Archimedes / Quadrature by parts

Before a birth of calculus, how to solve areas/volumes of a solid figure made up of smooth curves was referred to as *quadrature by parts*. Different from calculus, this method depends on figure types so as to successfully achieve clever solutions specific to the types.

In the times of the ancient Greek, *Archimedes* (278-212 B.C.) demonstrated "An area of a parabola is 4/3 times as large as an area of .

The demonstration is as follows.

Draw the two tangents to the parabola at points and , and the point of intersection between the two is defined as .

There is a third tangent parallel to side , and the point of intersection between this tangent and the parabola is defined as .

Produce line to , and the point of intersection between this and the side is defined as .

Then, point represents the middle point of side .

Therefore an area of

Next, Draw a tangent to the parabola at point , and the point of intersection between the two is defined as .

There is a third tangent parallel to side , and the point of intersection between this tangent and the parabola is defined as .

Produce line to , and the point of intersection between this and the side is defined as .

Suppose an area of is , then an area of .

Taking into account another triangle in the side of point yields .

Such procedure is to be endlessly repeated. However, only infinite times can cover all over the area .

Archimedes thought as follows:

Since is arbitrary, cannot be but .

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