**Hermann Minkowski**'s parents were Lewin Minkowski, a businessman, and Rachel Taubmann. Hermann was his parents' second son, the eldest being Oskar Minkowski who went on to become a famous pathologist. Lewin and Rachel Minkowski were Germans although their son Hermann was born while they were living in Russia. When Hermann was eight years old the family returned to Germany and settled in Königsberg where Lewin Minkowski conducted his business.

Minkowski first showed his talent for mathematics while studying at the Gymnasium in Königsberg. Already at this stage in his education he was reading the work of Dedekind, Dirichlet and Gauss. The outstanding abilities he showed at this time were noted in a letter that Heinrich Weber, then at Königsberg University, wrote to Dedekind in 1881 (see [13]). He studied at the University of Königsberg, entering the university in April 1880. He spent three semesters at the University of Berlin, for example spending the winter semester of the academic year 1882-83 there. His became close friends with Hilbert while at Königsberg, for Hilbert was an undergraduate at the same time as Minkowski. In 1884, while he was a student at Königsberg, Hurwitz was appointed to the staff. The student Minkowski soon became close friends with the newly appointed academic Hurwitz. He received his doctorate in 1885 from Königsberg for a thesis entitled *Untersuchungen über quadratische Formen, Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enthält.* Minkowski became interested in quadratic forms early in his university studies. In 1881 the Academy of Sciences (Paris) announced that the Grand Prix for mathematical science to be awarded in 1883 would be for a solution to the problem of the number of representations of an integer as the sum of five squares. Eisenstein had given a formula for the number of such representations in 1847, but he had not given a proof of the result. In fact the Academy of Sciences had set a problem for the Grand Prix which had already been solved, for Henry Smith had published an outline of a proof in 1867. However the Academy of Sciences were unaware of Smith's contributions when the prize topic was set.

Eisenstein had been studying quadratic forms in *n* variables with integer coefficients at the time he published his unproved formula in 1847 but as he was already ill by this time details were never published. Minkowski, although only eighteen years old at the time, reconstructed Eisenstein's theory of quadratic forms and produced a beautiful solution to the Grand Prix problem. Smith reworked his earlier proof, adding detail and submitted that to the Academy. The decision was that the prize be shared between Minkowski and Smith but this was a stunning beginning to Minkowski's mathematical career. On 2 April 1883 the Academy granted the Grand Prize in Mathematics jointly to the young Minkowski at the start of his career and the elderly Smith at the end of his. Minkowski's doctoral thesis, submitted in 1885, was a continuation of this prize winning work involving his natural definition of the genus of a form. After the award of his doctorate, he continued undertaking research at Königsberg.

In 1887, a professorship became vacant at the University of Bonn, and Minkowski applied for that position; according to the regulations of German universities, he had to submit orally to the faculty an original paper, as an Habilitationsschrift. Minkowski presented *Räumliche Anschauung und Minima positiv definiter quadratischer Formen* (Spatial visualization and minima of positive definite quadratic forms) which was not published at the time but in 1991 the lecture was published in [11]. Dieudonné writes:-

This lecture is particularly interesting, for it contains the first example of the method which Minkowski would develop some years later in his famous "geometry of numbers".

Minkowski taught at Bonn from 1887, being promoted to assistant professor in 1892. Two years later he moved back to Königsberg where he taught for two years before being appointed to the Eidgenössische Polytechnikum Zürich. There he became a colleague of his friend Hurwitz who had been appointed to fill Frobenius' chair after he left Zürich for Berlin in 1892. Einstein was a student in several of the courses he gave and the two would later become interested in similar problems in relativity theory. Minkowski married Auguste Adler in Strasburg in 1897; they had two daughters, Lily born in 1898 and Ruth born in 1902.

The family left Zürich in the year that their second daughter was born for Minkowski accepted a chair at the University of Göttingen in 1902. It was Hilbert who arranged for the chair to be created specially for Minkowski and he held it for the rest of his life. At Göttingen he became interested in mathematical physics gaining enthusiasm from Hilbert and his associates. He participated in a seminar on electron theory in 1905 and he learnt the latest results and theories in electrodynamics.

Minkowski developed a new view of space and time and laid the mathematical foundation of the theory of relativity. By 1907 Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-euclidean space. He considered space and time, which were formerly thought to be independent, to be coupled together in a four-dimensional 'space-time continuum'. Minkowski worked out a four-dimensional treatment of electrodynamics. His major works in this area are *Raum und Zeit* (1907) and *Zwei Abhand lungen über die Grundgleichungen der Elektrodynamik* (1909). Kline, reviewing [9] writes:-

A key point of the paper is the difference in approach to physical problems taken by mathematical physicists as opposed to theoretical physicists. In a paper published in1908Minkowski reformulated Einstein's1905paper by introducing the four-dimensional(space-time)non-Euclidean geometry, a step which Einstein did not think much of at the time. But more important is the attitude or philosophy that Minkowski, Hilbert - with whom Minkowski worked for a few years - Felix Klein and Hermann Weyl pursued, namely, that purely mathematical considerations, including harmony and elegance of ideas, should dominate in embracing new physical facts. Mathematics so to speak was to be master and physical theory could be made to bow to the master. Put otherwise, theoretical physics was a subdomain of mathematical physics, which in turn was a subdiscipline of pure mathematics. In this view Minkowski followed Poincaré whose philosophy was that mathematical physics, as opposed to theoretical physics, can furnish new physical principles. This philosophy would seem to be a carry-over(modified of course)from the Eighteenth Century view that the world is designed mathematically and hence that the world must obey principles and laws which mathematicians uncover, such as the principle of least action of Maupertuis, Lagrange and Hamilton. Einstein was a theoretical physicist and for him mathematics must be suited to the physics.

This space-time continuum provided a framework for all later mathematical work in relativity. These ideas were used by Einstein in developing the general theory of relativity. In fact Minkowski had a major influence on Einstein as Corry points out in [7]:-

In the early years of his scientific career, Albert Einstein considered mathematics to be a mere tool in the service of physical intuition. In later years, he came to consider mathematics as the very source of scientific creativity. A main motive behind this change was the influence of two prominent German mathematicians: David Hilbert and Hermann Minkowski.

We have mentioned several times in this biography that Minkowski and Hilbert were close friends. Less well known is the fact that Minkowski actually suggested to Hilbert what he should take as the theme for his famous 1900 lecture in Paris. Minkowski, in a letter to Hilbert written on 5 January 1900, writes:-

What would have the greatest impact would be an attempt to give a preview of the future, i.e. a sketch of the problems with which future mathematicians should occupy themselves. In this way you could perhaps make sure that people would talk about your lecture for decades in the future.

Time has certainly proved Minkowski correct!

Minkowski's original mathematical interests were in pure mathematics and he spent much of his time investigating quadratic forms and continued fractions. His most original achievement, however, was his 'geometry of numbers' which he initiated in 1890. *Geometrie der Zahlen* was first published in 1910 but the first 240 pages (of the 256) appeared as the first section in 1896. *Geometrie der Zahlen* was reprinted in 1953 by Chelsea, New York, and reprinted again in 1968. Minkowski published *Diophantische Approximationen: Eine Einführung in die Zahlentheorie* in 1907. It gave an elementary account of his work on the geometry of numbers and of its applications to the theories of Diophantine approximation and of algebraic numbers. Work on the geometry of numbers led on to work on convex bodies and to questions about packing problems, the ways in which figures of a given shape can be placed within another given figure.

At the young age of 44, Minkowski died suddenly from a ruptured appendix.

**Article by:** *J J O'Connor* and *E F Robertson*

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