LAST EDITED ON Oct-06-03 AT 04:04 PM (EST)

The first sketch shows how, most likely, Archimedes figured out the surface of a sphere: Put a sphere of radius R in a cylinder of the same radius R and height 2R. Cutting both figures into thin horizontal slices of height Dr, the surface of the cylinder slice wall is

DS_{CW} = 2pR·Dr

the same as the surface of the sphere slice wall

DS_{S} = 2pR cosq·Dr/cosq

Since the total area of the cylinder wall is

S DS_{CW} = 2pR·S Dr = 2pR·2R = 4pR^{2}

so is the surface of the sphere. Then he stated that the surface of a sphere is 4 times the area of its greatest circle - 4´pR^{2} - see Mathematics as a Language. Now, if you have a circle of radius R and you increase the radius by Dr, the area increase is (see the second sketch)

DA = 2pR·Dr

If you have 4 circles of radius R and do the same thing, the total area increase is

4·DA = 4·2pR·Dr = 8pR·Dr

and according to Archimedes' statement, this is the same surface increase as if you have a sphere of radius R and increase its radius by Dr.

Regards, Vladimir