**Salvatore Pincherle** was born in Trieste (part of Austria at the time) into a Jewish family. After the unification of Italy in 1860, Trieste remained the Imperial Free City of Trieste, the capital of the Austrian Littoral region, and the main Austrian commercial port. Salvatore's father was a business man and, after Salvatore had begun his education in Trieste, he moved to Marseilles taking his family with him. This move was made since the family were liberals, strongly in favour of Trieste becoming part of Italy and, since his father expressed these views openly, they were being oppressed by the local police [9]:-

Pincherle spent his childhood and adolescence in a small and cosy family atmosphere, where the intimacy of affection was strengthened by the sadness of exile and the passionate expectation of the historical events which, between the years1859and1870, were leading to the unification of his homeland.

Pinchele's mother, a very learned lady with excellent taste, was the young boy's first teacher. Then Pincherle began his school education at the Lycée Impérial (now the Lycée Thiers) in Marseilles. His initial interests were in the humanities, but the school in Marseilles specialised in science teaching and Pincherle soon became fascinated with mathematics through excellent teaching there. His childhood education, however, gave him a love for music and literature, and these interests continued throughout his life [5]:-

In his last years he still played piano nearly every day, and reread the works of his favourite author, Honoré de Balzac.

Leaving his father's home in Marseilles when his schooling was complete at the end of 1869, Pincherle went to Pisa where he won the competition for a place at the Scuola Normale Superiore. In 1870 he began his study of mathematics. He was taught by Enrico Betti who, after working in mathematical physics, was undertaking research in analysis and laying the foundations of general topology. Betti was a strong influence on Pincherle as was Ulisse Dini who, after working on differential geometry was, by this time, studying and teaching the foundations for the theory of functions of a real variable. Pincherle graduated with his laurea in 1874, also earning in the same year his right to lecture, the 'libera docenza', having submitted his two-part thesis *Sulle superficie di capillarità* Ⓣ and *Sulle costanti di capillarità* Ⓣ.

The quality of his work at the Scuola Normale Superiore of Pisa was very high and it was clear that Pincherle had shown that he had a very promising career in front of him as a university teacher. However, to go down that road would have meant that Pincherle would have had to have been supported by his father and he decided that he did not want to put any financial strain on the family. He decided, therefore, to seek a position as a secondary school teacher. He was appointed to the Regio Ginnasio Liceale of Pavia (today the Liceo Foscolo) where he taught the course *Sulle superficie d'area minima* Ⓣ. However, in addition to his high school teaching, he also made contact with Felice Casorati at the University of Pavia. In 1876 Eugenio Beltrami was appointed to the chair of mathematical physics at Pavia and he and Casorati had a major influence on Pincherle [9]:-

So, in that quiet place, Pincherle found new cultural support, new ideas and a new direction to his scientific work, and, with the freshness and energy of youth, he learned to impose severe discipline of work, which from then on he constantly observed throughout his long life, and it allowed him not to deviate from his research, even amid the cares of school teaching, to which he devoted himself with tireless fervour.

Casorati and Beltrami quickly had Pincherle interested in the new approach to analysis that Bernhard Riemann was taking. They encouraged Pincherle to apply for a postgraduate scholarship to enable him to study abroad for a year. He spent his year abroad in Germany, studying at the University of Berlin. During the academic year 1877-78 that he spent in Berlin, he attended lecture courses by Kummer, Kronecker and Weierstrass. However, it was Weierstrass who was the strongest influence on him and all his mathematical work from this time on shows the influence of the great mathematician. He returned to Pavia in the autumn of 1878 and resumed his work as a teacher at the Liceo. However, he was invited to give a course of lectures at the University of Pavia and he gave *Teorica delle funzioni analitiche secondo Weierstrass* Ⓣ, presenting, for the first time in Italy, Weierstrass's approach to analysis. He published these lecture notes as *Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principi del prof. C Weierstrass* Ⓣ in Giuseppe Battaglini's *Giornale di Matematiche* in 1880. This work is important both in the development of analysis and in particular the progress of mathematics in Italy.

Let us say a little about Pincherle's family at this point. He married Emma Morpurgo and their son, Maurizio Pincherle, was born in Pavia on 13 November 1879. Maurizio studied medicine and was appointed Professor in the Pediatric Clinic at the University of Bologna in 1929. Maurizio married Gilda Carneo and their son, Leo Pincherle (1910-1976) became a physicist, while another son, Mario Pincherle (1919-), studied both classics and engineering, and became a famous archaeologist. Salvatore and Emma Pincherle also had a daughter, Edvige Pincherle, who married Graziano Senigaglia. Graziano and Edvige had a daughter, Emma Senigaglia (1909-1991), who became a mathematician, studying for her doctorate at Bologna advised by Giuseppe Vitali.

Let us return to Salvatore Pincherle's biography. In the spring of 1880, following a competition, he was appointed to the chair of algebraic analysis and analytic geometry at the University of Palermo. It was a post he only held for a few months for he was offered a similar chair at the University of Bologna. He accepted the post which he continued to hold until he retired in 1928. At Bologna, Pincherle became a colleague of Luigi Donati (1846-1932), who was appointed to Bologna in 1877, and Cesare Arzelà who had been appointed to the chair of Infinitesimal Calculus in 1880. All three of these mathematicians were graduates of the Scuola Normale Superiore of Pisa and they quickly improved the department at Bologna which had somewhat lost its vigour [9]:-

Supported by exceptional learning, which never ceased to deepen and extend, even outside of his favourite field of work and beyond the boundaries of mathematics, he varied the topics of his courses from year to year, while always trying to illustrate the general theory of analytic functions in its historical setting both in its various orientations and in its relations with other branches of analysis.

We can only give a very few indications of the wide range of Pincherle's contributions since the list of his publications (see [15]) contains 271 items. His research mainly concerned functional equations and functional analysis. Together with Vito Volterra, he can claim to be one of the founders of functional analysis. Francesco Tricomi writes in [1]:-

Remaining faithful to the ideas of Weierstrass, he did not take the topological approach that later proved to be most successful, but tried to start from a series of powers of the D derivation symbol. Although his efforts did not prove very fruitful, he was able to study in depth the Laplace transform, iteration problems, and series of generalised factors.

Jean-Luc Dorier expresses a high opinion of Pincherle's contributions in [4]:-

From about1890, Pincherle published several papers in which he used the axiomatic approach with differential and integral equations. In1901, with one of his students, Ugo Amaldi, he published a book entitled 'Le Operazioni Distributive e le loro applicazioni all'analisi', in which the authors presented an axiomatic theory of functional operators. Although Pincherle referred to Grassmann and Peano, his approach went far beyond the framework of geometry and placed itself in quite a general context using, in particular, infinite-dimensional linear spaces. In this sense, his work was quite unusual for its time, since the use of axiomatic theory for infinite dimension was not much investigated until after1920...[However]his work had very little influence on the development of what would become functional analysis. The reasons for this lack of influence are partly due to the fact that such a general and formal axiomatic approach as proposed by Pincherle did not meet the concerns of most mathematicians in the first years of the20th century. Indeed, the conception of a function as a series of coefficients was still dominant at this time...

This book is quite remarkable in presenting ideas well ahead of its time. Pincherle defined the notion of dimension and basis, showing that in a set consisting of all linear combinations of n independent elements, n+1 elements are always dependent. He defined rank in an abstract way, then related it to the rank of a matrix. He then went on to look at linear operators, proving results which hold in both finite dimensional and infinite dimensional spaces. He also studied linear sets of numerical sequences and of Laurent series. He considered different operators, including differential operators and difference operators. Despite these far-sighted contributions, his work had little impact [4]:-

... even though he was the author of the article on functional equations and operators in the French version of the "Encyclopédie des mathématique pures et appliquées"(1912), in which he gave a very detailed historical account and referred to his own work, Pincherle's work itself did not have much influence.

In [18] (see also [17]) Francesco Mainardi and Gianni Pagnini discuss another very significant contribution by Pincherle:-

... the1888paper(in Italian)of S Pincherle on the 'Generalized Hypergeometric Functions' led him to introduce the afterwards named Mellin-Barnes integral to represent the solution of a generalized hypergeometric differential equation investigated by Goursat in1883. Pincherle's priority was explicitly recognized by Mellin and Barnes themselves ... In1907Barnes ... wrote: "The idea of employing contour integrals involving gamma functions of the variable in the subject of integration appears to be due to Pincherle, whose suggestive paper was the starting point of the investigations of Mellin(1895)though the type of contour and its use can be traced back to Riemann." In1910Mellin ... devoted a section(Proof of Theorems of Pincherle)to revisit the original work of Pincherle...

In 1915 Pincherle published his lecture notes as *Lezioni di Calcolo Infinitesimale Dettata Nella R. Universita di Bologna e Redatte per uso Degli Studenti*. These were very popular and, in 1919 he published a second edition in which he wrote [27]:-

The lectures on the infinitesimal calculus which I gave to the press at the end of1915, not without fear and trembling, have won favour beyond all expectations with the mathematical public...

A third edition appeared in 1926.

In addition to his remarkable research contributions and his university teaching, Pincherle was also involved in school level mathematics. He participated in the management of the teacher's college at Bologna and wrote a number of school level textbooks. For example *Gli elementi dell'Aritmetica* was first published in 1891 and ran to eighteen editions, the last being published in 1934. Similarly *Algebra elementare* ran to thirteen editions between 1883 and 1920, *Geometria pura elementare* ran to eight editions between 1881 and 1918, and *Geometria metrica e Trigonometria* ran to ten editions between 1882 and 1933. He also published problem books such as *Esercizi sull'Algebra elementare *(1896) and *Esercizi sulla Geometria elementare* (1897). He contributed several articles to encyclopaedias, one of which we have already mentioned above. Articles which appeared in the *Enciclopedia Italiana di Scienze, Lettere ed Arti* are *Determinanti* (1931), *Funzioni notevoli *(1932) and an obituary of Karl Weierstrass (1937). He served as an editor of the *Annali di matematica pura ed applicata* from 1918.

The Unione Matematica Italiana (Italian Mathematical Union) was established in Bologna by Pincherle on 7 December 1922. He became the Union's first President and the first editor of the *Bolletino dell'Unione Matematica Italiana*, a role he held until his death in 1936. Carlo Pucci discusses Pincherle's efforts to set up the Union in [26]. Pincherle was supported by Luigi Bianchi and Vito Volterra but many other leading Italian mathematicians did not support the idea, being quite happy with regional societies such as the Circolo Matematico di Palermo. The fact that the International Mathematical Union was founded in 1920 did much to help Pincherle persuade others for the need for the Unione Matematica Italiana.

He attended the International Congress of Mathematicians held at Toronto, Canada, in August 1924 where he was one of the plenary speakers giving the lecture *Sulle operazioni funzionali lineari*. At this Congress, he was elected president of the International Mathematical Union for the eight-year period 1924-32. He invited the Congress to Bologna for the 1928 Congress but it was decided to delay a decision on whether to accept. There were severe political problems, for neither the 1920 Strasbourg Congress nor the 1924 Toronto Congress had not been truly international since German mathematicians had not been allowed to attend due to World War I. Bologna was accepted for the 1928 International Congress of Mathematicians with Pincherle as president. It was through his efforts that German mathematicians were allowed to attend the Congress. Basically he achieved this by having the invitation to the 1928 Congress coming from the University of Bologna and inviting mathematicians directly. The International Mathematical Union's policy was directly opposed to this; they wanted invitations to be made only through mathematical societies. Since Pincherle was president of the International Mathematical Union he was in a very difficult position and it was with great political skill that he succeeded. Of course, some mathematicians refused to attend but it was a truly international meeting. In his opening address to the Congress as its President, Pincherle said:-

... if one does not ask those invited to which country or school they belong, but only if they support the progress of science and its benefits, who will be able to refuse their support, who will want to perpetuate the quarrels in that area when one seeks only the agreement of reason.

At a meeting of the International Mathematical Union which Pincherle chaired in September 1928, a resolution was unanimously adopted supporting all Pincherle's actions. Despite this strong support, he informed the delegates that he was stepping down as president since his efforts in that role had exhausted him. The authors of [7] write:-

The Bologna Congress stands as Pincherle's personal 'tour de force'. He judged the mood of the majority of mathematicians in the world correctly and through a series of hard negotiations, he found his way around roadblocks with patience yet with determination.

Pincherle received many honours for his contributions. He was elected to the Reale Accademia dei Lincei in 1887. He received the first Royal Prize for Mathematics, established by Umberto I in 1878 and awarded by a commission of the Reale Accademia dei Lincei, in 1889. Luigi Bianchi also received a Royal Prize at the same time. Pincherle was awarded the Sacchetti prize by the city of Bologna when he was appointed to the chair there in 1928. He was elected to the National Academy of Sciences of Italy (the "Academy of Forty"), the Reale Accademia delle Scienze of the Institute of Bologna, the Reale Accademia delle Scienze of the Institute of Turin, the Reale Istituto Lombardo di Scienze e Lettere, the Reale Istituto Veneto, the Accademia Pontaniana of Naples, the Royal Society of Edinburgh, the Bavarian Academy of Sciences, the Academy of Science of Coimbra, the Helvetic Society of Sciences, and made an honorary member of the Moscow Mathematical Society and the Calcutta Mathematical Society. He was awarded an honorary degree by the University of Oslo. He was a member of the National Research Council and the Governing Council of the Circolo Matematico di Palermo. He was frequently dean of the Faculty of Sciences at Bologna and president of the Academy of Sciences of Bologna.

Finally, we note that in 1925 the "Manifesto of Fascist Intellectuals", establishing the political and cultural foundations of Fascism, was published in Bologna. Pincherle was one of the signatories of the Manifesto but he died before the Fascist regime published its "Manifesto of Race" in 1938 which dismissed Jews from university positions. Pincherle's closest friend over the last years of his life had been his colleague Beppo Levi who had signed the "oath to Fascism" in 1931 but was dismissed under the Manifesto of Race.

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

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