> 1. Who was the first to give the correct formula for the volume of a
> sphere and how was the formula obtained?
The answer's 'really' Archimedes, who gave a proof by exhaustion in the
paper you describe below, although I wouldn't be surprised to hear that
some earlier person guessed the answer.
> 2. In his famous derivation of the relation between the volumes of the
> sphere, cylinder and the cone, Archimedes uses the formula for volume of
> a cone to find the volume of a sphere. My question is how was the volume
> of a cone found?
Archimedes explains how he found these results in the celebrated
"Discourse on Method", in the so-called "Heiberg manuscript" which was only
found early this century, and which was sold for $2,000,000 at Christie's
a few months ago. The proofs are in the paper above, which was part of the
same manuscript.
> 3. If the volume of the cone was found by the `devil's staircase' method
> (approximating with thin cylindrical slices of decreasing dia stacked on
> top of each other), then the same method could also have been applied to
> find the volume of a sphere using the Pythogoras theorem to obtain the
> relation for the radius at different heights. (The devil's staircase
> method also requires one to know the sum of a sequence of squares of
> integers and some idea of what happens in the limit.)
>
> 4. It seems to me then that the volume of a sphere could not have been
> any more difficult than the volume of a cone.
I don't think it was.
> 5. My last question is - which was found first - formula for the volume
> of a sphere or for the surface area of a sphere?
I doubt if we'll ever find out, and don't think it matters very much,
because the two formulae are so closely related that either can be
deduced from the other very simply.
John Conway