Born: c. 287 BC · Syracuse, Sicily (then a Greek city-state)
Died: c. 212 BC (aged around 75) · Syracuse, Sicily
Era: Hellenistic period
Region: Greek mathematics and science
School: Alexandrian mathematics
Main Interests: Mathematics · Physics · Engineering · Astronomy · Mechanics · Invention
Notable Works: On the Sphere and Cylinder · On Floating Bodies · The Method of Mechanical Theorems
Notable Ideas: Archimedean principle (buoyancy) · Archimedean screw · Calculus precursor · Laws of levers · Pi approximation · Inventive war machines

Major contributions of Archimedes include:

• Invented the sciences of mechanics and hydrostatics.

• Discovered the laws of levers and pulleys.

• Formulated principles like Archimedes’ Principle and the law of the lever.

• Invented devices such as the Archimedes’ screw for lifting water.

• Revolutionized geometry and anticipated integral calculus.

Problem/events that influence Archimedes' mathematical ideas:

Archimedes' mathematical ideas were greatly influenced by a mix of practical challenges and intellectual curiosity. His work on mechanics and engineering was driven by the need to solve real-world problems, particularly in the defense of his hometown, Syracuse. Archimedes applied his mathematical knowledge to design innovative war machines, such as catapults and cranes, and invented the Archimedean screw for raising water. His fascination with geometry, especially the properties of circles, led him to develop methods for calculating pi and determining areas and volumes of spheres and cylinders. Archimedes was also inspired by the mathematical traditions of Egypt and Babylon, refining their ideas on geometry and number approximation. His study of mechanics, particularly the law of levers, stemmed from practical experiences in construction and warfare, leading him to formalize principles of mechanical advantage. The intellectual environment of Alexandria, where mathematicians like Euclid were active, also influenced his approach to mathematical rigor. Finally, his famous discovery of the buoyancy principle, inspired by a problem posed by King Hiero II, revolutionized fluid mechanics. These combined influences shaped Archimedes' groundbreaking contributions to mathematics, engineering, and physics.

His Impact on the modern mathematics: 

Archimedes' impact on modern mathematics is immense, as his work laid the groundwork for many foundational concepts in geometry, calculus, and physics. His development of the method of exhaustion, a precursor to integral calculus, allowed for the precise calculation of areas and volumes, influencing later mathematicians like Newton and Leibniz. Archimedes' work on pi, particularly his approximation methods, contributed significantly to the mathematical understanding of irrational numbers. His principles of buoyancy and the law of the lever continue to be central in fluid dynamics and mechanical engineering today. Furthermore, his inventive spirit, which combined theoretical mathematics with practical application, set a precedent for modern mathematical problem-solving and engineering design. Archimedes' legacy remains vital in fields ranging from calculus and geometry to modern physics and engineering.