**Leonida Tonelli** was an Italian mathematician who worked on the theory of functions and the calculus of variations.

- Tonelli caught a serious infection while on an excursion to the mountains south of Bologna and this meant that he could not study for some considerable time.
- Cesare Arzelà (1847-1912), who had studied in Pisa with Ulisse Dini and Enrico Betti, was Tonelli's thesis advisor at Bologna.
- After the award of his doctorate, Tonelli was appointed as Salvatore Pincherle's assistant at Bologna to work in algebra and analytic geometry.
- This arrangement had been made while Tonelli was still an undergraduate, much to the disappointment of Arzelà who had been very keen to have Tonelli as his own assistant.
- He asked for an assistant to help him with his teaching duties but Pincherle refused to let Tonelli take on the role for he wanted to retain his collaboration.
- As a result, Tonelli ended up teaching three courses along with the corresponding exercise classes.
- Despite this heavy teaching load, Tonelli's mathematical research became deeper and wider and in 1910 he qualified as a lecturer in Infinitesimal Analysis.
- A competition for the Chair of Infinitesimal Analysis at the University of Parma was announced and Tonelli took part.
- However, despite not being appointed to the chair in Parma after winning the competition, Tonelli soon was competing for another chair, this time the Chair of Algebraic Analysis at University of Cagliari.
- Tonelli had a chair but he was not entirely happy.
- At first Italy kept out of the war and, while much of Europe became fought, Tonelli taught courses in rational mechanics at Parma.
- Later that year Tonelli decided to volunteer for war service.
- Subsequently Tonelli was with the Seventh Army when it held the front from the Stelvio Pass to Lake Garda, and on the morning of 4 November 1918 when the Italian flag was raised at Mezzolombardo.
- With the end of the war, the rector of the University of Parma requested that Tonelli might he allowed to return to take up his university duties.
- As one would expect, Tonelli has no publications appearing in 1917 or 1918, but by 1919 he had returned to his remarkably regular output of quality papers.
- On his retirement, Tonelli was appointed to his Chair of Infinitesimal Analysis.
- Around this time Tonelli received many honours recognising the high quality of his research contributions.
- As well as producing deep research, Tonelli was also involved in teaching Analysis courses at the University of Ferrara up to 1927, and later, at Bologna, he taught mathematics course for students of Chemistry and the Life Sciences.
- They offered Tonelli the Chair which he accepted in 1930.
- The Department of Mathematics at the University of Pisa is now named for Leonida Tonelli.
- The mathematicians that now constitute the faculty of the Department Leonida Tonelli are, in large part, the progeny of these outstanding mathematicians.
- His 1940 paper L'analisi funzionale nel calcolo delle variazioni Ⓣ(Functional analysis in the calculus of variations) is interesting in that it gives an insight into Tonelli's thinking on one of the topics for which he is most famed, namely the calculus of variations.
- We have already mentioned the remarkable number and quality of Tonelli's papers.
- The fourth, and final, volume Argomenti vari Ⓣ(Various topics), published in 1963, contains papers on trigonometric series, ordinary differential equations and integral equations (all published in or after 1924-25), and some miscellaneous work (from 1909 onwards), including Tonelli's biography of Salvatore Pincherle.
- Tonelli died at the age of 60 in 1946.

Born 19 April 1885, Gallipoli, Lecce, Italy. Died 12 March 1946, Pisa, Italy.

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Origin Italy

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