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 16th and early 17th centuries, 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
Nicolas Copernicus (1473-1543)

Copernicus published 'De Revolutionibus orbium coelestium' (1543) where he explained his heliocentric theory.

He gave several reasons why the sun would be at the center of the universe and not the Earth.

The heliocentric theory was defended by Kepler and Galileo and the theoretical evidence was provided by Newton's theory of universal gravitation.

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Tycho Brahe (1546-1601)

Tycho Brahe was an astronomer who made very accurate observations with improved astronomical instruments.

Brahe did not accept Copernicus's theory that the Earch moved around the sun. He proposed an astronomical model with the Earth in the center and with the planets rotating around the sun. Then he avoided the problem of a moving Earth. Brahe's model was accepted by most astronomers.

Kepler used Brahe's accurate observations to deduce his three laws of planetary motion.

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Galileo Galilei (1564-1642)

Galileo was an Italian physicist, mathematician, and astronomer. He studied falling bodies and insisted on testing theories by conducting experiments. To make his astronomical observations he built a telescope.

For him, the laws of nature are mathematical. He wrote: "The universe ... It is written in the language of mathematics."

When Galileo began publicly supporting the heliocentric views of Copernicus he was denounced to the Roman Inquisition.

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Johannes Kepler (1571-1630)

Kepler weas a German mathematician, astronomer, and astrologer. He supported the Copernican theory that the planets move around the sun and is best known for his laws of planetary motion.

Kepler studied the work of Archimedes and used infinitesimal techniques to calculate areas and volumes of bodies. He made contributions to the origin of integral calculus.

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Rene Descartes (1596-1650)

Descartes was a French philosopher, mathematician, and physicist.

As a mathematician he (and Fermat) invented the Cartesian coordinate system and founded analytic geometry, combining Geometry and Algebra. He identified each point of the plane with an ordered pair of real numbers. He expressed geometric properties using Algebra, then manipulated his algebraic expressions to get geometric results.

His work was crucial to the later developent of infinitesimal calculus by Newton and Leibniz.

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Bonaventura Cavalieri (1598-1647)

Cavalieri was an Italian mathematician.

He was one of the precursors of infinitesimal calculus. Cavalieri developed a "method of the indivisibles" which he used to determine areas and volumes.

We know his method as Cavalieri's principle: If two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids are also in this ratio.

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Pierre de Fermat (1601-1665)

Fermat was a French lawyer and mathematician.

He and Descartes were the founders of analytic geometry, which later played a major role in the development of infinitesimal calculus by Newton and Leibniz.

Fermat developed methods for determining maxima and minima for various curves using tangents to those curves.

Fermat is best remembered for his work in number theory, in particular for Fermat's Last Theorem.

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Isaac Newton (1643-1727)

The English mathematician and scientist Isaac Newton is considered to be one of the most influential mathematicians and scientists in history.

Using his theory of gravitation, he demonstrated Kepler's laws of planetary motion.

In mathematics, Newton shares credit with Gottfried Leibniz for the development of the differential and integral calculus.

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Gottfried Leibniz (1646-1716)

Leibniz was a German philosopher and mathematician.

He invented infinitesimal calculus independently of Newton, and his notation has been in general use since then.

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