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The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.

New Essays Concerning Human Understanding, IV, XII.

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# Kepler: The Volume of a Wine Barrel

## Kepler's Era

Johannes Kepler (1571–1630) was a German mathematician, astronomer, and astrologer, and a key figure in the 17th century scientific revolution. He lived after Copernicus and supported the heliocentric model of the universe. Kepler worked with Tycho Brahe and used Brahe's remarkable observational data to make his most famous discovery, the three laws of planetary motion now known as Kepler's Laws. Newton later showed that Kepler's laws could be deduced from Newton's laws of motion and universal gravitation law.

As a mathematician Kepler discovered two new regular polyhedra, worked on the problem of close packing of equal spheres, computed logarithms, and found volumes of solids of revolution. Our main purpose here is to understand Kepler's contributions to the development of the calculus.

Kepler lived before Newton and Leibniz. During the sixteenth century and early seventeenth century, the Greek mathematical masterworks, including Euclid's Elements, the Conics of Apollonius, and the works of Archimedes, were studied seriously. Numerous mathematicians refined the method of exhaustion and applied it to a wide variety of new quadrature (area) and cubature (volume) problems. Another point of interest was the determination of centers of gravity of solids. (On the importance of centers of gravity for the development of the calculus, see Baron, p. 90.) Renaissance mathematicians were more interested in new results and methods of discovery than in rigorous proofs. They freely used intuitive concepts of the infinite to produce infinitesimal methods for the solution of area and volume problems. Kepler and Cavalieri were two key mathematicians who helped invent these infinitesimal methods.

Kepler's Era Copernicus Brahe Galileo Kepler Descartes Cavalieri Fermat Newton Leibniz

Figure 1. Mathematicians who influenced Kepler or were influenced by Kepler, from Copernicus to Leibniz

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Cardil, Roberto, "Kepler: The Volume of a Wine Barrel," Loci (June 2010), DOI: 10.4169/loci003499