Knowing the result, Archimedes considered a secondary circle with the same area than the ellipse.

The radius of this circle is:

Archimedes wanted to probe that the area of the ellipse is equal to the area of this secondary circle

Archimedes considered an auxiliary polygon similar to P', the polygon inscribed in the auxiliary circle C'.

The relation between the areas of these two similar polygons is:

Archimedes estarted his double 'reductio ad absurdum'.

If posible, let the secundary circle greater than the ellipse.

"We can then inscribe in the secundary circle an equilateral polygon of 4n sides such that its area is greater than that of the ellipse. [cf. On the Sphere and Cylinder, I. 6.]" (Archimedes)

Then

Then we can consider a similar polygon in the auxiliary circle and the corresponding polygon in the ellipse.

"Supose that P' denotes the area of the polygon inscribed in the auxiliary circle, and P that of the polygon inscribed in the ellipse." (Archimedes)

We already know that

Then

"But this is imposible, because the later polygon is by hypothesis greater than the ellipse, and a fortiory greater than P.

Hence the secondary circle is not greater than the ellipse." (Archimedes)

If posible, let the secundary circle be less than the ellipse.

In this case we inscribe in the ellipse a polygon P with 4n equal sides such that

Archimedes consider polygon P' inscribed to the auxiliary circle and a similar polygon inscribed in the secondary circle.

As before

which is imposible

This completes the double reductio ad absurdum proof.

"Hence the secondary circle, being neither greater nor less than the ellipse, si equal to it; and the required result follows."(Archimedes)

"In esence, Archimedes has simply given a rigorous exhaustion proof of the intuitively evident fact that the area of the ellipse is b/a times the area of its auxiliary circle, corresponding to the observation that the circle is transformed into the ellipse by shrinking its vertical dimension by the factor b/a" (C. H. Edwards)

LINKS

	Archimedes and the area of an ellipse: an intuitive approach In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
	Equation of an ellipse Transforming a circle we can get an ellipse (as Archimedes did to calculate its area). From the equation of a circle we can deduce the equation of an ellipse.
	Ellipse and its foci Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.
	Plane developments of geometric bodies (4): Cylinders cut by an oblique plane We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
	Archimedes' Method to calculate the area of a parabolic segment Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
	Kepler: The Area of a Circle Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
	Kepler: Surface and volume of a sphere Kepler studied the volume and surface of the sphere. He thought the volume of the sphere as made up of small cones, then he sum all of these cones and get a relation between the surface of a sphere en its volume.