**Archimedes and the quadrature of the parabola**

**Archimedes of Syracuse** (c. 287–212 BC) was a Greek mathematician, scientist and engineer. He is widely regarded as one of the greatest mathematicians of all time.

One of Archimedes’ works was called *The Quadrature of the Parabola*. This proved various results about parabolas, and explained how to find the area of a *parabolic segment*, which is a finite region enclosed by a parabola and a line. This is easy to do nowadays using the well-known theory of integral calculus, but this was not developed until the 17th century, about 1900 years after the time of Archimedes.

Integral calculus calculates areas by approximating the area to be measured by a union of geometric shapes whose exact areas are known, and then applying a limiting process. Archimedes’ technique was very similar to this. The key to his idea was to inscribe into the parabolic segment a triangle with the same base and height. In other words, the triangle had the original line segment as its base, and touched the curved part of the parabola at the point where the tangent line to the parabola was parallel to the line segment. Archimedes proved that if the triangle has area T, then the area A of the parabolic segment was given by 4T/3.

Archimedes described a method of filling up the rest of the parabolic segment by exhaustion, using smaller and smaller triangles. The graphic shows two lighter blue triangles, four yellow triangles, eight (barely visible) red triangles, and so on. There are twice as many triangles of each successive colour as there were of the previous colour. Archimedes proved that the area of a triangle of each successive colour is 1/8 of the area of the previous type of triangle, although this is not an obvious result. For example, each light blue triangle has an area of T/8.

These observations reduce the problem of finding the area A to evaluating the sum at the bottom of the picture, which is a geometric series. Nowadays, there is a well-known formula that applies in this situation, but Archimedes summed the series using a clever ad hoc geometric argument instead.

Archimedes made some other very significant discoveries using integration-like methods. He proved that the area of a circle of radius r is equal to πr^2, and he also discovered the formulae for the surface area and volume of a sphere, and for the volume and area of a cone. Archimedes is also known for inventing the *Claw of Archimedes* and the *Archimedes heat ray*, both of which were weapons to defend the city of Syracuse. The claw was a kind of mobile grappling hook that could lift enemy ships out of the water, and modern experiments suggest that this would have been a workable device. The heat ray was a system of mirrors to focus reflected sunlight on to enemy ships, thus setting them on fire. Modern attempts to reproduce the heat ray have concluded that it would not have worked quickly enough in typical weather conditions to be able to burn enemy ships.

**Relevant links**

Wikipedia on Archimedes: http://en.wikipedia.org/wiki/Archimedes

Wikipedia on *The Quadrature of the Parabola* (including the graphic here): http://en.wikipedia.org/wiki/The_Quadrature_of_the_Parabola

Picture of Archimedes from http://totallyhistory.com/archimedes/

I stole the joke in the picture from Dan McQuillan on Twitter.

Here’s another good joke about Newton and Leibniz developing calculus in the 17th century, which someone in my department has on their office door: http://xkcd.com/626/

#mathematics #sciencesunday

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