Archimedes

Archimedes was born in the Greek city-state of Syracuse on the island of Sicily. His father, Phidias, was an astronomer, which likely influenced young Archimedes' early interest in mathematics and the natural sciences. Syracuse was a thriving center of Greek culture and learning, providing an intellectually stimulating environment. The city was ruled by King Hieron II, who would later become Archimedes' patron and friend. Growing up in this prosperous port city, Archimedes was exposed to both theoretical Greek mathematics and practical engineering challenges, a combination that would define his life's work.

Archimedes traveled to Alexandria to study at the famous Library and Museum, the greatest center of learning in the ancient world. There he studied with the successors of Euclid and likely met Conon of Samos and Eratosthenes, who became lifelong friends and correspondents. The Library's vast collection of scrolls and the vibrant community of scholars exposed him to the latest developments in mathematics, astronomy, and engineering. This period was formative in developing his rigorous mathematical approach and his ability to apply abstract principles to practical problems. The intellectual ferment of Alexandria shaped his methodology of combining theoretical proofs with empirical observations.

After completing his studies, Archimedes returned to Syracuse where King Hieron II appointed him as the court mathematician and engineer. This position gave him the freedom to pursue both theoretical research and practical inventions. He established a workshop where he could conduct experiments and build mechanical devices. His reputation for solving complex problems quickly spread throughout Syracuse. The king frequently consulted him on matters ranging from military defense to verifying the purity of gold crowns, showing the high regard in which his expertise was held.

According to legend, King Hieron asked Archimedes to determine if a crown was pure gold without damaging it. While taking a bath, Archimedes noticed the water level rise as he entered the tub. He suddenly realized that the volume of water displaced equals the volume of the object submerged, which could be used to determine the crown's density. Excited by this discovery, he allegedly jumped from the bath and ran naked through the streets shouting 'Eureka!' (I have found it!). This principle of buoyancy, now known as Archimedes' Principle, became fundamental to fluid mechanics and demonstrated his genius for connecting everyday observations to universal physical laws.

Archimedes invented the screw pump, now known as the Archimedes Screw, a device for raising water from a lower level to a higher level. The device consists of a screw inside a hollow pipe, which when turned, lifts water up through the tube. This invention was possibly developed during his time in Egypt to help irrigate fields from the Nile River, or to remove water from the holds of ships in Syracuse's harbor. The elegant simplicity and effectiveness of the design meant it remained in use for centuries and is still used today in some applications. This invention exemplified his ability to create practical solutions based on mathematical principles.

Archimedes made groundbreaking discoveries about levers, fulcrums, and mechanical advantage. He famously claimed, 'Give me a place to stand, and I shall move the Earth,' illustrating his understanding that with a long enough lever and a fulcrum, even enormous weights could be moved with small forces. To demonstrate this principle to King Hieron, he allegedly single-handedly launched a fully loaded ship using a compound pulley system, a feat that would normally require many men. His systematic study of simple machines laid the foundation for the science of mechanics. He proved mathematically that the distance from the fulcrum is inversely proportional to the weight, establishing the law of the lever.

Archimedes developed a rigorous method for calculating pi by inscribing and circumscribing polygons around a circle and calculating their perimeters. By using polygons with 96 sides, he proved that pi lies between 3 1/7 and 3 10/71 (approximately 3.1408 and 3.1429), which was the most accurate approximation of pi for over a thousand years. This work appeared in his treatise 'Measurement of a Circle.' His method of exhaustion, using progressively more sides on the polygons to approach the true value, was a precursor to modern integral calculus. This achievement demonstrated his mastery of both geometry and the concept of limits.

Archimedes completed his masterwork 'On the Sphere and Cylinder,' which contained his proudest discovery: that the volume and surface area of a sphere are two-thirds that of the cylinder in which it is inscribed. He provided rigorous geometric proofs for these relationships, considering this theorem so important that he requested a sphere inscribed in a cylinder be engraved on his tombstone. The work also contained his method of exhaustion to find areas and volumes, and included solutions to problems that would later require calculus. This treatise demonstrated the height of Greek geometric reasoning and remained influential throughout the medieval period and Renaissance.

Archimedes wrote 'The Sand Reckoner,' a treatise addressed to King Gelon II, in which he devised a system for expressing extremely large numbers, specifically to calculate how many grains of sand would fit in the universe. At a time when the Greek numerical system made large numbers difficult to express, he created a notation system capable of representing numbers up to 8×10^63. Beyond its mathematical innovation, the work also contained his estimate of the size of the universe based on the heliocentric model of Aristarchus of Samos. This work showed his interest in both pure mathematics and cosmology, and demonstrated that no quantity, however large, was beyond mathematical expression.

As Rome besieged Syracuse during the Second Punic War, Archimedes designed an array of ingenious war machines to defend his city. These included catapults of various sizes that could launch massive stones at ships, large cranes with grappling hooks (the 'Iron Hand' or 'Claw of Archimedes') that could lift enemy ships out of the water and capsize them, and allegedly a system of mirrors to focus sunlight and set Roman ships on fire. The Roman commander Marcellus reportedly said that Archimedes was using his ships as cups to ladle water from the sea. These machines terrified the Roman soldiers and delayed the capture of Syracuse for two years, earning Archimedes legendary status as an engineer.

When Syracuse finally fell to the Romans under General Marcellus, Archimedes was killed by a Roman soldier despite Marcellus' orders to capture him alive. According to legend, Archimedes was so absorbed in studying a mathematical diagram drawn in the sand that he told the soldier, 'Do not disturb my circles!' The irritated soldier killed him on the spot. Another account says he was killed while carrying mathematical instruments, which the soldier mistook for valuable items. Marcellus was reportedly deeply upset by Archimedes' death and ensured he received an honorable burial. As Archimedes had requested, his tomb was marked with a sphere inscribed in a cylinder, representing his greatest mathematical achievement. The death of Archimedes marked the end of an era of Greek mathematical brilliance.