Archimedes of Syracuse (c. 287 – c. 212 BCE) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from Syracuse, Sicily. His life details are sparse, but his surviving works establish him as a prominent figure in classical antiquity and one of the greatest mathematicians. Archimedes made significant contributions to geometry, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola, employing the method of exhaustion and concepts of infinitesimals. He approximated pi (π), defined the Archimedean spiral, and developed a system for expressing large numbers through exponentiation.In the realm of physics, Archimedes applied mathematics to physical phenomena, formulating the law of the lever and the principle of buoyancy, known as Archimedes' principle. He conducted astronomical measurements, including the apparent diameter of the Sun, and is credited with constructing a planetarium device to model celestial movements. Archimedes also designed innovative machines, such as the screw pump and various war machines to defend Syracuse from Roman invasion.Archimedes died during the siege of Syracuse, reportedly killed by a Roman soldier despite orders for his protection. Cicero later described visiting his tomb, which featured a sphere and cylinder, symbols of his mathematical achievements. Although his inventions gained recognition, his mathematical writings were less known in antiquity. They were first compiled comprehensively around 530 CE by Isidore of Miletus, with Eutocius' commentaries later broadening their readership. Archimedes' works were translated into Arabic in the 9th century and into Latin in the 12th century, influencing scientists during the Renaissance and the Scientific Revolution. The discovery of the Archimedes Palimpsest in 1906 revealed lost works and insights into his methodologies.Archimedes was born around 287 BCE, likely in Syracuse, a self-governing colony in Magna Graecia. His father, Phidias, was an astronomer, but little else is known about his family. Plutarch noted a possible connection to King Hiero II of Syracuse, though other sources suggest he may have come from humble origins. His educational background remains uncertain, but he maintained correspondence with scholars in Alexandria, indicating a network of intellectual exchange.One notable story involves the 'wreath problem,' where Hiero II suspected a goldsmith of fraud. Archimedes discovered a method to determine the purity of a crown by measuring water displacement, famously exclaiming 'Eureka!' upon his realization. Another account describes his involvement in the design of the Syracusia, a large ship, where he demonstrated the principle of leverage to move heavy objects.During the Siege of Syracuse in 214 BCE, Archimedes reportedly oversaw the defense of the city using advanced war machines, including improved catapults and cranes. Accounts of his use of 'burning mirrors' to set Roman ships ablaze are less substantiated and debated among historians. His death occurred amidst the chaos of the siege, with various accounts detailing his final moments, often emphasizing his focus on mathematical problems even in peril.Archimedes' mathematical contributions include the method of exhaustion, which he used to derive the area of a circle and the volumes of various geometric shapes. He also developed a system for representing large numbers in his work, The Sand Reckoner, and posed complex problems, such as the Cattle Problem, which involved solving simultaneous equations. His treatises, including On the Sphere and Cylinder and Quadrature of the Parabola, showcase his innovative approaches to geometry and mechanics.Many of Archimedes' works have been lost or exist only in fragments. The Archimedes Palimpsest, discovered in 1906, contains several of his treatises, including the only surviving copy of On Floating Bodies. This palimpsest has provided valuable insights into his mathematical techniques and theories. Archimedes' legacy endures as a foundational figure in mathematics and science, influencing generations of scholars and researchers.