Archimedes was an ancient Greek mathematician, physicist, engineer, inventor, and astronomer who lived in Syracuse, a Greek city-state in Sicily, during the 3rd century BC. He is considered one of the greatest and most influential mathematicians of all time. Some key details about his life and work:


- Archimedes was born around 287 BC in Syracuse, and died in 212 BC during the Second Punic War when Syracuse was conquered by Roman general Marcus Claudius Marcellus. Archimedes was killed by a Roman soldier despite orders to capture him unharmed due to his valuable scientific knowledge.


- His father, Phidias, was an astronomer which likely influenced Archimedes' interest in mathematics and mechanics from a young age. Not much else is known about his family background or early life.


- Archimedes studied in Alexandria, Egypt which was a great center of scholarship and the foremost Greek city for mathematics and science at the time. He likely studied with successors of Euclid and learned a great deal of geometry and other mathematics while there.


- Once back in Syracuse, Archimedes devoted his life to research and made many groundbreaking discoveries in math and physics. However, he viewed practical engineering applications with as much interest as pure mathematical inquiries.


- His key contributions include calculating an approximate value for pi, developing a method for determining volumes and surface areas of solids from immersing them in fluids, formulating explanations for the principles of levers and pulleys, creating innovative machines of war to defend Syracuse, and making observations and theories about astronomy.


- The story goes that Archimedes discovered the concept of water displacement and buoyant forces when he stepped into a bath and noticed the water level rising. This inspired him to formulate the famous Archimedes Principle regarding buoyancy which proved very useful in his later work.


- Another famous legend is when the king asked Archimedes to determine whether a newly crafted crown was pure gold without damaging it. Weighing the crown was inconclusive, but while stepping into a bath, Archimedes supposedly realized he could submerge the crown and measure the precise amount of water displacement to calculate its density and composition. He was so excited that he immediately ran home naked yelling "Eureka!" (I have found it!).


- Archimedes made many important discoveries in geometry as well, establishing formulas for calculating the volumes of spheres, cylinders, pyramids and cone-shaped figures, as well as the surface areas of spheres and cylinders. He showed great ingenuity in enclosing solids between circular and triangular boundaries to better approximate their areas.


- He was also a pioneer in infinitesimals and limits, applying the method of exhaustion to effectively integrate areas and volumes by approximation through summing an infinite sequence of areas or volumes of enclosing shapes. This was a key step toward integral calculus.


- In physics, Archimedes elucidated the principles behind levers, pulleys and compound pulley systems. He analyzed balances and centers of gravity. One treatise entitled On Floating Bodies laid out what later came to be known as Archimedes' Principle regarding the mechanics of buoyancy.


- Fascinated by astronomy as well, Archimedes formulated theories to explain the movement of the planets which may have influenced Copernicus's later heliocentric model. He also studied the nature of spheres and circles related to astronomical phenomena.


- Apart from his brilliant mind as a theoretician, Archimedes also had a profoundly inventive and practical side. During the Second Punic War when Rome attacked Syracuse by sea, Archimedes developed a wide array of mechanical war machines such...