Betti studied mathematics and physics at the University of Pisa, winning a place as a student in one of the grand-ducal colleges where he supported himself by private tutoring. At the university he was taught by Ottaviano Fabrizio Mossotti (1791-1863) and Carlo Matteucci (1811-1868). Mossotti had been exiled from Italy for his liberal views and, after a while in Switzerland and then England, he had been professor of experimental physics at the University of Buenos Aires before returning to Italy in 1835. He taught at the University of Pisa from 1840 where he gave courses on mathematical physics, celestial mechanics and geodesy which Betti attended. Betti learnt about experimental physics from Matteucci who had studied in Paris under François Arago. It was Arago who had recommended his appointment to Pisa in 1840. We note that Matteucci was awarded the Copley Medal by the Royal Society of London in 1844. Betti graduated with a laurea in pure and applied mathematics in 1846 having been advised by the professor of algebra Giuseppe Doveri (1792-1857). Doveri's early education had been in Florence and he had obtained a degree in mathematics from the University of Pisa.
Following the award of his degree, Betti was appointed as an assistant at the University of Pisa. He worked at the university at a time when political and military events in Italy were intensifying as the country came nearer to unification. There were not only the internal politics of unification but there were problems with Austria and France, both countries having their own agendas. Mossotti strongly supported the fight for independence and led a Tuscany University Battalion in an attempt to achieve this aim. Betti joined this battalion led by Mossotti and, with the rank of corporal, he fought in the battle of Curtatone and Montanara on 29 May 1848. This battle, part of the War of Independence of Italy, was fought between Austrian troops who had been stationed in the fortified town of Mantua, and Tuscan soldiers who were supported by young volunteers like those of the Tuscany University Battalion. These young men, like Betti, had no experience of battle but were filled with enthusiasm for their cause and proved to be excellent fighters. Betti and others had been promoted to officers merely for the occasion. They spent fifteen days carrying out military training before the battle took place. The odds were heavily in the favour of the Austrian troops when they left Mantua to attack the Tuscans for there were 20,000 of them against 7,000 Tuscans. The Tuscany University Battalion waited for orders from the Tuscan army but, when no orders arrived and they could hear the sounds of battle about 2 km away, they charged into the battle and fought with great bravery beside the regular Tuscan soldiers. Eventually the Tuscans were forced to retreat with heavy losses but the Austrians had themselves received even greater losses and did not advance. Betti was extremely fortunate to survive the battle which proved to be a vital one in a campaign which would eventually be successful. After this battle, Betti returned to the University of Pisa.
After working as an assistant at the University of Pisa, Betti returned to his home town of Pistoia where he became a teacher of mathematics at the Forteguerri secondary school in the town in 1849. This, of course, was the school at which Betti had studied. He took on these teaching positions not because he wanted to spend his life as a school teacher but rather because he had to earn his living while he undertook research which, he hoped, would gain him a university appointment. Capecchi writes in :-
The relative cultural insulation determined the original character of his research on the solution by radicals of algebraic equations. Though Galois' works originated in the 1820s, still in the second half of the nineteenth century they were found hard to be understood even in France.In 1854 he moved to Florence where again he taught in a secondary school. During these years when Betti was a secondary school teacher, he was undertaking research with Mossotti as his advisor. In  the extant correspondence between Betti and Mossotti between 1847 and 1857 is published. Mossotti gives continual research advice to his pupil but it is clear from the correspondence that the relation between the two is not simply that of teacher and pupil but the two are also friends. Betti explains his ideas about research to Mossotti, in particular he was working to give satisfactory proofs of many propositions which Galois had simply stated without giving any proof. In fact Betti became the first to publish observations and demonstrations on Galois theory with his papers of 1851-1852. These papers, published in the Annali di Scienze fisiche e matematiche, are: Sopra la risolubilità per radicali delle equazioni algebriche irriduttibili di grado primo Ⓣ (1851); Un teorema sulle risolventi dell'equazioni risolubili per radicali Ⓣ (1851); and Sulla risoluzione dell'equazioni algebriche Ⓣ (1852). We should note, however, that these are not Betti's first publications for he had published the paper on mathematical physics Sopra la determinazione analitica dell'efflusso dei liquidi per una piccolissima apertura Ⓣ in 1850.
Betti was appointed as professor of higher algebra at the University of Pisa in 1857. In the following year he, along with Francesco Brioschi and Felice Casorati, visited the leading mathematical centres of Europe. They visited Göttingen, Berlin and Paris making many important mathematical contacts. In particular in Göttingen Betti met and became friendly with Riemann. Back in Pisa in 1859 he moved to the chair of analysis and higher geometry. He gave his inaugural professorial address in 1860 which was not published but details of it survive and are discussed in . Mossotti, who held the chair of mathematical physics, died in 1863 and Betti was appointed to that chair in addition to the chair of analysis and higher geometry.
We have already explained Betti's involvement in the 1858-59 war with Austria in which the French at first joined the Italians against the Austrians. However, by 17 March 1861, the Kingdom of Italy was formally created. Rome and Venice were not part of Italy at this stage, however, and there continued high levels of political activity as the government structure was discussed. Betti served in the government of the new country when he became a member of Parliament in 1862, representing Pistoia, continuing in this role until 1867.
In an attempt to improve his health, Riemann made an Italian visit in the autumn of 1863 and renewed his friendship with Betti. In a letter to his friend and colleague P Tardy, written from Florence on 6 October 1863, Betti writes (see ):-
I have newly talked with Riemann about the connectivity of spaces, and have formed an accurate idea of the matter.Over quite a number of years Betti mixed political service with service for his university. He served a term as rector of the University of Pisa and he became director of its teachers' college, the Scuola Normale Superiore, in 1864 holding this post until his death. Under his leadership the Scuola Normale Superiore in Pisa became the leading Italian centre for mathematical research and mathematical education. The creation of the new Kingdom of Italy led to a renewed interest in mathematics and its teaching throughout the country and Betti played a major role in this. He had strong views on how mathematics should be taught in schools. He :-
... loved classical culture, and with Brioschi he championed the return to the teaching of Euclid in secondary schools, for he regarded Euclid's work as a model of discipline and beauty.He furthered these aims by collaborating with Brioschi in making a translation of Euclid's Elements. Betti, without Brioschi's assistance, also translated another school text, namely Joseph Bertrand's Algebra elementare Ⓣ. We have already noted that in 1864 Betti succeeded Mossotti when he was appointed to the chair of mathematical physics and he continued to hold this chair for the rest of his life. Carlo Matteucci, another of his teachers, had founded the journal Nuovo Cimento in 1844 and, in 1863 Betti became its editor-in-chief. He published ten papers in this journal. In 1871 Betti founded the Annali della Scuola Normale - Sezione della classe di scienze fisiche e matematiche, designed as a place where students could publish dissertations or habilitation theses. In 1870 he moved from the chair of analysis and geometry to the chair of celestial mechanics. He also held this chair for the rest of his life.
Political events continued to build the new country of Italy, with the Treaty of Vienna bringing Venice into the Italian Kingdom in 1866. Rome was attacked by Italian troops the following year but France defended the city with its troops against the attack. There was widespread unrest in Italy due to dissatisfaction with the government and it was far from clear that the newly unified country would not split apart again. In 1870, however, Italian troops captured Rome and it became the capital of the Kingdom of Italy. Betti continued to undertake political roles in the developing country. He served as an undersecretary of state for education for eighteen months from October 1874 to March 1876 but he :-
... longed, however, for the academic life, solitary meditation, and discussion with close friends.He served as a senator in the Italian Parliament in 1884 but again he missed the academic life :-
His principal aim, however, was always pure scientific research with a noble philosophical purpose.As Ulisse Dini notes in , Betti worked in many, very different, mathematical areas such as: the theory of algebraic equations; the theory of elliptic functions, algebraic functions of a complex variable, on spaces of many dimensions etc in analysis and on the applications of this to geometry; he published many works on mathematical physics and celestial mechanics, the theory of Newtonian forces, the theory of heat, the theory of electricity, magnetism, elasticity, capillary, hydrodynamics, the motion of systems of particles, and the extension of the principles of dynamics. However, he is particularly known for his contributions to algebra and topology. In his early work in the area of equations and algebra, as we have already seen, Betti extended and gave proofs relating to the algebraic concepts of Galois theory. These had been previously published without proofs and in this task Betti thus made an important contribution to the transition from classical to modern algebra. He published these important contributions in several works starting in 1851 and he was the first to give a proof that the Galois group is closed under multiplication. In 1854 Betti showed that the quintic equation could be solved in terms of integrals resulting in elliptic functions.
However, we should not give the impression that Betti was the first to clarify all the difficulties in Galois' work. Although Jordan, in his Traité des substitutions et des equations algebriques Ⓣ (1870) credits Betti with having filled the gaps in Galois' arguments and with having been the first to establish the sequence of Galois' theorems rigorously, the fact is that Betti's work contains substantial obscurities and errors. Mammone, in , brings these points out very clearly, yet the paper  itself contains group theoretical errors as was pointed out by Peter Neumann when he reviewed it. Betti's errors appear to relate to normal subgroups of groups and he makes the false assumption that (in modern terms) every extension splits.
We have already mentioned above that Riemann visited Betti in Pisa in 1863. Influenced by discussions with his friend Riemann, Betti was inspired to do important work in theoretical physics, in particular in potential theory and elasticity. He also published papers on the theory of functions, concentrating on elliptic functions. In fact this change of direction by Betti towards mathematical physics led to him substituting chairs at Pisa in 1870 as we remarked above. Dini, who Betti had taught earlier, was appointed to fill his chair of analysis and higher geometry.
Let us look in a little more detail at Betti's work on mathematical physics and in particular on elasticity. We give a description based on a somewhat modified version of that given in :-
Betti explored several aspects of mathematical physics; one of the most important was that regarding classical mechanics. In his early work he assumed a mechanistic approach, where force and not energy is the funding concept and virtual work the regulating law. In his work on capillarity, 'Memoria sopra la teoria della capillarità' Ⓣ published in the 'Annali delle Università toscane (Pisa)', Betti assumes bodies as formed by molecules which attract each other at short distance and repel at very short distance, and which do not practically interact at larger, but still very short distances. In his memoirs 'La teorica delle forze che agiscono secondo la legge di Newton e sue applicazioni all'elettrostatica' on Newtonian forces, Betti declared his Newtonian ideaology. Betti changed his attitude in his second memoir on capillarity, 'Teoria della capillarità' published in 'Nuovo Cimento', by giving the potential an energetic meaning and a founding role, on the basis of William Thomson's studies. This change was definitive in the 'Teoria della elasticità' Ⓣ (1874), where no reference is made to internal forces, even avoiding the explicit mention of stress. When Betti wrote 'Teoria della elasticità', the theory of elasticity was already mature with known principles, though not completely shared. The exposition developed there, like modern handbooks, follows an axiomatic approach. Betti's principles are on the one hand the concepts of potential energy and strains, and on the other hand the principle of virtual work. Though the book is not particularly original for its theoretical aspects, it is important for the exposition of a general procedure to evaluate the displacements on a three-dimensional elastic continuum and for the solution of specific problems. Moreover, the manner in which the single arguments are presented became paradigmatic for most handbooks on the theory of elasticity.Betti published a memoir on the theory of elliptic functions La teorica delle funzioni ellitiche Ⓣ (1860), containing results which were developed further by Weierstrass some years later. He published an important paper on topology in 1871 which contained what we now call the "Betti numbers". This was his famous Sopra gli spazi di un numero qualunque di dimensioni Ⓣ published in the Annali di matematica pura ed applicata. The Betti numbers were so named by Henri Poincaré who was inspired to study topology through Betti's work on the subject. Another of his papers is Sopra una estensione della terza legge di Keplero Ⓣ (1888). This paper, in the area of celestial mechanics, generalises Lagrange's studies on the three-body problem.
We should also mention at this point the impressive list of students that studied with Betti including: Ernesto Padova (1845-1896), Eugenio Bertini, Cesare Arzelà, Guido Ascoli, Ulisse Dini, Gregorio Ricci-Curbastro, Vito Volterra, Valentino Cerruti (1850-1909), Giuseppe Lauricella (1867-1913), Carlo Somigliana (1860-1955), Salvatore Pincherle, Mario Pieri, Federigo Enriques and Luigi Bianchi.
Following his appointment to the chair at Pisa in 1857 he gave lectures on a wide range of different mathematical topics in each academic year up to the academic year 1890-91. An enthusiastic and clear lecturer, he was greatly loved by his students. By November 1890, however, he was already struggling to undertake his teaching commitments as he slowly became paralysed. This paralysis gradually became more severe making it increasingly difficult for him to continue to work at all. However, his death in his villa at Soiana came unexpectedly. Although he died in the middle of the summer vacation when most of the professors were absent from the university, nevertheless he was given a solemn funeral with many honours. It was attended by many of his colleagues and former pupils as well as many citizens of Pisa. He was buried in the Camposanto monumentale of Pisa.
Betti received many honours including election to the Accademia dei Lincei in Rome (1851), the National Academy of Sciences of Italy (the "Academy of Forty") (1860), the Academy of Sciences, Letters and Arts of Modena (1860), the Royal Society of Naples (1863), the Lombard Institute of Science and Letters in Milan (1864), the Academy of Sciences of Turin (1864) as well as the Berlin Academy of Sciences, the Göttingen Academy of Sciences, the Royal Society of London and the Royal Swedish Academy of Sciences in Stockholm. In  Bottazzini briefly describes the contents of twelve boxes in the library of the Scuola Normale Superiore in Pisa that contain Betti's unpublished notes and letters. There does not appear to be any further research published on the contents of these papers.
Article by: J J O'Connor and E F Robertson