Volume of Sphere/Proof by Archimedes/Historical Note

Historical Note on Archimedes' Proof of Volume of Sphere

The volume of the sphere was proved by Archimedes in around $250$ BCE.

His proof appears as Proposition $33$ in his On the Sphere and Cylinder.

In the version given here, the notation has been brought up to date.

He also describes in detail how he arrived at this result in The Method.

In his words:

I am persuaded that this method will be of no little service to mathematics. For I can see that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.

-- Archimedes: The Method

It can be argued that this was the point integral calculus was first invented.

Tradition has it that he asked his friends to mark his tombstone with the representation of a sphere inscribed within a cylinder, to commemorate this achievement.

This was done at the order of Marcus Claudius Marcellus, the Roman general who finally defeated the City of Syracuse in $212$ BCE.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.6$: How Archimedes Discovered Integration
• 2004: Ian Stewart: Galois Theory (3rd ed.) ... (previous) ... (next): Historical Introduction