- #1
murshid_islam
- 457
- 19
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:
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?
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?