What did Archimedes mean by the area of a spherical segment matching a cylinder segment?

[murshid_islam]

I was reading "Significant Figures: The Lives and Work of Great Mathematicians" by Ian Stewart. The following is an excerpt from its chapter on Archimedes:

On the Sphere and Cylinder contains results of which Archimedes was so proud that he had them inscribed on his tomb. He proved, rigorously, that the surface area of a sphere is four times that of any great circle (such as the equator of a spherical Earth); that its volume is two thirds that of a cylinder fitting tightly round the sphere; and that the area of any segment of the sphere cut off by a plane is the same as the corresponding segment of such a cylinder.

So, the first one was that he proved the surface area of a sphere $= 4\pi r^2$.
The second one is the result that the volume of a sphere $= \left(\frac{2}{3}\right)2\pi r^3 = \frac{4}{3}\pi r^3$.
But what is the third one? What does "the area of any segment of the sphere cut off by a plane is the same as the corresponding segment of such a cylinder" mean?

 

[mfb]

The phrasing is strange, but I assume this is the area of a spherical cap, 2 pi r h where h is the "height" of the cap: This is the same area of the cylinder from the first statement within the same height.
In other words, cut the cylinder orthogonal to its axis anywhere, look for the area of the sphere encased within and it will match the outer surface of the cylinder element (assuming you cut in places that intersect the sphere).

 

[hutchphd]

A work of the ancients that unerringly delights is Ptolemy's theorem about cyclic polynomials. It just seems amazing to me.

Fixed link...

 

[Lnewqban]

... What does "the area of any segment of the sphere cut off by a plane is the same as the corresponding segment of such a cylinder" mean?

Copied from
https://mathworld.wolfram.com/ArchimedesHat-BoxTheorem.html

Copied from
http://mathcentral.uregina.ca/QQ/database/QQ.09.99/wilkie1.html

 

[murshid_islam]

A work of the ancients that unerringly delights is Ptolemy's theorem about cyclic polynomials. It just seems amazing to me.

Fixed link...

I'm not sure I understand how that is relevant to Archimedes' works on spheres and cylinders. Am I missing something?

 

[murshid_islam]

Thank you, mfb and Lnewqban.