended to be, a sacred offering, and it would be placed in the temple as planned. So it was important that Hiero find out the truth quickly, before the day fixed for the ceremony, and without damaging the crown in any way.
Hiero believed there was only one man in Syracuse capable of discovering the truth and solving his problem. This was his cousin, Archimedes, a young man of 22, who was already renowned for his work in mathematics, mechanics and physics.
Deep in thought, pondering how best to solve the king’s problem, Archimedes walked to the public baths for his daily bath. Still thinking about the golden crown, he went through the rituals of cleansing and washing, and stepped into a tub of cool water for his final dip. As he began to lower himself into the water, the water in the tub began to spill out over the sides. Curious, Archimedes continued to lower himself slowly into the water, and he noticed that the more his body sank into the water, the more water ran out over the sides of the tub. He realised that he had found the solution to Hiero’s problem.
He was so excited by his discovery that he jumped out of the tub at once, and ran all the way home without remembering to put his clothes on, and shouting ‘Eureka, Eureka!’ – which in Greek means, ‘I have found it! I have found it!’
What Archimedes had found was a method for measuring the volume of an irregularly-shaped object. He realised that an object, when immersed in water, displaced a volume of water equal to its own volume, and that by measuring the volume of the displaced water, the volume of the object could be determined, regardless of the object’s shape. So, he could measure the volume of the crown by measuring the volume of the water spilled from a container filled with water to the brim when the crown was fully dipped in it.
How then, would this realisation help him to answer Hiero’s question – had the goldsmith mixed silver in the golden crown or not? Let us see how Archimedes used his discovery to solve the king’s problem.
In physics, when we speak of the density of an object, we are comparing its mass with its volume, or, in simpler words, considering how heavy it is in relation to its size. For example, iron is denser than cork. So a lump of iron is much heavier than a piece of cork of the same size, or much smaller than a piece of cork of the same weight.
Archimedes knew that gold was denser than silver – so a piece of gold weighing a certain amount would be smaller than a piece of silver weighing the same:
Thus, if the goldsmith had stolen some of the gold the king had given him, and replaced it with an equal weight of silver in the crown, then the total volume of the gold+silver crown would be greater than the volume of the original amount of gold.
So now, all that remained for Archimedes to do was to compare the volume of the crown to the volume of the amount of gold that Hiero had given the goldsmith.
The simplest method of determining the volume of the crown would have been to melt it down, shape it into a cube and measure its volume. But Hiero had given strict instructions that the crown was not to be damaged in any way. So how was the volume to be determined? This is where Archimedes’ discovery came in useful.
First, Archimedes took a lump of gold and a lump of silver, each weighing exactly the same as the crown, and filled a large vessel with water to the brim, precisely measuring how much water was contained in the vessel.
He then gently lowered the lump of silver into it. This caused as much water to spill out over the sides of the vessel as was equal in volume to the lump of silver. Archimedes took the lump of silver out of the water and carefully measured the amount of water left in the vessel, thus arriving at the amount of water that had been displaced by the silver.
He again filled the vessel with water to brim, taking care to fill it with exactly the same amount of water as before. He then lowered the lump of gold into the water, and let the water displaced by it spill out over the sides. Then, doing as he had done with the lump of silver, Archimedes took out the lump of gold from the water, and arrived at the amount of water that had been displaced by the gold.
He found that a smaller quantity of water had been displaced by the gold than the silver, and the difference was equal to the difference in volume between a lump of gold and a lump of silver of the same weight.
He filled the bowl with water to the brim a final time, taking care to fill it with exactly the same amount of water as before. This time he lowered the crown into the water. He knew that if the crown was pure gold, its volume would be the same as that of the lump of gold (which he had made sure weighed the same as the crown), regardless of shape, and that it would displace the same amount of water as the gold. But, if the goldsmith had replaced some of the gold with silver, then the volume of the gold+silver crown would be greater than the volume of the gold, and so the crown would displace more water than the gold.
Archimedes found that the crown did, in fact displace more water than the lump of gold of equal weight. Thus he came to the conclusion that the crown was not pure gold, and that the goldsmith had indeed mixed some silver (or other, lighter metal) into the gold in an attempt to cheat the king.
This story of Archimedes and the golden crown is found in De Architectura or The Ten Books of Architecture, written by the Roman architect Marcus Vitruvius Pollo some time during the first century BCE. This story is not found anywhere among the known works of Archimedes, though in his book, On Floating Bodies, he gives the principle known as Archimedes’ Principle, which states that a body partially or completely immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.
The method that Vitruvius says was used by Archimedes, though correct in theory, has been criticised by scientists as too difficult to implement with the amount of accuracy that would be needed to detect a component of silver or other lighter metal in the crown. This is because that the amounts of gold and silver in the case of a crown would be so small that the difference in their volumes, and the consequent difference in the amount of water displaced, would be too small to measure with precision with the measurement methods available to Archimedes.
More than eighteen hundred years after Archimedes is said to have helped King Hiero detect the goldsmith’s fraud, another young man, also twenty-two years old at the time, pondered the same problem. This young man was Galileo Galilei, the Italian mathematician, physicist and astronomer. In 1586, Galileo wrote a short treatise called La Bilancetta, or The Little Balance, in which he expressed his scepticism of Vitruvius’ story and presented his own theory of how Archimedes might actually have detected the goldsmith’s dishonesty. He based his theory on the Archimedes Principle, and on Archimedes’ work on levers.
Galileo’s method is simple, yet precise and detailed, even determining the exact quantity of gold and silver (or a lighter metal) in the alloy. It is very likely that Archimedes detected the goldsmith’s fraud by a method similar to that described by Galileo. While not detailing Galileo’s treatise here, let me give a method, based on what Galileo says, that Archimedes might have used:
Instead of immersing the crown and an equal weight of gold in a vessel filled with water, Archimedes could have suspended the crown from one end of a pair of scales, balancing it with an equal amount of gold on the other end. Once equally balanced, he would have immersed the suspended crown and lump of gold into a vessel of water. Now, since a body immersed in water is buoyed up by a force equal to the weight of the water displayed by the body, the denser body, which has a smaller volume for the same weight, would sink lower in the water than the less dense one.
So, if the crown was pure gold, the scales would continue to balance even when immersed in the water. If the crown was not pure gold, and silver or a lighter metal had been mixed with the gold thus increasing its volume, then the scales would tilt towards the denser gold. And thus it would have been possible for Archimedes to find out quickly and simply, without damaging Hiero’s golden wreath in any way, whether the goldsmith had cheated the king or not.
More about Archimedes
Archimedes was a Greek mathematician, scientist and engineer, who lived in the ancient Greek city-state of Syracuse.
Very little is known of his personal life. He was born about 287 BCE in Syracuse. In one of his works, The Sand Reckoner, Archimedes says that his father was Phidias, an astronomer.
Except for a period spent in Alexandria, Egypt, where he studied under the followers of the mathematician Euclid, Archimedes spent his life in Syracuse. According to Plutarch, the ancient Greek historian and biographer, Archimedes was a distant cousin of Hiero II, the ruler of Syracuse. Hiero’s long reign was a period of peace and stability in Syracuse, and gave Archimedes the opportunity to pursue his work in peace. Hiero often turned to Archimedes for advice on military and other matters.
Archimedes is regarded as the greatest mathematician and scientist of his age, though only a few of his writings have survived into modern times. According to the Encyclopaedia Britannica, there are only nine known extant treatises in Greek by Archimedes.
Of these treatises, five are of particular interest:
On the Sphere and Cylinder (two volumes): this contains his discovery that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed, and that the surface area of a sphere is four times that of its greatest circle.
On the Measurement of the Circle: a short work which contains his approximation for the value of Pi. Archimedes showed that Pi lies between 223/71 and 22/7. The latter value was used throughout the Middle Ages and it is still used today when a rough calculation is required. This work also contains accurate approximations of the square roots of various numbers.
On Floating Bodies (two volumes): this is the first known work in hydrostatics (the study of liquids at rest), a branch of science of which Archimedes is considered the founder. This is the work that leads to the Archimedes Principle, which states that a body partially or completely immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.
The Method of Mechanical Theorems: this describes the process of discovery in mathematics. In it, Archimedes describes some of the ‘mechanical’ techniques he used to arrive at the values he proved mathematically in On the Sphere and Cylinder.
The Sand Reckoner: this is a small work, written for the layman. It deals with the inadequacies of the Greek numerical notation system, by showing how to express a very large number – the number of grains of sand that it would take to fill the universe. In doing this, he, in effect established a place-value system, with a base of 100,000,000. In this work, he also describes the theory put forward by the Greek astronomer Aristarchos of Samos, that the Sun is at the centre of the Universe, but dismisses it as ‘impossible’.
Archimedes’ other extant treatises are:
On the Equilibrium of Planes (two volumes)
On Spirals
On Conoids and Spheroids
The Quadrature of the Parabola
From references to him in the writings of other authors, we know that Archimedes wrote several more works, which have not survived. Among these is what is known as the Cattle Problem, which asks us to calculate the number of cattle owned by the Sun God Helios, starting from a few simple relations. Though no original work by Archimedes regarding this problem has survived, the problem is believed to have originated, at least partially, from him. This problem was finally fully solved in 1965, with the help of computers.
Other works attributed to Archimedes survive in Arabic translation. It is not certain though, whether they really were written by Archimedes, or are merely derived from or based on his work. These include the Stomachion (parts of which also survive in Greek), which describes a game or puzzle in which a square is divided into fourteen pieces of different shapes, that have to be rearranged to form other interesting shapes.
In the 10th century CE, several of Archimedes works were copied by an unknown scribe, most probably a monk, into a parchment codex or book. In the 12th century CE, the book was unbound and washed, and the parchment on which Archimedes’ works were copied was reused and rebound. The erasure was partial, and today, thanks to the power of modern technology, we can read what the monk had copied. This parchment is called the Archimedes Palimpsest (a palimpsest is a manuscript page, that has been written on, cleaned, and used again). This is not a separate work of Archimedes, but a collection of some of his works discussed above.
Archimedes is also called the father of integral calculus and of mathematical physics.
In his own time, Archimedes was famous not so much for his work in mathematics as for his inventions, which were many.
They included compound pulley systems, a planetarium showing the motions of the sun, moon, and planets as viewed from the earth, and a mechanism known as the Egyptian or Archimedes Screw, for raising water, which was used for by the Egyptians to raise water from streams and canals to irrigate their fields and by the Romans to pump water out of mines and the holds of ships. He also designed and built war machines that were used in the defence of Syracuse against its enemies.
Archimedes died in 212 BCE in Syracuse, during the sack of Syracuse by Roman forces who had finally captured the city after a two-year long siege. The Romans were led by Marcus Claudius Marcellus. Marcellus had given strict orders that Archimedes was not be harmed, but was to be brought to him with honour. But despite those orders, Archimedes was killed by a Roman soldier.
The precise details of his last moments are not known, though various accounts exist. According to some Roman historians, a Roman soldier, intent on looting, broke into his house, and demanded to know who he was. Archimedes, oblivious of the chaos around him, and absorbed in some diagrams he had traced in the dust, did not give his name, but shielding his drawings with his hands, begged the soldier not to disturb his work. The Roman soldier disregarded his plea and killed him.
Plutarch gives a slightly different account. He writes that a Roman soldier came up to Archimedes and commanded him to follow him to Marcellus. But Archimedes, in the middle of a mathematical problem, refused to follow until he had solved the problem. The soldier, enraged, ran him through with his sword. Plutarch offers an alternative version as well: he says that Archimedes, on his way to see Marcellus, and carrying with him his mathematical instruments, was killed by soldiers who thought he was carrying gold.
It is said that Archimedes last words were ‘Don’t disturb my circles.’
Plutarch goes on to say that Marcellus was greatly disturbed when he heard of Archimedes death, and declared the soldier who had killed him ‘a murderer’.
In 75 BCE, a hundred and thirty-seven years after the death of Archimedes, another Roman searched for Archimedes in Syracuse. This was Cicero, later famous as a statesman, lawyer, orator, writer and philosopher. Cicero managed to locate Archimedes’ grave, which he found overgrown with thorns and brambles. Archimedes, says Plutarch, had requested his friends that, when he died, to mark his tomb with a sphere inscribed inside a cylinder. Cicero searched for this, and sure enough, he located a grave marked by a little column surmounted by a sphere and a cylinder. He writes that the verses engraved on it were still visible when he found Archimedes tomb, though part of each line had been worn away.
And more about Galileo
Galileo Galilei was an Italian mathematician, astronomer and physicist.
He was born on February 15, 1564 in Pisa. His father, Vincenzio Galilei was a musician.
Galileo was educated at a monastery near Florence before he went on to study medicine at the University of Pisa. He also studied mathematics with a private tutor. At the age of twenty-five, he became a lecturer in mathematics at the University of Pisa. Later he went to the University of Padua as a professor in mathematics.
A significant part of Galileo’s work is related to mechanics (the study of motion and the forces producing motion). He was the first to apply mathematics to mechanics.
Galileo also suggested the use of the pendulum for clocks, and proposed the law of uniform acceleration for falling bodies. He also developed the astronomical telescope, with which he discovered craters on the Moon, sunspots, the phases of Venus, and the satellites of Jupiter. He also showed that the Milky Way was made up of stars.
His work in astronomy led him to support the theory put forward by the astronomer and mathematician, Nicolaus Copernicus, that the earth and planets revolved round the Sun. This was against the belief of the Roman Catholic Church, which held that the earth was the centre of the Universe and that the Sun revolved around the earth. The Church forced Galileo to recant, and placed him under house arrest for the last eight years of his life, for having believed and taught Copernicus’ theory. It was only in 1992 that the Church formally acknowledged its error.
Galileo’s major works were Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican (1632) and Dialogue Concerning Two New Sciences (1638).
Galileo died on January 8, 1642.
