**Siegfried Aronhold**'s father was the merchant M Aronhold, and his mother was W Saaling from an Angerburg family. The Aronhold family were Jewish, and when Siegfried Aronhold matriculated at university he gave his religion as Jewish. However, when he produced a handwritten CV later in his life (at age 44 years) he gave his religion as Protestant. This change was almost certainly not due to religious motives but rather to improve his prospects in the anti-Semitic environment in which he lived. He was brought up in Angerburg and studied first at the elementary school there before moving to the Gymnasium at Rastenburg (now Ketrzyn, Poland). His father died while he was studying at the Gymnasium and, following this tragedy, his mother moved to Königsberg where Aronhold attended the Altstädtischen Gymnasium, graduating in 1841.

Aronhold then entered the University of Königsberg to study mathematics and natural sciences, matriculating on 25 October 1841. There he was taught by Friedrich Wilhelm Bessel, Friedrich Julius Richelot, Ludwig Otto Hesse and Franz Ernst Neumann but he was most influenced by Carl Gustav Jacob Jacobi. At the University, he joined the Mathematical Seminar run by Bessel, Jacobi and Neumann and twice received awards for producing the best work. When Jacobi was appointed to the University of Berlin in June 1844, Aronhold followed him and he continued his studies of mathematics in Berlin under Lejeune Dirichlet and Jakob Steiner. He also studied physics at the University of Berlin, taught by Heinrich Gustav Magnus and Heinrich Wilhelm Dove. However, by this stage in his education he was able to work independently on mathematical research problems.

Things were certainly not easy for the young Aronhold. He did not have any enthusiasm for school teaching and as a consequence he never made the time to take the state examinations to qualify him as a high school teacher. This was not his only problem, however, for he was well aware that as a Jew his prospects of a normal academic career were limited. Therefore, although he loved pure mathematics, he spent much time studying the applied side of the subject to give him better prospects to teach users of mathematics, physicists and engineers. He wrote from Berlin to Hesse in Königsberg on 1 October 1849 [4]:-

Unfortunately my external circumstances are not such that I could achieve a solid position in life, and the daily effort to earn a living produces a depressing influence on my scientific endeavours, making it utterly impossible to meet those obligations to which I consider necessary for entering into a better situation.

In February 1850 he wrote to Hesse saying that he was too busy with his research to find time to take the examinations to qualify as a Gymnasium teacher, see [4]. He wrote to Richelot in September 1850 making the same comments, see [6]. For many years he made his living by giving private mathematics lessons. His research had, however, progressed extremely well and he had published a highly influential paper *Über die homogenen Funktionen dritter Ordnung von drei Veränderlichen* (On the third-order homogeneous functions of three variables) in Crelle's Journal in 1849. In the autumn of 1850 he was offered a post as tutor in a respectable family in Vienna.

His doctorate from the University of Königsberg was awarded on 5 April 1851. Although he had not been studying at Königsberg, Richelot supported the award of the degree based partly on his thesis *Über ein neues algebraisches Prinzip* (On a new algebraic principle) and partly on the paper he had published in 1849 in Crelle's Journal. He resigned from his post as tutor in Vienna later in 1851 and returned to Berlin to take up a post.

He taught at the Royal Academy of Architecture in Berlin, an institution which later became part of the Technical University of Berlin, from 1851. From 1852 until 1854 he also taught at the Artillery and Engineer's School in Berlin. Aronhold continued to teach at the Academy of Architecture as a dozent for several years. He was appointed to a permanent position at the Academy of Architecture in 1860 and later in the same year to a permanent position at the Industrial Institute, a second institution which would later become part of the Technical University of Berlin. Until this time he had not held a sufficiently secure position to allow him to support a wife and family but that all changed with the permanent positions in 1860. In that year he married Marie Julie Friederike Hayn; they had three children, Henriette Marie (born 1861), Emilie Margarethe (born 1866) and Maximilian Ignatz (born 1869). Maximilian Aronhold became an assistant judge in 1900.

Karl Weierstrass also taught at the Industrial Institute where he had been a professor since 1856 but he collapsed in December 1861 and was too ill to teach during 1862. At this stage Aronhold took over Weierstrass's teaching and, two years later, succeeded to his chair at the Industrial Institute. He was also appointed professor at the Royal Academy of Architecture in 1863. Although he now had a high status in good institutions, nevertheless these Institutions did not have the power to confer doctorates. This meant that any students he advised necessarily had to have formal supervisors in other institutions. The most famous of these students was Arthur Moritz Schönflies, who was one of Aronhold's students in the early 1870s, but he had also taught Schönflies's older brother Samuel Martin Schoenflies over ten years earlier. Aronhold was well respected for his excellent teaching [1]:-

He was considered an enthusiastic and inspiring teacher, and was held in high esteem everywhere.

In fact he was so happy teaching at the Industrial Institute that he turned down offers from prestigious institutions; the University of Giessen (1868), the University of Zürich (1868), the University of Dresden (1874) and the University of Heidelberg (1874). In 1879 the Industrial Institute and the Academy of Architecture were combined to form the Technical University of Berlin. Aronhold became Vice President of the new Technical University but was forced to step down from this post on 1 July 1880 because of a serious illness. He never fully recovered from this illness and died four years later.

Aronhold made important contributions to the theory of invariants. The topic was also being intensively studied by Sylvester, Cayley, Clebsch and Hesse but Aronhold was the first German to work on this topic. Certain linear partial differential equations which he came across in his work are characteristic of invariant theory and are named after him. His most important work on the topic was *Über eine fundamentale Begründung der Invariantentheorie *published in Crelle's Journal in 1863 [1]:-

In this treatise Aronhold offers solid proof of his theory, which he had welded into an organic entity. His method refers to functions that remain unchanged under linear substitutions. He stresses the importance of the logical development of a few basic principles so that the reader may find his way through other papers. Aronhold established his theory in general and does not derive any specific equations. He derives the concept of invariants from the concept of equivalency for the general linear group of invariants. ... His efforts to obtain equations independent of substitution coefficients led to linear partial differential equations of the first order, which have linear coefficients. These equations, which are characteristic of the theory of invariants, are known as 'Aronhold's differential equations'.

Although others were working in the same area at this time, Aronhold states that he arrived at his principles in 1851 and so has priority. There is certainly no reason to doubt that this is in fact the case and letters he wrote to Hesse during the years 1849 to 1851 support his claim. Before publishing this important memoir Aronhold [1]:-

... had worked on plane curves. The problem of the nine points of inflection of the third order plane curve, which had been discovered by by Plücker, was brought to completion by Hesse and Aronhold. Aronhold explicitly established the required fourth degree equations and formulated a theorem on plane curves of the fourth order. Seven straight lines in the plane determine one, and only one, algebraic curve of the fourth order, in that they are part of their double tangents and that among them there are no three lines whose six tangential points lie on a conic section.

Finally we note that Aronhold was honoured with election to the Göttingen Academy of Sciences in 1869 and he was appointed as an editor of *Annali di Matematica* in 1867.

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

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