Kepler wrote a book in 1615 about areas and volumes of surfaces and solids: Nova stereometria doliorum vinariorum (New solid geometry of wine barrels). In this book Kepler thought of the sphere as being composed of an infinite number of infinitesimal pyramids or cones each having its vertex at the center of the sphere and whose bases made up the surface of the sphere.

The volume of a pyramid or a cone is one-third the product of the area of the base and the altitude (in this case the altitude is the radius of the sphere). Then Kepler was able to deduce the relation between the volume of the sphere and its surface:

Then, if we know the volume of the sphere (as Archimedes knew more than 2200 years before us):

we can deduce the area of the surface of the sphere:

Or viceversa, if we know the surface we can deduce the volume of the sphere.

This is one example of a theorem about the sphere in one original Kepler's Nova stereometria doliorum vinariorum. We can read this page, and the full book, in The Posner Memorial Collection,Carnegie Mellon University Libraries.

A translation could be: The surface area of a sphere is four times the area of a great circle of the sphere.

LINKS

Kepler: The Area of a Circle

Kepler used an intuitive infinitesimal approach to calculate the area of a circle.

Kepler: The volume of a wine barrel

Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.

Kepler: The best proportions for a wine barrel

Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Cavalieri: The volume of a sphere

Using Cavalieri's Principle we can calculate the volume of a sphere.

Archimedes and the area of an ellipse: an intuitive approach

In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.

Archimedes and the area of an ellipse: Demonstration

In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.