**Gregorio Ricci-Curbastro**'s father was Antonio Ricci-Curbastro and his mother was Livia Vecchi. It was a family of high status known throughout the province of Ravenna. Antonio Ricci-Curbastro, although certainly never achieving anything close to the fame achieved by his son Gregorio, nevertheless was himself well known as an engineer. Neither Gregorio nor his brother Domenico attended school. All their education prior to entering university was carried out at home where their parents employed private tutors.

In 1869 Ricci-Curbastro entered the University of Rome with the intention of studying mathematics and philosophy. He was only sixteen years old at the time and, although he had not attended school, he was well prepared academically. Political events, however, conspired to make Rome a somewhat unfortunate choice, although a very natural one given his place of birth. When Ricci-Curbastro began his studies in Rome, although the Kingdom of Italy had been created a few years earlier, Rome was not part of that Kingdom being part of the Papal States in which Ricci was born and brought up. Rome had been attacked by Italian troops in 1867 but France had defended the city and employed its troops against the attack. In 1870, however, Italian troops captured Rome and it became the capital of the Kingdom of Italy. Ricci-Curbastro studied at Rome for one year from 1869 to 1870 and then returned to his parents home where he remained for two years before beginning a second university career.

This time he went, not to Rome but to the University of Bologna. He studied there during the years 1872-73, then moved to Pisa where he attended the Scuola Normale Superiore which, under Betti's leadership, was becoming the leading Italian centre for mathematical research and mathematical education. As well as attending lectures by Betti in Pisa, Ricci-Curbastro also attended lectures by Dini. In 1875 Ricci-Curbastro was awarded a doctorate for his thesis *On Fuchs's research concerning linear differential equations.* He remained at Pisa working on a paper which he presented the following year to fulfil the requirements necessary to teach. The paper was *On a generalisation of Riemann's problem concerning hypergeometric functions.* Neither this paper, nor his doctoral thesis, have been published.

A perceptive reader will have noticed that both of these first two works by Ricci-Curbastro were based on works by German, rather than by Italian, mathematicians. The next pieces of work which he undertook were, likewise, not based on ideas by Italian mathematicians. The first was a series of articles on Maxwell's theory of electrodynamics and the work of Clausius which Betti asked him to write. Three of these articles appeared in *Nuovo Cimento* in 1877 and, in the same year, an article appeared in *Giornale di matematiche di Battaglini* which Dini had asked him to write on Lagrange's problem on a system of linear differential equations.

Ricci-Curbastro now competed for a scholarship and he won one which allowed him to spend the year 1877-78 abroad. That he chose to go to Germany should be no surprise and in fact he chose to study at the Technische Hochschule in Munich where Klein had been appointed to the chair two years earlier. As well as Klein, Brill worked at the Technische Hochschule in Munich and Ricci-Curbastro attended lectures by both these famous mathematicians. As Speziali writes in [1]:-

Ricci greatly admired Klein, and his esteem was soon reciprocated; nevertheless, Ricci does not seem to have been decisively influenced by Klein's teaching. It was, rather, Riemann, Christoffel, and Lipschitz who inspired his future research. Indeed, their influence on him was even greater than that of his Italian teachers.

Returning to Pisa in 1879, Ricci-Curbastro became Dini's assistant. Then, from 1880 until his death in 1925 he was professor of mathematical physics at the University of Padua. He did not only teach mathematical physics, however, for from 1891 he also taught courses on advanced algebra at Padua. It was only after he was appointed to the chair at Padua that he had the security that would allow him to marry and, in 1884, he married Bianca Bianchi Azzarani. They had three children; two sons and a daughter.

Ricci-Curbastro's early work was in mathematical physics, particularly on the laws of electric circuits and differential equations. He changed area somewhat to undertake research in differential geometry and was the inventor of the absolute differential calculus between 1884 and 1894. The initial contributions had been made by Gauss, then the ideas had been developed in Riemann's 1854 *Probevorlesung* and in an 1861 paper which he wrote for a prize contest of the Paris Académie des Sciences. However, it was a paper of Christoffel, published in *Crelle's Journal* in 1868, which was the main influence on Ricci-Curbastro to begin his investigations in 1884 on quadratic differential forms. He first systematically presented the important ideas in 1888 in a paper written for the 800^{th} anniversary of the University of Bologna. Speziali writes in [1]:-

The method he used to demonstrate[the invariance of the quadratics]led him to the technique of absolute differential calculus, which he discussed in its entirety in four publications written between1888and 1892.

Much of Ricci-Curbastro's work after 1900 was done jointly with his student Levi-Civita. In a fundamental joint paper that year *Méthodes de calcul différentiel absolu et leurs applications* he used (for the only time) the name Ricci instead of his full name. This paper had been requested five years earlier by Klein. The authors state their aims in the preface to their important seventy-seven page paper:-

The algorithm of absolute differential calculus, the instrument matériel of the methods ... can be found complete in a remark due to Christoffel. But the methods themselves and the advantages they offer have their raison d'être and their source in the intimate relationships that join them to the notion of an n-dimensional variety, which we owe to the brilliant minds of Gauss and Riemann. ... Being thus associated in an essential way withV_{n}, it is the natural instrument of all those studies that have as their subject, such a variety, or in which one encounters as a characteristic element a positive quadratic form of the differentials ofnvariables or of their derivatives.

In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations. The main ideas of this paper are discussed in [1]. Ricci-Curbastro's absolute differential calculus became the foundation of tensor analysis and was used by Einstein in his theory of general relativity.

The paper [7], written by Ricci-Curbastro's student Levi-Civita, lists sixty-one of his publications. However, he found time to also contribute to local government as did many of the Italian mathematicians of his time. He served as a councillor for his home town of Lugo and in this capacity was involved in many projects relating to the supply of water and to swamp drainage (an activity which many Italian mathematicians became involved with over several centuries). Later he served as a councillor for Padua and there his interests included school education and finance. Offered the position of mayor of Padua, however, he declined.

Ricci-Curbastro received many honours for his outstanding contributions, although one would have to say that the importance of his work was not fully understood at the time when he produced it, but rather it was realised some time later. He was honoured with membership of several academies such as the Istituto Veneto which he was admitted to in 1892 and which he served as president in 1916-18. He also was a member of the Accademia dei Lincei from 1899, the Accademia di Padua from 1905, the Academy of Sciences of Turin from 1918, the Società dei Quaranta from 1921, the Reale Accademia di Bologna from 1922 and the Accademia Pontifica from 1925.

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

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