A pioneering genius of pure and applied mathematics, Hermann Minkowski (1864 1909) founded the geometry of numbers and wrote extensively about his researches into the field. Until the distinguished American mathematician Harris Hancock interpreted Minkowski's writings, they were accessible only to a few specialists. Hancock elaborated on the master's writings, placing them in clear, readable form. This classic two-volume edition returns Hancock's brilliant exposition to the mathematics community after a long hiatus.
Development of the Minkowski Geometry of Numbers concerns itself primarily with geometric problems involving integers and with algebraic problems approachable through geometrical insights. In addition to demonstrating that geometric proofs and theorems in number theory are often simpler and more elegant than arithmetic proofs, the author illuminates many other algebraic and geometric topics. Starting with preliminary background and historical remarks, Volume 1 examines surfaces that are nowhere concave; the volume of bodies; linear forms; the arithmetical theory of a pair of lines; algebraic numbers; and the theory of continuous fractions. Some of Minkowski’s shorter papers are discussed as well. Topics featured in the subsequent volume include approximations of algebraic numbers and of real quantity through rational numbers, the arithmetic of the ellipsoid, and extreme standard bodies.
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