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Archimedes: the story on HearLore | HearLore
Archimedes
Archimedes of Syracuse, born around 287 BC, was a figure so formidable that a Roman general later called him a geometrical Briareus, a hundred-handed giant of intellect. While history remembers him for the golden crown and the burning mirrors, his true legacy lies in a mind that bridged the gap between abstract thought and physical reality centuries before such a connection was deemed possible. He was not merely a mathematician but a physicist, engineer, and astronomer who lived in the self-governing colony of Syracuse on the island of Sicily. His father, Phidias, was an astronomer, suggesting a household steeped in the stars, yet Archimedes himself remains a shadowy figure whose personal life is obscured by time. We do not know if he married or had children, nor can we confirm if he ever traveled to Alexandria, Egypt, despite his correspondence with scholars there. What we do know is that he maintained a collegial relationship with the great minds of his day, writing letters to Dositheus of Pelusium and Eratosthenes of Cyrene, men who were the intellectual giants of the Hellenistic world. His life spanned seventy-five years, ending in 212 BC, but his influence would stretch far beyond the borders of ancient Greece, shaping the trajectory of science for two millennia.
The Crown And The Bath
The story of the golden wreath is the most famous anecdote of his life, yet it may be a fabrication of later centuries. King Hiero II of Syracuse had commissioned a golden wreath for a temple, providing the pure gold himself, but suspected the goldsmith had substituted some silver and kept the rest for profit. Unable to make the craftsman confess, Hiero turned to Archimedes. According to the Roman architect Vitruvius, writing two centuries after the event, Archimedes stepped into a bath and noticed the water level rise as he sank lower. In a moment of epiphany, he realized that the volume of water displaced equaled the volume of his submerged body, and thus the volume of the crown. So excited was he that he ran naked through the streets of Syracuse crying Eureka, meaning I have found it. He then compared the crown to lumps of gold and silver of equal weight, showing that the crown displaced more water than the gold but less than the silver, proving the mixture. However, a different account from a fifth-century Latin poem suggests a more rigorous method involving a hydrostatic balance, where the crown and the gold were weighed in water to reveal the difference in density. Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes, considered this second method probable, noting its accuracy and its basis in the principles Archimedes himself had demonstrated in his treatise On Floating Bodies. The bathtub story, while enduring, may be a simplification of a complex scientific investigation into buoyancy and density.
Archimedes of Syracuse was born around 287 BC and died in 212 BC at the age of seventy-five years. His life spanned from the late 4th century BC to the early 3rd century BC during the Hellenistic period.
What happened to Archimedes during the Roman siege of Syracuse in 214 BC?
Archimedes constructed war machines including improved catapults and cranes to defend Syracuse against the Roman army under Marcus Claudius Marcellus starting in 214 BC. He was killed by a Roman soldier in 212 BC while drawing figures in the dust and refusing to leave his work.
How did Archimedes prove the goldsmith cheated King Hiero II of Syracuse?
Archimedes determined the volume of the golden wreath by measuring the water displaced when the object was submerged in a bath. He compared this displacement to that of pure gold and pure silver of equal weight to prove the goldsmith had substituted silver.
What mathematical method did Archimedes use to approximate the value of pi?
Archimedes employed the method of exhaustion in his work Measurement of a Circle to approximate pi by drawing polygons with 96 sides inside and outside a circle. This technique determined that pi lay between 3 and 3 1/7, a remarkably accurate approximation for his time.
When was the Archimedes Palimpsest discovered and what did it contain?
The Danish professor Johan Ludvig Heiberg discovered the Archimedes Palimpsest in 1906 at the Walters Art Museum in Baltimore, Maryland. The 174-page goatskin parchment contained 10th-century copies of previously lost treatises by Archimedes including On Floating Bodies and The Method of Mechanical Theorems.
When the Roman army under Marcus Claudius Marcellus laid siege to Syracuse in 214 BC, Archimedes stepped out of his study to defend his home. He had constructed war machines for King Hiero II during his lifetime but had never been given the chance to use them until the Romans arrived. Plutarch, Livy, and Polybius all testify to the effectiveness of these machines, which included improved catapults and cranes that could lift Roman ships out of the water with iron claws and drop them back in to sink them. The Romans were so terrified of these devices that they avoided the city walls, and Archimedes is said to have delayed their capture for years. A more improbable account, appearing centuries later, claims he used burning mirrors to focus sunlight and set ships on fire, a device known as Archimedes' heat ray. While modern researchers have attempted to recreate this effect with mixed results, the historical consensus leans toward the mechanical machines described by the earliest sources. The siege ended in 212 BC when the city finally fell. According to the oldest account from Livy, Archimedes was killed by a Roman soldier who did not know his identity. Plutarch adds that the soldier demanded Archimedes come with him, but the mathematician declined, saying he had to finish working on a problem. Another version has him carrying mathematical instruments that the soldier thought were valuable. His last words, often quoted as Do not disturb my circles, do not appear in any ancient source, but the sentiment is clear: he was killed while drawing figures in the dust, a moment of tragedy that marked the end of a life dedicated to the study of the infinite.
The Method Of Exhaustion
Archimedes' mathematical genius was not limited to physical machines; he pioneered techniques that anticipated calculus by two thousand years. In his work Measurement of a Circle, he employed the method of exhaustion, a technique previously used by Eudoxus of Cnidus to find the volume of a tetrahedron, cylinder, cone, and sphere. Archimedes took this method further, approximating the value of pi by drawing a larger regular hexagon outside a circle and a smaller one inside, then progressively doubling the number of sides of each polygon. After four steps, when the polygons had 96 sides each, he determined that pi lay between 3 and 3, a remarkably accurate approximation for his time. He also used this technique to prove that the area enclosed by a parabola and a straight line is 4/3 the area of a corresponding inscribed triangle, expressing the solution as an infinite geometric series. In On the Sphere and Cylinder, he obtained the result of which he was most proud: the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume of the sphere is 2/3 that of the cylinder, and the surface area is also 2/3. This proof was so significant to him that he requested a sphere inscribed in a cylinder be placed on his tomb, a request Cicero later confirmed when he visited the tomb in Syracuse. Archimedes' ability to calculate the area of an ellipse, the volume of a segment of a paraboloid of revolution, and the area of a spiral demonstrated a mastery of geometry that would not be surpassed until the development of calculus.
The Sand And The Stars
In The Sand Reckoner, Archimedes devised a system of counting based on the myriad, the Greek term for 10,000, to calculate a number greater than the grains of sand needed to fill the universe. He proposed a number system using powers of a myriad of myriads, concluding that the number of grains of sand required to fill the universe was 8 in modern notation. This treatise also mentions the heliocentric theory of the Solar System proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. Archimedes determined the Sun's apparent diameter by describing the procedure and instrument used to make observations, applying correction factors to account for observational error. He is the first known Greek to have recorded multiple solstice dates and times in successive years, a feat that would not be matched for centuries. In the Cattle Problem, he challenged the mathematicians at the Library of Alexandria to count the numbers of cattle in the Herd of the Sun, which involves solving a number of simultaneous Diophantine equations. A more difficult version of the problem requires some answers to be square numbers, and the answer is a very large number, approximately 7.760271. These works demonstrate Archimedes' ability to apply mathematics to physical phenomena, from the vastness of the universe to the smallest grains of sand, and his willingness to tackle problems that seemed impossible to his contemporaries.
The Lost Codex And The Palimpsest
For centuries, Archimedes' most revolutionary work, The Method of Mechanical Theorems, was thought to be lost forever. It was only in 1906 that the Danish professor Johan Ludvig Heiberg discovered the Archimedes Palimpsest, a 174-page goatskin parchment of prayers written in the 13th century. Heiberg confirmed that it was indeed a palimpsest, a document with text that had been written over an erased older work. The older works in the palimpsest were identified as 10th-century copies of previously lost treatises by Archimedes, including On Floating Bodies, The Method of Mechanical Theorems, and Stomachion. The palimpsest was sold at auction in 1998 for $2.2 million and stored at the Walters Art Museum in Baltimore, Maryland, where it was subjected to modern tests including the use of ultraviolet and infrared light to read the overwritten text. The discovery provided new insights into how Archimedes obtained mathematical results, revealing that he used a novel method, an early form of Cavalieri's principle, to rederive the results from his treatises sent to Dositheus. He used the law of the lever to find the center of gravity of an object first, and reasoned geometrically from there to more easily derive the volume of an object. This treatise was thought lost until the discovery of the Archimedes Palimpsest, which has since returned to its anonymous owner, but its contents have changed our understanding of Archimedes' genius.
The Legacy Of The Giant
Archimedes is often called the father of mathematics and mathematical physics, and historians of science and mathematics almost universally agree that he was the finest mathematician from antiquity. Gauss's heroes were Archimedes and Newton, and Moritz Cantor reported that he once remarked in conversation that there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein. Alfred North Whitehead said that in the year 1500 Europe knew less than Archimedes who died in the year 212 BC. The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri, meaning Rise above oneself and grasp the world. His influence extends to the modern world, from the SS Archimedes, the world's first seagoing steamship with a screw propeller, to the state motto of California, Eureka. There is a crater on the Moon named Archimedes, as well as a lunar mountain range, the Montes Archimedes. His work was translated into Arabic by Thābit ibn Qurra in the 9th century and into Latin via Arabic by Gerard of Cremona in the 12th century, and were an influential source of ideas for scientists during the Renaissance and in the Scientific Revolution. The discovery in 1906 of the Archimedes Palimpsest has provided new insights into how he obtained mathematical results, confirming that he was the most important scientist who ever lived.