What were the contributions of Pythagoras and Archimedes?

Pythagoras is best known for the Pythagorean theorem, which establishes the relationship between the sides of a right triangle, and for his contributions to mathematics, philosophy, and Music Theory, emphasizing the importance of numbers in understanding the universe. Archimedes, a renowned mathematician and inventor, made significant advancements in geometry, particularly in calculating areas and volumes of shapes, and is famous for the Archimedes' principle regarding buoyancy. Both figures laid foundational concepts that influenced later developments in mathematics and science.