1, Archimedes' calculation method for determining the area of parabola bow, helix and circle, and the surface area and volume of complex geometric bodies such as ellipsoid and paraboloid. In the process of deriving these formulas, he founded the "exhaustive method", which is similar to the method of gradually finding the limit in modern calculus.

He is the first person to study pi scientifically. He proposed to calculate pi by increasing the number of sides and approaching the areas of inscribed polygons and circumscribed polygons. He calculated the range of pi: 223/7 1

3. Facing the tedious numerical representation in ancient Greece, Archimedes also pioneered the method of memorizing large numbers, which broke through the restriction that the number of Greek letters at that time could not exceed 10,000, and used it to solve many mathematical problems.

4. The famous Archimedes axiom is put forward and expressed in modern mathematical language. Archimedes principle means that for any natural number (excluding 0)a and B, if ab.

Astronomy:

1, he invented the planetarium driven by water conservancy, and used it to simulate the movements of the sun, planets and the moon and perform an eclipse of the sun and the moon;

2. He thinks that the earth is spherical and revolves around the sun, which is earlier than Copernicus' Heliocentrism 1800 years. Limited by the conditions at that time, he did not make a thorough and systematic study on this issue.