Archimedes' Quadrature of Parabolic Segments

Archimedes was an ancient Greek mathematician, scientist and inventor. He lived from circa 287 BC – 212 BC but is still considered one of the greatest mathematicians of all time. Portrait of Archimedes

The Quadrature of the Parabola was one of his proofs. It states that the area of a parabolic segment (a region bound by a parabola and a line) is 4/3 that of a certain inscribed triangle.

Interestingly, Archimedes made this discovery over 2,000 years before Isaac Newton and Gottfried Leibniz developed integral calculus. Furthermore, finding the area under a curve solved the problem of finding volumes of cylinders and spheres, which was essential for fair trade.

Archimedes used the Method of Exhaustion, which finds the area of a curved shape by inscribing successively smaller triangles until the shape is filled.

A Diagram Illustrating Archimedes' Quadrature of Parabolic Segments

To create a parabolic segment, we draw a line through a parabola, such that it intersects it in two points.

Here you can see that the filled in area is the parabolic segment, bound by the parabola and the line

A triangle is inscribed such that the third point of the triangle is halfway between the other two points

We then repeat this process by inscribing smaller triangles until the curved shape is filled

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The Definition of this Principle above