11.2. Johannes Kepler and the development of mathematics

The name of Kepler heads the list of those associated with the development of a whole new period in mathematics—the period of variable quantities. At the very beginning of the seventeenth century his investigations of the laws of planetary motion led him into the difficult and important area of integral calculation. Though he was not able to perform integrations as such, he treated the new concept in terms of the area bounded by a curve, and developed procedures that come near to the modern methods of numerical integration. Nor was this all; he developed these methods and used them in his Nova Stereometria to determine the volume of various quite complicated solids of revolution. The questions of differential calculation were not ignored either: in the same book Kepler deals with extrema and the isoperimeter problem. In his work on optics, Kepler was the first to introduce into geometry the principle of continuity and the concept of a point at infinity—an important step towards the creation of projective geometry. His work on the theory of polygons and polyhedra was also noteworthy, particularly his discovery of the regular star-shaped polyhedra and of the space-filling property of these and other regular convex solids. The very scope of the calculations which he performed in his many years of search for the planetary laws is sufficient to earn Kepler his place as the greatest master of calculation of his times. While carrying out this arduous task he turned his creative ability to the perfection of methods and aids that would speed up the work. He was among the first to work out an original theory of logarithm construction, and he prepared logarithmic tables not unlike those of Napier, though not identical with them. He was indeed the first to use logarithmic calculation in important astronomical researches. His tables outlived those of Bürgi and Napier, and were reissued in 1700. In the 12,700 pages of Kepler manuscript material preserved in the archives of the USSR Academy of Sciences it can be seen that Kepler was not only helper but very probably initiator in the invention of the first calculating machine by his friend, the Tübingen Professor W. Schickard. In the same manuscript collection his German text on the logarithmic calculation, his "Weinvisierbüchlein", and other writings all show his substantial contribution to the development of German-language scientific literature. Kepler's views on the nature of mathematical concepts are reflected in his manuscript (also in the Academy's collection) "De quantitatibus Libelli". Parts of this also supplement our knowledge of Kepler's contribution to the development of infinitesimal methods and other problems.