**Ferdinand Minding**'s father was Gottlieb Minding (1781-1816) from Breslau who, at the time of Ferdinand's birth, was a lawyer in Kalisz but also a musician who was a librettist. His mother was Modesta (died 1814), the daughter of Johann Valentin von Holst (1758-1841), a lawyer in Riga, and Caroline Colins (1758-1841).

The family moved from Kalisz to Hirschberg in Prussian Silesia when Ferdinand was one year old since his father, Gottlieb Minding, was appointed as a judge in that city. Hirschberg is now known as Jelenia Góra and today it is a town in Poland near the Polish-Czech border. The American statesman John Quincy Adams, who later became the sixth President of the United States, visited Hirschberg in 1800 and wrote:-

Ferdinand attended Hirschberg Gymnasium, graduating with his Abitur in 1824. From there he went to the University of Halle where he studied philology and philosophy for two semesters. At this time it was typical for university students in this part of the world to spend parts of their undergraduate years at different universities and so, after his two semesters at Halle, Minding went to the University of Berlin where he continued his studies attending courses by Georg Wilhelm Friedrich Hegel (1770-1831) who at that time was giving courses on aesthetics, the philosophy of religion, the philosophy of history, and the history of philosophy. He also attended lectures by the historian Leopold von Ranke, who was appointed to Berlin in 1825. Ranke opposed the views of Hegel on the philosophy of history so Minding received an interestingly different approach from these two teachers. He also listened to some lectures on natural sciences and mathematics but he concentrated his studies on philosophy.Nothing can be more beautiful than the location of Hirschberg, a beautifully built city with numerous splendid buildings, in a valley surrounded by hills on all sides, with the magnificent view of the Giant Mountains

He graduated from Berlin in 1827 and then taught in secondary schools. In 1828-29 he taught mathematics, history and German at the Gymnasium in Elberfeld. Except for attending a few lectures, Minding had not studied mathematics at university, so how did he become a mathematics teacher? The answer is that he was self taught in mathematics having studied the subject on his own while pursuing other topics at university. While he was a school teacher, he studied for his doctorate in mathematics which was awarded by the University of Halle for his thesis *De valore integralium duplicum quam proxime inveniendo* Ⓣ on approximating the values of double integrals. For someone to reach the level of a doctoral thesis without having been formally taught mathematics is a quite remarkable achievement. Minding published his thesis, having made some minor changes to it, in Crelle's *Journal für die reine und angewandte Mathematik*, as *Über die Berechnung des Näherungswertes doppelter Integrale* Ⓣ (1830).

In 1830 Minding became a mathematics lecturer at the University of Berlin where he taught the barycentric calculus as presented in the works of August Möbius and published *Auflösung einiger Aufgaben der analytischen Geometrie mittels des barycentrischen Calculs* Ⓣ (1830). He also gave lectures on number theory which he wrote up as a textbook *Anfangsgrunde der hoheren Arithmetik* Ⓣ (1832). Minding announced the publication of his book in Crelle's *Journal für die reine und angewandte Mathematik* in 1831 and in his announcement he [4]:-

In his book Minding [4]:-... described pure number theory as both a necessary foundation for algebra(whose basic notion is that of number)and a paradigm for a rigorously developed, autonomous branch of mathematics.

In addition to teaching at the University, in 1834 Minding began teaching at the School of Architecture in Berlin, taking over courses which had been taught up to that time by Lejeune Dirichlet. Minding gave courses of lectures on the theory of curves, on analytical dynamics, and on analysis at the School of Architecture. In 1836 he married Auguste Regler (1810-1889) in Berlin. Auguste was the daughter of Carl August Regler and Henriette Kempsky. Ferdinand and Auguste Minding had one son, Karl Bernhard Minding (born 1839), and two daughters. In 1842 Lejeune Dirichlet proposed Minding for election to the Berlin Academy of Science but he was not elected at this time and this may have prompted Minding to seek a position away from Berlin. An additional factor in deciding to leave Berlin must have been the fact that he had, on two occasions, attempted to gain promotion to extraordinary professor, both attempts having ended in failure. In the following year, 1843, he left Berlin when he was appointed as professor of mathematics at the University of Dorpat, a post he held for 40 years. We note that Dorpat is today known as Tartu. This university was in a slightly unusual position since Estonia had been controlled by Sweden and by Russia at different periods and, at this time, it was controlled by Russia. However, teaching at the university was in the German language and, although the finance and administration of the university was from Russia, its academic leanings were towards Germany with the majority of the professors being German.... presents shortened and expurgated version of[Gauss's 'Disquisitiones Arithmeticae']focusing on the basic parts of the various sections ... Minding dropped the more delicate part of Gauss's theory of forms(genera, composition), but added things expected from the perspective of a textbook, for instance, linear Diophantine equations or continued fractions. He did, however, identify the quadratic reciprocity law as "the most remarkable theorem of higher arithmetic," and, in an historical endnote[on page198], stressed rigour: "Gauss's 'Disquisitiones Arithmeticae' offers a presentation of arithmetic conducted with ancient rigour and distinguished by new discoveries. Among other things an excellent merit of the work lies in the rigorous proof of the reciprocity theorem. Since then the science has been enriched by a non-negligible number of new proofs and results, some of which are to be found in the memoirs of various learned societies, others in mathematical journals, and especially in Crelle's Journal for Mathematics."

Moving to Dorpat meant that Minding became a colleague of Karl Eduard Senff (1810-1849). Senff had studied at Dorpat under Martin Bartels, then was appointed to teach there in 1834, and was promoted to professor in 1837. In fact the Frenet-Serret formulas should be named after Senff who had them in his thesis *Theoremata principalia e theoria curvarum et superficierum* Ⓣ in 1831, sixteen years before Jean Frenet's thesis containing the formulae and Joseph Serret's work on the topic. Sadly the collaboration between Minding and Senff was relatively short since Senff died in December 1849. However, they were able to jointly supervise the undergraduate studies of Karl Mikhailovich Peterson who later wrote his Candidate's Thesis *Über die Biegung der Flächen* Ⓣ advised by Minding. While mentioning Minding's famous students we should include Axel Harnack who studied with Minding while an undergraduate but went to Germany for his doctorate.

At Dorpat, Minding taught algebra, analysis, geometry, probability, mechanics and physics. He [6]:-

From 1851 until 1855 he was Dean of the Faculty of Physicomathematics at Dorpat [1]:-... wrote many important works not only on differential geometry but also in the theory of ordinary differential equations, in analytic mechanics(anticipating the geometrical treatment later developed by Beltrami and Lipschitz), in the calculus of variations(especially on the isoperimetric problem for curved surfaces), etc.

In 1864 Minding and the other members of his family became Russian citizens and, in the same year, he was elected to the St Petersburg Academy of Sciences.In1850the Faculty of Philosophy[of the University of Dorpat]was divided into that of physicomathematics and that of history-philology, and in1851Minding was elected to a four-year term as dean of the former division.

His work, which continued Gauss's study of 1828 on the differential geometry of surfaces, greatly influenced Karl Peterson. Adolf Kneser said of Minding:-

For example, in 1830 Minding published on the problem of the shortest closed curve on a given surface enclosing a given area. He introduced the 'geodesic curvature' although he did not use the term which was due to Bonnet who discovered it independently in 1848. In fact Gauss had proved these results in 1825 before either Minding or Bonnet, but he had not published them. Minding also studied the bending of surfaces proving what is today called Minding's theorem in 1839. The following year he published in Crelle'sMinding may be described as the first successor of Gauss, who, though moving along the master's path, has gone in essential points beyond Gauss.

*Journal für die reine und angewandte Mathematik*a paper giving results about trigonometric formulae on surfaces of constant curvature. Lobachevsky had published, also in Crelle's Journal, related results three years earlier and these results by Lobachevsky and Minding formed the basis of Beltrami's interpretation of hyperbolic geometry in 1868. Victor Katz writes [5]:-

Minding also worked on differential equations, algebraic functions, continued fractions and analytic mechanics. In differential equations he used integrating factor methods. This work won Minding the Demidov prize of the St Petersburg Academy of Sciences in 1861. It was further developed by A N Korkin. Darboux and Émile Picard pushed these results still further in 1878.Gauss's studies were continued by Minding, who was particularly interested in surfaces of constant negative curvature and, in an article in1939, found the three surfaces of revolution to which they can be applied, among them the surface generated by the revolution of the tractrix around its own asymptote, i.e., Beltrami's 'pseudosphere'. In a later article(1940), Minding arrived at another interesting result, though without perceiving its important implications. He observed that the trigonometric relations in geodesic triangles of a surface of constant negative curvature could be obtained from the corresponding formulas of spherical geometry on a sphere of radius R by multiplying R by √-1. While Minding failed to notice that these formulas agree with those for the hyperbolic plane, established by Lobachevsky in his 'Imaginary geometry'(1837), Beltrami was aware of this fact, which he developed in his 'Attempt'.

For a list of papers written by Minding, see THIS LINK.

Papers published by Minding from 1868 until the end of his life include (we give English translations of the German or French titles): *Proof of a theorem in statics* (1868); *A rule of forming denominators and numerators in the representation of a continued fraction by an ordinary fraction* (1869); *On a problem in the probability theory originating from meteor observations* (1869); *On the method of least squares* (1871); *On the mean curvature of surfaces* (1875); *On the curves of shortest circumference on surfaces of revolution* (1876); *On some isoperimetric problems* (1877); *Theory of shortest curves on curved surfaces* (1878); and *On the Theory of the shortest curves over rings with given area on curved surfaces* (1878).

You can see the German or French titles in the list of Minding's papers at THIS LINK.

In 1882, with a rise in nationalistic policies in Russia, a move began to change the University of Dorpat to a Russian institution, both in terms of the language of instruction and the nationality of the professors. It was renamed the University of Yuryev in 1898 and, except for a period of German occupation during World War II, it became the University of Tartu in 1919 with teaching in Estonian. Minding retired in 1883, near the start of this change forced on the university. He died two years later.

As we mentioned above, Minding was awarded the prestigious Demidov Prize by the St Petersburg Academy of Sciences in 1861. He was elected to the St Peterburg Academy of Sciences in 1864, the same year in which he took Russian citizenship. He was further honoured by the St Petersburg Academy of Sciences in 1879 when they made him an honorary member.

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

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