Archimedes: Greek Genius of the Ancient World

Archimedes of Syracuse, one of the greatest mathematicians and inventors of antiquity, profoundly influenced fields ranging from geometry to hydrostatics. Born in 287 BCE in Syracuse, a Greek city-state on the island of Sicily, Archimedes lived during a period of scientific discovery and political turbulence. His contributions not only advanced the knowledge of his time but also laid the foundations for future innovations, impacting fields as diverse as engineering, physics, and military strategy.

Terry Bailey explains.

Portrait of a scholar (Archimedes?). Domenico Fetti, 1620.

Early life and education

Archimedes was born into a family connected to the ruling elite of Syracuse. In his work 'The Sand-Reckoner', Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known, it is assumed this is what likely sparked Archimedes' early interest in the sciences. Little is known about Archimedes' early years, but it is believed he travelled to Alexandria in Egypt, the intellectual hub of the Mediterranean world, to study under the successors of Euclid. This connection with Alexandria placed Archimedes within the vibrant mathematical community of his time.

Upon returning to Syracuse, Archimedes dedicated himself to a life of inquiry and discovery. His surviving written work indicates a keen desire to explore both the abstract principles of mathematics and their practical applications. He had a deep love for pure mathematics but was equally committed to solving real-world problems.

Mathematical contribution

Archimedes' work in mathematics established him as a towering figure in the discipline. One of his most celebrated achievements is the approximation of pi (π). Archimedes developed a method of calculating the value of π by inscribing and circumscribing polygons around a circle, finding their perimeters and using this data to estimate π with remarkable precision. His method, known as the method of exhaustion, is considered an early form of integral calculus and remained the most accurate approximation of π until modern times.

Another significant contribution is Archimedes' principle of the lever. In his treatise 'On the Equilibrium of Planes', Archimedes laid down the laws governing levers and balance. His famous statement, "Give me a place to stand, and a lever long enough, and I will move the Earth, (World)," captures the essence of his discovery that a small force, applied correctly using a lever, can move large objects. This principle revolutionized mechanics and influenced engineering practices for centuries.

Archimedes also made strides in understanding the concept of centroids in geometry. In his work, 'On the Quadrature of the Parabola', he calculated the area of a parabolic segment and demonstrated that the centroid of a parabola lies along its axis. His ability to combine abstract reasoning with physical insight was unparalleled in the ancient world.

Hydrostatics and the principle of buoyancy

Archimedes is perhaps best known for his work in hydrostatics, particularly his discovery of the principle of buoyancy. This principle states that any object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. The story of Archimedes' Eureka, (Greek: εὕρηκα, I have found), moment is legendary, even if not exactly what occurred.

King Hiero II of Syracuse asked Archimedes to determine whether a crown made for him was pure gold or adulterated with silver. While bathing, Archimedes realized that the water displaced by his body in the bath provided a way to measure the volume of the crown and thus its density. Leaping from his bath in excitement, Archimedes is said to have run through the streets shouting Eureka!, I have found it!

Archimedes' principle of buoyancy became a cornerstone of fluid mechanics, underpinning theories of flotation and sinking. This discovery had practical applications in shipbuilding, as it allowed engineers to calculate whether ships would float based on their weight and the water they displaced.

Inventions and engineering feats

Archimedes' inventive genius was not limited to theoretical work; he also designed practical machines that were ahead of their time. One of his most famous inventions is the Archimedean screw, a device used to raise water. The screw consists of a helical surface inside a cylinder, and when turned, it would lift water from a lower to a higher elevation. Originally designed for irrigation in Egypt, the Archimedean screw is still used today in some parts of the world for pumping water.

In the realm of warfare, Archimedes applied his knowledge of mechanics to develop powerful machines that helped defend Syracuse during the Second Punic War. His military devices were so effective that they became the stuff of legend. Among these were massive catapults capable of hurling projectiles at enemy ships, and large cranes or "claws" that could grab ships, lift them out of the water, and then smash them back down, sinking them. Another invention, known as the Archimedes' heat ray, allegedly used mirrors to focus sunlight onto Roman ships, setting them on fire. While the historical accuracy of the heat ray remains debated, it highlights the blend of scientific theory and military necessity in Archimedes' work.

The method of mechanical theorems

One of Archimedes' lesser-known yet most significant contributions to mathematics is the method of mechanical theorems, detailed in his work of the same name. In this treatise, Archimedes explored the use of mechanical reasoning to discover mathematical theorems. By imagining geometric figures as physical bodies with weight, he was able to "balance" them and deduce geometric relationships. This approach was revolutionary because it introduced an early form of integral calculus long before Newton and Leibniz developed the field in the 17th century.

Archimedes' spiral—a curve traced by a point moving uniformly away from a central point while the point revolves around that point—was another breakthrough. He used this spiral to square the circle, showing that the area under one turn of the spiral was equal to a quarter of a circle's area. His work on spirals remains relevant in modern mathematics, particularly in polar coordinate systems.

The Sand Reckoner, challenging the size of the universe

Archimedes' curiosity extended beyond the Earth and into the cosmos. In his treatise 'The Sand Reckoner', he set out to determine whether it was possible to calculate the number of grains of sand that could fill the known universe. While this might seem like a trivial exercise, Archimedes used it as a way to propose a new system for expressing large numbers, challenging the conventional Greek numeral system. His estimate pushed the boundaries of ancient cosmology and provided a method for handling astronomical magnitudes, illustrating his unique ability to connect mathematics with natural philosophy.

Archimedes' death and legacy

The legacy of Archimedes is indelibly tied to his tragic death during the Siege of Syracuse in 212 BCE. The Romans, led by General Marcus Claudius Marcellus, laid siege to the city, and Archimedes played a critical role in designing machines that delayed the Roman advance. According to Plutarch, Archimedes was so engrossed in his work during the siege that he was oblivious to the Roman army entering the city. A Roman soldier found Archimedes working on a mathematical problem, allegedly Archimedes responded, "Do not disturb my circles," referring to the geometric figures he was drawing in the sand, the soldier, either misunderstood or ignored the orders, killing Archimedes on the spot, despite orders from Marcellus that Archimedes was to be spared.

Archimedes' contributions to science and engineering left an enduring legacy. His surviving works were studied, copied, and transmitted through the centuries, influencing Islamic scholars in the medieval period and later the European Renaissance. His method of approximation and mechanical theorems paved the way for the development of modern calculus, while his inventions demonstrated the power of applied science. Archimedes' influence can be seen in the works of later astronomers and mathematicians such as Galileo Galilei, Isaac Newton, and even physicists mathematicians and engineers of today.

Rediscovery of Archimedes' works

Many of Archimedes' original writings were lost over time, but his ideas and concepts survived through Latin and Arabic translations. One of the most exciting discoveries related to Archimedes' work came in 1906 when the Archimedes Palimpsest was uncovered. This medieval manuscript contained previously unknown works by Archimedes, including The Method of Mechanical Theorems. The palimpsest was produced when monks in the 13th century scraped off the original Archimedean text and reused the parchment for religious writings. Modern technology has allowed scholars to recover the erased text, providing new insight into Archimedes' mathematical genius.

Archimedes stands as a beacon of human ingenuity and intellectual curiosity. His profound insight into mathematics, physics, mechanics, and applied science such as hydrostatics continues to resonate today, demonstrating how a single individual's quest for knowledge can shape the course of human history.

From his invention of the Archimedean screw to his principles of levers and buoyancy, Archimedes' achievements spanned both theoretical brilliance and practical applications. His life and work exemplify the synergy between scientific inquiry and the real-world problems that science seeks to solve, making him one of the most remarkable figures of the ancient world.

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Notes:

Surviving writing of Archimedes:

·       Measurement of a Circle

·       The Sand Reckoner

·       On the Equilibrium of Planes

·       Quadrature of the Parabola

·       On the Sphere and Cylinder

·       On Spirals

·       On Conoids and Spheroids

·       On Floating Bodies

·       Ostomachion, (Loculus of Archimedes or Archimedes' Box)

·       The cattle problem

·       The Method of Mechanical Theorems

·       Book of Lemmas or Liber Assumptorum is a treatise with 15 propositions on the nature of circles

·       Archimedes' Palimpsest described above includes treatises:-

·       On the Equilibrium of Planes

·       On Spirals

·       Measurement of a Circle

·       On the Sphere and Cylinder

·       On Floating Bodies

·       The Method of Mechanical Theorems

·       Stomach-ion

·       Speeches by the 4th century BC politician Hypereides

·       A Commentary on Aristotle's Categories

·       Associated other works

Euclid

Euclid (Εὐκλείδης) was a Greek mathematician, often referred to as the "father of geometry." He lived around 300 BCE, during the reign of Ptolemy I in Alexandria. His most famous work, Elements, is a comprehensive compilation of the mathematical knowledge of his time and became the foundational text for geometry for centuries.

Elements consist of 13 books covering geometry, number theory, and mathematical logic. Euclid also contributed to optics, astronomy, and the study of conic sections. His work laid the groundwork for deductive reasoning in mathematics, a method still used today.

What was the 213-212 BC Siege of Syracuse?

The Siege of Syracuse (213–212 BCE) was a pivotal event during the Second Punic War (218-201 BCE), marking one of the most famous confrontations between the Roman Republic and the Kingdom of Syracuse in Sicily. This siege is notable not only for the Roman military efforts but also for the significant role that the famed Greek mathematician and inventor Archimedes played in the city's defense. The strategic importance of Syracuse and the unique defensive technologies invented by Archimedes made this a defining moment in history.

Terry Bailey explains.

Hiero of Syracuse calls Archimedes to fortify the city. By Sebastiano Ricci.

The Context of the Siege

By 213 BCE, the Second Punic War was raging, and Rome was embroiled in a life-or-death struggle with the Carthaginian general Hannibal, who had invaded Italy. Syracuse, a wealthy and strategically important city-state, had been a Roman ally but shifted its allegiance to Carthage after the death of its ruler, King Hiero II. His grandson, Hieronymus, had initially taken the throne and allied Syracuse with Carthage. However, after a brief reign, Hieronymus was assassinated, leading to a power struggle and the eventual rise of a pro-Carthaginian faction within the city.

Rome, determined to bring Syracuse back into its fold, sent a large force under the command of Marcus Claudius Marcellus in 213 BCE. Marcellus' goal was to lay siege to the city, retake it for Rome, and neutralize its potential as a Carthaginian stronghold in the Mediterranean.

Archimedes, widely considered one of the greatest scientific minds of antiquity, played a crucial role in the defense of Syracuse. Although his reputation today rests largely on his contributions to mathematics and physics, during the siege, he demonstrated his genius in military engineering. According to historical accounts, most notably from the Roman historian Livy and the Greek historian Polybius, Archimedes designed a series of war machines that helped defend the city against the Roman attacks.

The Roman forces initially underestimated the difficulty of capturing Syracuse. They planned to use a combination of a naval blockade and a land assault to breach the city's formidable defenses. However, Archimedes' inventions dramatically hindered these efforts.

Archimedes' War Machines

1.   The Claw of Archimedes ("The Iron Hand"), one of the most famous of Archimedes' inventions was a massive crane-like device known as the "Claw of Archimedes" or the "Iron Hand." It was designed to defend the city against naval attacks. The Claw consisted of a long arm attached to a series of ropes and pulleys, which extended over the city walls. When Roman ships approached the city, the Claw could be lowered into the water to grasp the hull of the ship. Once securely attached, the mechanism would lift the ship out of the water and violently shake or drop it, often capsizing the vessel. This machine struck fear into the Roman sailors, as ships were unexpectedly lifted and destroyed by an invisible force.

2.   Catapults and Ballistae, Archimedes also designed advanced versions of traditional siege weapons like catapults and ballistae. These machines were used to launch massive stones and projectiles at Roman ships and troops from a great distance. What made these weapons especially effective was their accuracy and the ability to fire at varying ranges, depending on the size of the projectile. Archimedes reportedly calculated the optimal angles and trajectories for launching these missiles, maximizing their impact.

3.   The Burning Mirrors (Archimedes' Heat Ray) perhaps the most legendary of Archimedes' supposed inventions was a device designed to set Roman ships ablaze using concentrated sunlight. Ancient sources, particularly later accounts from writers like Galen and Anthemius of Tralles, describe Archimedes using large, polished bronze or copper mirrors to focus sunlight onto the sails of Roman ships, causing them to catch fire. Although this story has long been debated among historians and scientists—some suggesting it was more myth than reality—it has endured as part of the lore surrounding Archimedes' genius. Modern experiments have attempted to recreate the "heat ray" with varying degrees of success. While it is unlikely that Archimedes' mirrors were responsible for destroying entire ships, they may have played a psychological role in the defense by intimidating Roman forces, even if simply projecting concentrated sunlight onto the Roman ships.

4.   Defensive Walls and Siege Countermeasures, Beyond his more dramatic inventions, Archimedes also contributed to the reinforcement of Syracuse's defensive walls and the city's overall preparedness for siege warfare. He designed mechanisms for rapidly reinforcing weak points in the walls and designed traps that could be triggered when Roman forces attempted to scale or breach them. These countermeasures significantly delayed Roman progress and prevented the besieging army from quickly overwhelming the city's defenses.

The Roman Response and the Fall of Syracuse

Despite the effectiveness of Archimedes' war machines, the siege dragged on for two years. Marcellus and his legions were frustrated by their inability to break through the city's defenses. Over time, the Roman commander recognized that a direct assault would continue to be costly, so he shifted his tactics. He tightened the naval blockade and waited for a moment of opportunity.

In 212 BCE, that opportunity came. The Roman forces exploited a lapse in vigilance among the defenders. During a festival honoring the goddess Artemis, part of the Roman army managed to scale the city walls under cover of night and opened the gates from within. Roman soldiers poured into the city, and Syracuse fell to the invaders.

Tragically, Archimedes' life ended during the sacking of Syracuse. According to historical accounts, Marcellus had given explicit orders that the scientist was to be captured alive, likely due to his immense knowledge and the potential value he held for Rome. However, during the chaos of the city's capture, a Roman soldier encountered Archimedes, who was reportedly engrossed in a mathematical problem. When the soldier ordered Archimedes to follow him, the mathematician allegedly responded, "Do not disturb my circles," referring to the geometric figures he was drawing in the sand. The soldier, either misunderstanding or ignoring the orders, killed Archimedes on the spot.

Marcellus was reportedly dismayed upon learning of Archimedes' death, and he ensured that the scientist received full funeral honors.

Aftermath and Legacy

The fall of Syracuse was a significant victory for Rome during the Second Punic War. It removed a major ally of Carthage in the Mediterranean and secured Sicily as a Roman province. Marcellus' triumph over the city earned him the nickname "The Sword of Rome." However, the legacy of the siege is forever intertwined with the genius of Archimedes. His war machines, whether fully real or partially mythologized have captured the imagination of generations.

Archimedes' contributions to science, engineering, and mathematics continue to be studied to this day. His work in mechanics, particularly the understanding of levers and pulleys, laid the foundation for centuries of technological development. The siege of Syracuse demonstrates not only the practical applications of his genius but also the tragic loss of knowledge that sometimes accompanies war.

In the centuries following the siege, Archimedes became a symbol of the intersection between science and warfare, showing how intellect could alter the course of battle. The inventions attributed to him are part of the broader history of ancient military engineering, influencing Roman and later Byzantine defensive technologies.

In conclusion, the Siege of Syracuse stands as a testament to both the military prowess of Rome and the extraordinary intellect of Archimedes. While the Roman victory secured Syracuse and furthered their dominance in the Second Punic War, it was Archimedes' remarkable contributions to the city's defense that left an indelible mark on history.

His innovative war machines not only delayed the inevitable fall of Syracuse but also showcased the power of scientific ingenuity in warfare. The ultimate capture of Syracuse by Rome marked a turning point in the Second Punic War, weakening Carthaginian influence and securing Sicily as a critical province for the Roman Republic.

However, the siege also highlights the tragedy of war, symbolized by the death of Archimedes, one of antiquity's greatest minds. His mathematical and engineering brilliance, demonstrated during the siege, became a source of fascination for later generations, blending fact and legend.

Archimedes' inventions—whether fully realized or part of myth—illustrate the potential for scientific thought to shape the course of history.

His understanding of mechanics, optics, and physics laid the groundwork for future developments in science and engineering, while his role in the defense of Syracuse remains a symbol of how intellect can influence the outcomes of even the most formidable military struggles. Ultimately, the Siege of Syracuse serves as a reminder of the enduring legacy of Archimedes and the often bittersweet intersection of science, war, and human endeavor.

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Notes

Sicily was originally taken from Carthaginian control during the First Punic War (264–241 BCE) and became the first province of the Roman Republic. The Kingdom of Syracuse was an allied independent region in the southeast of the island and a close ally of Rome during the long reign of King Hiero II

Point of interest:

One of the most interesting anecdotes concerning Archimedes and the King of Syracuse is the story of him running down the street naked shouting Eureka, (Greek: εὕρηκα, I have found).

Archimedes noticed while taking a bath that the level of the water in the tub rose as he got in, thus realizing that this effect could be used to determine the golden crown's volume, which was a problem the king had set him to discover if the metal worker who had made the crown was cheating the king out of gold by mixing fewer precious metals.

The water is incompressible, therefore when the crown was submerged it would displace an amount of water equal to its volume, thereby, dividing the mass of the crown by the volume of water displaced, and its density could be obtained.

Needless to say, if less precious and less dense metals had been added, the density would be lower than that of gold. This is what Archimedes discovered thus proving that silver had been mixed in with the gold and the metal worker was indeed cheating the king out of gold.