**Ferdinand Joachimsthal**'s parents were David Joachimsthal and Friederike Zaller, the daughter of Bezallel Cohn from Glogau. The family was Jewish. David Joachimsthal, the son of Lazarus Joachimsthal from Glogau, was a merchant living in the town of Goldberg which was at that time part of the Kingdom of Prussia. Goldberg, meaning 'gold mountain' in German, had been a centre of gold and copper mining since the Middle Ages. The small town is about 18 km southeast of the much larger city of Liegnitz. Ferdinand attended the Gymnasium at Liegnitz (now Legnica, Poland) where he was taught by Eduard Kummer. Appointed to the Liegnitz Gymnasium in 1832, Kummer taught Joachimsthal mathematics and physics. Under Kummer's guidance, Joachimsthal began undertaking mathematical research while at the school in Liegnitz. In 1836 he entered the University of Berlin where he was taught by Lejeune Dirichlet and Jakob Steiner. Dirichlet had been appointed to a chair in Berlin in 1828 but he also had a heavy teaching load at the Military Academy. Steiner had been appointed as an extraordinary professor of geometry two years before Joachimsthal began his studies and he had already made a name for himself as a leading expert on projective geometry. Joachimsthal remained at Berlin for three semesters before, taking the route that most students took at that time, he moved on to study at a second university.

From 1838 he studied at the University of Königsberg where his teachers included Carl Jacobi and Wilhelm Bessel. Jacobi had made important contributions to geometry and at the time that Joachimsthal was studying in Königsberg, Jacobi was working on his theory of determinants. Bessel was an astronomer who was the first to measure to parallax of a star. He made his announcement of the distance to 61 Cygni in the year that Joachimsthal began studying at Königsberg. After taking his first degree at Königsberg, Joachimsthal went to Halle to undertake research at the University. His thesis advisor at Halle was Otto Rosenberger (1800-1890), and he obtained a doctorate from there in July 1840 for his thesis *De lineis brevissimis in superficiebus rotatione ortis*. He passed the oral examination "with distinction". His thesis advisor Otto Rosenberger had been a student of Bessel and was appointed to Halle as a professor in 1831 on Bessel's recommendation. At Halle, Rosenberger taught mathematics and astronomy but, although he continued to undertake research, he published nothing after 1836. Now, almost certainly to improve his chances of getting an academic post, Joachimsthal converted from Judaism to the Protestant Christian faith after the award of his doctorate.

At this time it was necessary for school teachers to take the 'Examen pro facultate docendi' to qualify to teach in a Gymnasium. This examination tested the candidate on philosophy, history and mathematics. It also required that a practical teaching test be taken. Joachimsthal passed the 'Examen pro facultate docendi' and, from 1844, taught in Berlin at the Konigliche Realschule. From 1847 he taught at the Collège Royal Français in Berlin where he was appointed a professor in 1852. This school had been founded by the Huguenots in 1689, four years after Louis XIV revoked the Edict of Nantes and declared Protestantism to be illegal in France. Joachimsthal also habilitated at the University of Berlin on 13 August 1845, becoming the first Jew to habilitate there. He was examined by Martin Ohm and the astronomer Johann Franz Encke (1791-1865). He gave his sample lecture *Über die Untersuchungen der neueren Geometrie, welche sich der Lehre von den Brennpunkten anschliessen* Ⓣ to the whole of the department in Berlin on 7 August 1845 and also gave a public lecture *De curvis algebraicis* Ⓣ on the 13 August following which the 'Venia legendi' (the permission to teach) was conferred on him.

At the University of Berlin Joachimsthal taught courses on analytic geometry and calculus, giving more advanced courses on the theory of surfaces, the calculus of variations, statics and analytic mechanics. He is famed for the high quality of his lectures and he also introduced novel methods of teaching by organising examples classes. Although teaching in examples classes was new to Berlin, Joachimsthal had realised the benefits of this style of teaching while he was a member of Jacobi's seminar in Königsberg. At Berlin, his colleagues included many famous mathematicians who all contributed to his development of mathematical ideas, in particular Gotthold Eisenstein, Lejeune Dirichlet, Carl Jacobi, Jakob Steiner and Carl Borchardt. Joachimsthal's abilities as a lecturer are illustrated in [1]:-

His lectures attracted more students than did those of Eisenstein, his brilliant colleague in the same field.

While teaching as a docent at the University of Berlin, Joachimsthal also continued teaching at the Konigliche Realschule and he published the 21-page article *Über die Bedingung der Integrabilität* Ⓣ in the *Jahresbericht* of the Konigliche Realschule of Berlin in 1844. In 1846 he published the important memoir *Über die Bedingung der Integrabilität * Ⓣ in Volume 33 of Crelle's Journal. This work, which studied the integrability of differential equations with more than two variables, gave a complete answer to a question that had been considered by Lagrange. He appended a note to the article, dated April 1846, saying that Joseph Raabe had published a similar result in an article in Volume 31 of Crelle's Journal dated September 1844 but Joachimsthal had announced his result in the *Jahresbericht* of the Konigliche Realschule in April 1844 and so claimed priority. After 1847 he left the Konigliche Realschule taking up an appointment at the Collège Royal Français and, based on the courses he was giving there, he published the textbook *Cours de géométrie élémentaire à l'usage des élèves du Collège Royal Français* Ⓣ in 1852. This book [1]:-

... demonstrated his talent as a textbook writer through its clear logical structure and insight(rarely found in accomplished mathematicians)into the difficulties facing beginners...

Despite his outstanding contributions, Joachimsthal did not climb the academic ladder in the way that one would have expected and, despite frequent pleas from his colleagues who gave him the highest recommendations, he remained a docent at the University of Berlin. It is hard to attribute this to anything other than the general level of anti-Semitism that pervaded the university system at this time. His attempts to improve his position included submitting a second habilitation in 1850, this time to the University of Breslau with the thesis *De duabus aequationibus quarti et sexti gradus in theoria linearum et superficierum secundi ordinis occurrunt*. However, he was appointed by the Prussian Ministry of Culture to a chair in Halle on 7 May 1853. The chair had become vacant due to the death of Ludwig Adolph Sohncke (1807-1853). Joachimsthal did not hold the chair in Halle for long since he was offered the chair of mathematics at Breslau (now Wroclaw in Poland) in 1855. This came about since Kummer, who had held the chair at Breslau since 1842, was offered a chair in Berlin. Kummer realised that Karl Weierstrass would be a strong contender to fill the Breslau chair but he wanted Weierstrass as a colleague in Berlin (although a position was not available straight away). Kummer, therefore, strongly recommended Joachimsthal, who he regarded extremely highly, to fill the chair in Breslau. The scheme worked, with Joachimsthal ranked first and Weierstrass third, so Joachimsthal became Kummer's successor at Breslau in 1855. Here, as he had earlier, he acquired a high reputation for teaching, in fact even higher than the brilliant lecturer Kummer who he replaced. At Breslau [1]:-

...[Joachimsthal]taught, among other things, analytic geometry, differential geometry, and the theory of surfaces, in which - exceptional for the time - he operated with determinants and parameters. He gave special lectures on geometry and mechanics for students of mining engineering and metallurgy. The average number of his listeners exceeded that of Kummer, who with Weierstrass was later to become one of the most sought-after teachers of mathematics. In1860, for example, Joachimsthal had an audience of sixty-six attending his mechanics lectures. By the time of his death, at the age forty-three, he had acquired a wide reputation as an excellent teacher and kind person.

His high quality teaching extended to the textbooks which he wrote; these are famed for their clarity of exposition. Jacobi, who was very impressed with Joachimsthal's *Cours de géométrie élémentaire*, persuaded him to write *Elemente der Analytische Geometrie der Ebene* which Jacobi wanted to be part of the book *Geometrie des Raumes* that he was planning to write. However, Jacobi died before completing this project. Joachimsthal wrote *Cours de géométrie élémentaire* but he died before the work could be published and Oswald Hermes (1826-1909), who had been a student of Kummer graduating in 1849, edited Joachimsthal's notes and published them in 1863. A second edition was published in 1871. Notes from a lecture course Joachimsthal gave in Breslau in the winter semester of 1856-57 was published as the book *Anwendung der Differential- und Integralrechnung auf die allgemeine Theorie der Flächen und der Linien Doppelter Krümmung* in 1872. The editor who converted the lecture notes into a book was Karl Heinrich Liersemann (1835-1896). A second edition of the book appeared in 1890. Liersemann explains in a Preface how he had attended Joachimsthal's lectures in 1856-57 and, having been asked by Joachimsthal, he took careful notes with the aim of their being the basis for a book. He writes:-

After the lecture course finished, Joachimsthal received the manuscript that I had prepared and had thoroughly examined it and found nothing substantially wrong with it. However, he laid it aside probably because other areas of mathematics, namely the elements of analytic geometry, occupied him. If the undersigned has only now, after so many years have passed, decided to publish the lectures, the reason is mainly that it has taken me a long time to put the finishing touches to prepare the work for printing. But once I set about the task, the high quality of the lectures gave me the enthusiasm to complete it, believing that the experts would forgive my efforts because of the merits of the original lectures.

Influenced by the work of Jacobi, Dirichlet and Steiner, Joachimsthal wrote on the theory of surfaces where he made substantial contributions, particularly to the problem of normals to conic sections and second degree surfaces. We should particularly mention his 20-page paper *Mémoire sur les surfaces courbes* (1848). The motivation for many of his works were problems posed by other mathematicians, for example he answered questions posed by Pierre Bonnet, Philippe de la Hire, Carl Johann Malmsten (1814-1886), Heinrich Schröter, Jakob Steiner, and Jacques Charles-François Sturm.

Joachimsthal applied the theory of determinants to geometry. He made the important step of introducing oblique coordinates. Joachimsthal surfaces are named after him, these have a family of plane lines of curvature within the plane of a pencil. He has a theorem named after him which concerns the intersection of surfaces. He is also remembered for another theorem on the four normals to an ellipse from a point inside it. One significant aspect of his work is the clever notation which he introduced. Often one fails to recognise the importance of good notation, which can aid understanding and help to point the user towards significant results. Joachimsthal introduced a notation that can be used to write down the equations of tangents and polars of plane and projective conics. The various notations introduced by Joachimsthal in the area of second order equations and conic sections have an influence that has extended far beyond these areas, for example into the important work of Frank Morley.

Of course his determination that his works should be of the very highest standard meant there was a price to pay. As stated in [1]:-

Joachimsthal's contributions were substantial and lucid. His marked predilection for mature, polished exposition was expressed in constant recasting, revising and rewriting, so that many planned works never reached completion.

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

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