Archimedes was a Greek mathematician who grew up in the city of Syracuse. He was born in 287 BC to Phidias the astronomer. When Archimedes was a teenager, Phidias sent his son to Alexandria to learn from the greatest minds of the ancient Hellenistic world. He is famous for discovering how to calculate the approximation of pi, (π), showing that it is greater than 223/71 and less than 22/7 and defining and investigating a spiral that now bears his name.
The infinite series[]
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola. In Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and whose third vertex is where the line that is parallel to the parabola's axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.