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To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can be moare equalle.

In G. Simmons, Calculus Gems, New York: McGraw Hill Inc., 1992.

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# Archimedes' Method for Computing Areas and Volumes

## Cylinders, Cones, and Spheres

Recall the following information about cylinders and cones with radius r and height h:

Volume Center of Gravity
Cylinder $$\pi$$r2h
 On the cylinder's axis, half-way between top and bottom

Cone ($$\pi$$r2h)/3
 On the cone's axis, three times as far from the vertex as from the base

Suppose a sphere with radius r is placed inside a cylinder whose height and radius both equal the diameter of the sphere. Also suppose that a cone with the same radius and height also fits inside the cylinder, as shown below.

We place the solids on an axis as follows:

For any point S on the diameter AC of the sphere, suppose we look at a cross section of the three solids obtained by slicing the three solids with a plane containing point S and parallel to the base of the cylinder. The cross-sections are all circles with radii SR, SP, and SN, respectively. What Archimedes discovered was that if the cross-sections of the cone and sphere are moved to H (where |HA| = |AC|), then they will exactly balance the cross section of the cylinder, where HC is the line of balance and the fulcrum is placed at A.

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