**Axel Harnack** was an Estonian-born German mathematician who contributed to potential theory.

- Theodosius, born in St Petersburg, Russia, was appointed as a docent in church history at the University of Dorpat in 1843 and, by the time Axel was born, he was Professor of Practical and Systemic Theology there.
- We note that Adolf and Axel were twins.
- Axel completed his schooling in Dorpat then, in 1869, he began his university studies in mathematics and physics at the University of Dorpat.
- He had been appointed Professor of Physics at Dorpat in 1868, the year before Harnack began his university career.
- Harnack was an outstanding student and, in 1872 he was awarded a prize for a geometrical dissertation which has never been published.
- Having spent thirteen years growing up in Erlangen, it was a natural choice for Harnack that, after graduating from Dorpat, he went to Erlangen to write his doctoral thesis.
- Although Erlangen was not a major mathematical centre at this time, the young Felix Klein had been appointed there as professor in 1872 and he was a natural adviser for Harnack.
- We note that although Klein was a professor, he was only two years older than his student Harnack! The research project that Klein gave Harnack was to further investigate the parametric representation of the third-order curves which had been introduced by Alfred Clebsch.
- Harnack did excellent work and his thesis Über die Verwerthung der elliptischen Functionen für die Geometrie der Curven dritten Grades Ⓣ(On using elliptic functions for the geometry of curves of the third degree) was published in Mathematische Annalen in 1875.
- Harnack published four papers in 1875, the other three being Zur Theorie der ternären cubischen Formen Ⓣ(On the theory of ternary cubic forms), Über einen Beweis des Abel'schen Theorem Ⓣ(On a Proof of Abel's Theorem), and Über eine Behandlungsweise der algebraischen Differentiale in homogen Coordinaten Ⓣ(On a method of treatment of algebraic differentials in homogeneous coordinates).
- The last mentioned of these papers was, in fact, Harnack's habilitation thesis which he submitted to the University of Leipzig where he earned his right to teach and was appointed as a docent in 1875.
- After only a year at the University of Leipzig, Harnack was called to a chair of mathematics at the Polytechnic in Darmstadt in the autumn of 1876 but he moved again after a year, this time being called to a chair of mathematics at the Polytechnic in Dresden in 1877.
- Harnack's book is not simply a translation since he added several long and significant comments.
- The time that Harnack spent in Davos, sadly, did not lead to a permanent cure, and the illness recurred in the summer of 1887.
- Harnack had published the book Die Elemente der Differential- und Integralrechnung.
- Introduction to the study) in 1881 and George L Cathcart had approached Harnack in March 1888 to see if he would be happy for him to publish an English translation of the text.
- Harnack replied to Cathcart in letter dated 15 March 1888 about the English translation, wishing to make some changes in the material in the book for the new English edition.
- Harnack had been hoping to spend the Easter vacation in the north of Italy and Cathcart replied saying that he too would be in north Italy for the Easter vacation and suggested they meet there during the vacation.
- However, on 16 March Harnack collapsed when in the middle of giving a lecture at the Polytechnic.
- They never managed to meet in Italy since Harnack died in Dresden on 3 April.
- Harnack's mathematical contributions are significant and, although some of the results he published were false since he made an assumption that two concepts were the same when, in fact, they were subtly different, nevertheless his work is important in the development of integration theory.
- An indication of the importance of Harnack's contributions can be seen from the fact that (in March 2017) MathSciNet lists 738 papers with "Harnack" in the title.
- Most of these refer to the Harnack inequality or to the Harnack principle (sometimes known as Harnack's theorem).
- If one looks at how many papers in MathSciNet have a review which contains Harnack's name, one finds there are 1926.
- The Harnack inequality appears in his 1887 book Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunction in der Ebene Ⓣ(The foundations of the theory of the logarithmic potential and the unique potential function in the plane.).
- This principle, also in his 1887 book, together with the inequality form the starting point for a very rich and beautiful theory which explains the large number of publications with "Harnack" in the title.

Born 7 May 1851, Dorpat, Russian Empire (now Tartu, Estonia). Died 3 April 1888, Dresden, Germany.

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Origin Estonia

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive