**Gianfranco Cimmino**'s father was Francesco Cimmino (1864-1939), a historian and poet from Naples who taught history at the Liceo Genovesi and taught Sanskrit at the University of Naples. He translated Indian poems and dramas, published his own poems, and wrote the music for two libretti. Benedetto Croce (1866-1952), the famous historian, humanist and philosopher, described Francesco Cimmino as:-

Gianfranco's mother Olimpia was from an aristocratic family, the Gibellini Tornielli Boniperti, from Novara in the Piedmont region of northwestern Italy. Olimpia and Francesco married in 1903 and their first two children were Maria Luisa and Eugenia. Maria Luisa died on 2 January 1907 when she was still a baby. Gianfranco, born on Thursday 12 March 1908, was baptized with the name of Gianfranco Luigi Giuseppe but only used the given name Gianfranco.... an expert connoisseur of oriental literature and a skilful author of verses for music ...

In 1923, when Cimmino was fifteen years old, he graduated from high school with the classical certificate. He studied mathematics at the University of Naples and graduated with his laurea in 1927. His thesis advisor had been Mauro Picone who had moved to the University of Naples in 1925. Cimmino was only nineteen years old when he graduated with his thesis on approximate methods of solution for the heat equation in 2-dimensions, but he was appointed as an assistant to Picone who held the chair of analytical geometry at the University of Naples. He held this position during the academic year 1927-28. He published material from his thesis in his first paper *Nuovo metodo d'approssimazione per la soluzione dei problemi di valori al contorno relativi all'equazione del calore* Ⓣ (1928). In 1929 he published two papers: *Estensione della identità di Picone alla più generale equazione differenziale lineare ordinaria autoaggiunta* Ⓣ and *Su una questione di minimo* Ⓣ. He was awarded a scholarship to study abroad and spent the academic year 1930-31 at the University of Monaco and the University of Göttingen.

In Naples, Cimmino was a member of an impressive team led by Picone. He was one of four assistants, the other three being Renato Caccioppoli, Carlo Miranda and Giuseppe Scorza Dragoni. Caccioppoli had graduated from Naples in 1925 where he had been a student of Ernesto Pascal, before becoming Picone's assistant. Carlo Miranda entered the University of Naples in 1927 and studied with Picone, graduating in 1931, and then becoming an assistant. These four assistants are referred to as "Mauro Picone's four musketeers" by Scorza Dragoni in his obituary of Miranda [4]:-

They worked in the "Istituto per le Applicazioni del Calcolo", attached to the chair of Mathematical Analysis, which had been founded by Picone in Naples in 1927 with financial assistance from the Banco di Napoli. This Institute was perhaps the earliest Institute for numerical analysis in the world. Picone left Naples in 1932, together with Miranda, but Cimmino remained there until 1938. After Picone left, Cimmino became a lecturer in charge of the courses of Higher Analysis between 1932 and 1935. Then he was in charge of the courses of Analytic Geometry from 1935 to 1938 [3]:-All four members of this remarkable group were bound by a strong sense of camaraderie and friendship.

His two most important papers on this topic wereSome of Cimmino's most remarkable papers date to the period1937-38and concern the theory of partial differential equations of elliptic type. In particular, Cimmino was the first mathematician to study the Dirichlet problem with generalized boundary conditions.

*Nuovo tipo di condizioni al contorno e nuovo metodo di trattazione per il problema generalizzato*Ⓣ (1937-38) and

*Sul problema generalizzato di Dirichlet per l'equazione di Poisson*Ⓣ (1940). Between these papers, however, he published the paper

*Calcolo approssimato per le soluzioni dei sistemi lineari*Ⓣ (1938) which actually gave details of work on numerical analysis that he had done a few years earlier. The paper contains the following introductory note by Picone:-

While in Naples, Cimmino entered the competition for the chair of Mathematical Analysis at the University of Cagliari, in Sardinia. He was ranked first by the panel and took up the position in 1938. However, he did not remain long at Cagliari for, in November 1939, he moved to Bologna when appointed to the chair of Mathematical Analysis at the University. He held this chair for the rest of his career. From 1965 to 1972 he was Dean of the Faculty of Mathematical, Physical and Natural Sciences of the University of Bologna. He was a special commissioner to the National Institute of Advanced Mathematics from 1973 to 1977 and he was director of the Institute from 1978 to 1982.Professor Gianfranco Cimmino can be considered, among other things, as one of the founders of the Istituto per le Applicazioni del Calcolo, to which he lent his continuing and productive assistance during the embryonic stages of the Institute itself, in Naples, in the laboratory annexed to that university's Calculus chair, from July1928to October1932. Towards the end of that period, Professor Cimmino devised a numerical method for the approximate solution of systems of linear equations that he reminded me of in these days, following the recent publication by Dr Cesari ..., which provides a systematic treatment of the above mentioned computing methods but which, however, does not consider the one by Cimmino, a method which, in my opinion, is most worthy of consideration in the applications because of its generality, its efficiency and, finally, because of its guaranteed convergence which can make the method practicable in many cases. Therefore, I consider it useful to publish in this journal Professor Cimmino's note on the above mentioned method, a note that he has agreed to write on my insistent invitation.

In 1940 he married Maria Rosaria Martinez (1914-1998); they had seven children born between 1941 and 1952, namely Francesco, Olimpia, Giovanni, Ortensia, Giuseppe, Maria Beatrice and Pietro. Now, also in 1940, various members of the family adopted the name Cimmino Gibellini Tornielli Boniperti, combining his father's surname with his mother's family name.

We have seen that Cimmino made contributions to partial differential equations of elliptic type and to computing approximate solutions to systems of linear equations. However, he also made important contributions to many other areas of mathematics, for example the calculus of variations; differential geometry; conformal and quasi-conformal mappings; topological vector spaces; and the theory of distributions. However, as Michele Benzi points out in [3] (and [4]), it was his work on numerical analysis, particularly the 1938 paper we mentioned above, that:-

Strangely, this important 1938 paper has been omitted from Cimmino's collected works [2]. Benzi discusses Cimmino's contributions to numerical analysis in the interesting paper [3] and gives the following summary of the impact of his mathematics:-... has had a remarkable, if belated, impact and has proven to be of lasting importance for many areas of applied scientific computing.

Cimmino published seven books, most of which were based on notes from his lecture courses. He also wrote some popular articles, includingWhile Cimmino did not live to see how extensive the influence of his numerical work would be, he did know that algorithms based on his1938paper were being developed and applied in the medical field. Indeed, one of his last reviews for the 'Zentralblatt', appeared in1988, concerned a paper on the use of Cimmino's method in radiation therapy planning. He must have been pleased to learn that his elegant, youthful intuition had been found to be useful in the medical field. The story of Cimmino's method is interesting for several reasons. It shows that good mathematical ideas may take many years to become fully appreciated and to be brought to fruition. Major technological changes may completely transform the perception of an algorithm, from useless curiosity to brilliant invention. If an idea is indeed good, it is almost certain that it will eventually come to the forefront(perhaps through re-discovery), no matter how obscure the original publication or how long the period of oblivion. ... It is also interesting to observe that Cimmino's impact on his main research area, the theory of partial differential equations, while non-negligible, has not been as great and as lasting as his work in numerical mathematics. The reasons for this are complex, but may be due at least in part to the state of relative isolation in which Cimmino(as well as other members of the Neapolitan mathematical school)conducted his research, publishing his most important papers in Italian and in journals of rather limited circulation. Moreover the techniques(and even the notation)he used were somewhat different from those adopted by many of the leading mathematicians of the time.

*Dante and Mathematics*. He received many honours for his contributions. He was awarded the Gualtiero Sacchetti prize from the city of Bologna in 1964-65 [Gualtiero Sacchetti (1836-1917) was an engineer and Senator of the Kingdom of Italy]. In 1965 he received the Gold Medal for Merit from the School of Culture and Arts. He was elected a member of the Academy of Sciences of the Institute of Bologna, of the National Academy of Lincei (in 1969), of the National Academy of Sciences, Arts and Literature of Modena, and of the National Society of Sciences, Arts and Literature of Naples.

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