Borchardt's doctoral work, on non-linear differential equations, was supervised by Jacobi and submitted in 1843. However, the thesis was not published and has since been lost. Jacobi was in poor health and it was agreed that he could spend a year in Italy convalescing. He went with Borchardt and they spent time in both Rome and Naples. Dirichlet and Steiner were also in Rome at the same time and it proved a useful time for Borchardt. The year 1846-47 he spent in Paris where he met Chasles, Hermite and Liouville. He attended a course by Liouville on doubly periodic functions and although Liouville intended to publish the notes which Borchardt took of his lectures, in the end they were not published due to a priority dispute between Liouville and Hermite. Borchardt married Rosa Oppenheim and recently there has been speculation that after Borchardt's death, Rosa had a child with Weierstrass.
Borchardt taught at the University of Berlin from 1848 when he was appointed as a Privatdozent. He quickly became a close personal friend of Weierstrass and was one of the privileged few, along with Sofia Kovalevskaya, whom Weierstrass addressed with the familiar 'Du' form. He succeeded Crelle as editor of Crelle's Journal in 1856, a task he undertook until 1880 despite not being in very good health. The correct title of the Journal was the Journal für die Reine und Angewandte Mathematik Ⓣ but it had been known as Crelle's Journal up to the time Borchardt took over as editor. The journal was then often referred to as "Borchardt's Journal" or in France as "Journal de M Borchardt". After Borchardt's death, the Journal für die Reine und Angewandte Mathematik again became known as Crelle's Journal.
He did important research on the arithmetic geometric mean continuing work in this area which had been begun by Gauss and Lagrange. In 1881 Borchardt published an algorithm for the arithmetic-geometric mean of two elements from (two) sequences, although it was actually first proposed by Gauss in a letter to Pfaff written in 1800. Although Gauss's letter is lost we know its contents through Pfaff's reply which was published in Gauss's Complete Works and indicates that Gauss had discovered the result. From this 1881 paper by Borchardt the name "Borchardt algorithms" has come into use to describe algorithms of this type. Borchardt also generalised results of Kummer on equations determining the secular disturbances of the planets. A secular disturbance is one which is not periodic, but continually acts in the same direction. In fact this was his first contribution and was published in his first paper of 1846. In this work he used determinants and Sturm functions :-
In several further papers Borchardt applied the theory of determinants to algebraic equations, mostly in connection with symmetric functions, the theory of elimination, and interpolation.After Jacobi's death there was considerable speculation as to the exact role he had played in the theory of elliptic functions. This was partially answered when Jacobi's letters to Lagrange were published by Bertrand in 1869 but the position was still somewhat confused as the letters from Lagrange to Jacobi were not included in this work. Borchardt completed publishing the remaining parts of the correspondence in 1875 and Jacobi was then able to get full recognition for his contributions to the theory of elliptic functions made independent of those of Abel. Borchardt contributed to spreading the mathematical ideas introduced by Jacobi but he also spread Jacobi's ideas on the way that universities should be organised, namely in a research oriented way.
Borchardt's complete works, published in 1888, contains 25 papers and, in addition to the topics discussed above, contains papers on maxima and on the theory of elasticity. Finally we note that the first of the eight volumes of Jacobi's Collected Works was edited by Borchardt and published in 1881. Borchardt died before being able to edit further volumes which were edited by Weierstrass.
Article by: J J O'Connor and E F Robertson