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Enrico Betti's father died when he was very young and his mother had to bring him up and educate him on her own.
Betti studied mathematics and physics at the University of Pisa and while there he was taught by Mossotti. Betti graduated with a degree in mathematics in 1846 and, following this, he was appointed as an assistant at the university. 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 the battalion led by Mossotti and he fought in two battles, being those of Curtatone and Montanara.
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 a secondary school in the town in 1849. In 1854 he moved to Florence where again he taught in a secondary school. He was appointed as professor of higher algebra at the University of Pisa in 1857. In the following year Betti, along with Brioschi and Casorati, visited the 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 he moved in 1859 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 [5].
In 1859 there was a 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. However, Betti served in the government of the new country when he became a member of Parliament in 1862.
Riemann made an Italian visit in the autumn of 1863 and renewed his friendship with Betti. 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 teacher's 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 [1]:
... 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 Bertrand's Algebra elementare.
In 1864 Betti succeeded Mossotti when he was appointed to the chair of mathematical physics. His final move was to substitute the chair of celestial mechanics for his chair of analysis and geometry in 1870.
Political events in Italy continued, 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 a few months in 1874 but he [1]:
... longed, however, for the academic life, solitary meditation, and discussion with close friends.
Again he served as a senator in the Italian Parliament in 1884 but again he missed the academic life [1]:
His principal aim, however, was always pure scientific research with a noble philosophical purpose.
Betti is noted for his contributions to algebra and topology. His early work is in the area of equations and algebra. Betti extended and gave proofs relating to the algebraic concepts of Galois theory. These had been previously published without proofs. 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. 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 [8], brings these points out very clearly, yet the paper [8] 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 did 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.
Betti published a memoir on topology in 1871 which contained what we now call the "Betti numbers". These were so named by Poincaré who was inspired to study topology through Betti's work on the subject.
In [4] 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
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List of References (9 books/articles)
 
A Poster of Enrico Betti
 Mathematicians born in the same country

Honours awarded to Enrico Betti (Click below for those honoured in this way)  
LMS Honorary Member  1871 
Crossreferences in MacTutor
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School of Mathematics and Statistics University of St Andrews, Scotland  
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