**Leonida Tonelli**'s family were Venetian. He spent his childhood in a number of different Italian cities, living most of the time in Gemona, Pavia and Pesaro. He completed his schooling in Pesaro, where he undertook technical studies, and then entered the University of Bologna in 1902 at the age of seventeen. His lecturers at Bologna included Salvatore Pincherle, who taught him analysis, Federigo Enriques, who taught him projective and descriptive geometry, and Cesare Arzelà who taught function theory. 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. His university studies, as a consequence, took longer than might be expected.

Cesare Arzelà (1847-1912), who had studied in Pisa with Ulisse Dini and Enrico Betti, was Tonelli's thesis advisor at Bologna. Tonelli was awarded a doctorate in 1907 for his thesis *I polinomi di approssimazione di Tchébychev* Ⓣ. This work on Chebyshev polynomials was published in a 72-page paper in the *Annali di Matematica pura e applicata* in 1908. This was not his first publication for *Sulle funzioni derivate* Ⓣ was published in 1907, with *Sulla rettificazione delle curve* Ⓣ (1908) and *Discontinuità di 1a specie e gruppi di punti* Ⓣ (1908) appearing in the following year. 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. The tension between Pincherle and Arzelà was a rather sad situation particularly when, after a few years, Arzelà became ill and could not give all his courses. 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. He published two major papers in 1910, namely *Sull'iterazione* Ⓣ and *Sulla rappresentazione analitica delle funzione di piť variabili reali* Ⓣ.

A competition for the Chair of Infinitesimal Analysis at the University of Parma was announced and Tonelli took part. He was the unanimous winner of the competition in 1911 but problems arose which prevented the appointment being made. First Arzelà became ill and then, in March 1912, he died. The final decision was taken by the Board of Governors at its meeting in June 1913 when they decided to cancel the competition. 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. Again he was placed first and, on 16 October 1913, took up the Chair at the University of Cagliari.

Tonelli had a chair but he was not entirely happy. He had two problems, one being that he was not too happy to be in Sardinia and would much prefer to be on the Italian mainland, the other being that he would much rather have a Chair of Infinitesimal Analysis to a Chair of Algebraic Analysis. He therefore took part in the new competition for the Chair of Infinitesimal Analysis at the University of Parma and again was successful. He took up his chair in Parma in 1914, at about the time that World War I was beginning in Europe.

At first Italy kept out of the war and, while much of Europe became fought, Tonelli taught courses in rational mechanics at Parma. However, in May 1915 Italy entered the war, declaring war on Austria-Hungary. However, their offensive was not successful and the Austrians began a counter-offensive in March 1916. Later that year Tonelli decided to volunteer for war service. He served in the artillery in the Seventh Army, fighting with great bravery in the mountains in Macedonia near the town of Monastir (now called Bitola) where he contracted malaria. 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. He received many decorations for his bravery during the fighting. In addition to the Croce al Merito di Guerra (War Merit Cross) and the bronze Medaglia Al Valore Militare (Military Medal for Valour) for bravery in the military action at Mezzolombardo, he also received two Commendations and two Promotions of War Merit. The artillery commander of the Seventh Army described Tonelli as follows [1]:-

An officer of truly exceptional quality, capable of carrying out any assignment with skill, alertness and decision, he was one of the most intelligent performers under my command. He is also a brave soldier, who worthily honoured his chair.

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. He had taken up again the duties of his chair at Parma in 1918 but in the spring of 1922 he was called to the Chair of Higher Analysis at the University of Bologna. There he taught analysis courses, alternating his teaching with that of Pincherle until Pincherle retired in the autumn of 1928. 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. He received the Gold Medal from the National Academy of Sciences of Italy (the "Academy of Forty"), and in 1927 the Accademia dei Lincei awarded him their Grand Prize for his mathematical achievements in 1925. Over these years his publication record is truly remarkable with eight papers appearing in 1925, including important ones on the calculus of variations, and eleven papers in 1926. He was an invited speaker at the International Congress of Mathematicians in Toronto in August 1924, giving the lecture *Sul calcolo delle variazioni* Ⓣ. Four years later, he was a plenary speaker at the International Congress of Mathematicians held in Bologna in September 1928, giving the lecture *Il contributo italiano alla teoria delle funzioni di variabili reali* Ⓣ. In addition to his plenary talk, he also gave the lecture *Sulla semicontinuità degli integrali doppi* Ⓣ at the Congress. Again, at the International Congress of Mathematicians in Zürich in 1932, he gave the invited lecture *Sul calcolo delle variazioni* Ⓣ.

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. However, the School of Mathematics at the University of Pisa was going through hard times and they looked for an outstanding mathematician to revive its fortunes. They offered Tonelli the Chair which he accepted in 1930. He remained at Pisa for the rest of his career, although during the years of 1939-42 he was officially transferred to the University of Rome. However, he did not spend time in Rome, treating the appointment as only nominal. The Department of Mathematics at the University of Pisa is now named for Leonida Tonelli. The Department's web page explains Tonelli's influence:-

Tonelli taught in Pisa for many years. After his premature death in1946, it fell to his students to carry forward his demanding heredity during the extremely difficult years following the end of the Second World War. One of these, Alessandro Faedo, for many years rector of the University of Pisa, was largely responsible for the subsequent flowering of Pisan mathematics. It began with his bringing to Pisa Federico Cafiero, a student of Renato Caccioppoli, and reached its peak in the1960's with the arrival of Aldo Andreotti, Jacopo Barsotti, Enrico Bombieri, Sergio Campanato, Gianfranco Capriz, Ennio De Giorgi, Giovanni Prodi, Guido Stampacchia, and Edoardo Vesentini. 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* Ⓣ 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. H H Goldstine writes:-

This paper is an exposition of the reasons that led the author to develop his approach to the calculus of variations and of the results obtained by himself and others. He discusses at some length the classical calculus of variations before showing how the functional calculus was brought to bear on the calculus of variations by means of the concept of semi-continuity. Some of the more important results of his work are mentioned in passing, and the essay closes with a few remarks on extensions of the calculus of variations to abstract spaces.

The interesting paper *Su alcuni concetti dell'analisi moderna* Ⓣ (1942) is, Tibor Radó writes:-

... concerned with the concepts of bounded variation and absolute continuity for functions of one variable and functions of several variables, with particular emphasis upon the work of Italian mathematicians.

We have already mentioned the remarkable number and quality of Tonelli's papers. He published 137 papers, all single authored except one in 1915 written in collaboration with** **Guido Fubini, and a number of important books including *Fondamenti di calcolo delle variazioni* Ⓣ (2 volumes) (1921, 1923), *Serie trigonometriche* Ⓣ (1928), and (with E Lindner) *Corso di matematica per la Scuola media* Ⓣ (3 volumes) (1941, 1942). His large number of publications meant that it was decided to produce four volumes of *Selected Works*. The first volume *Funzioni di variabile reale* Ⓣ, published in 1960, contains 46 papers on the theory of functions of a real variable. The second volume *Calcolo delle variazioni* Ⓣ, published in 1961, contains 27 articles, namely, all those dealing with the calculus of variations which appeared between 1911 and 1924. The third volume *Calcolo delle variazioni* Ⓣ, published in 1962, contains all papers on the calculus of variations that appeared after 1926, including a posthumous paper of 1950. The fourth, and final, volume *Argomenti vari* Ⓣ, 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. He was buried alongside Ulisse Dini and Luigi Bianchi in the Monumental Cemetery in Pisa.

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