**Enrico Betti** is noted for his contributions to algebra and topology. Betti also did important work in theoretical physics, in particular in potential theory and elasticity.

- Enrico's schooling was at the Forteguerri school in Pistoia where he had a classical education.
- 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.
- 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.
- Betti graduated with a laurea in pure and applied mathematics in 1846 having been advised by the professor of algebra Giuseppe Doveri (1792-1857).
- Following the award of his degree, Betti was appointed as an assistant at the University of Pisa.
- 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.
- 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.
- 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.
- During these years when Betti was a secondary school teacher, he was undertaking research with Mossotti as his advisor.
- 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.
- Betti became the first to publish observations and demonstrations on Galois theory with his papers of 1851-1852.
- 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 Ⓣ(On the analytical determination of the flow of fluids through a very small opening) in 1850.
- Betti was appointed as professor of higher algebra at the University of Pisa in 1857.
- In particular in Göttingen Betti met and became friendly with Riemann.
- 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.
- 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.
- 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.
- Betti, without Brioschi's assistance, also translated another school text, namely Joseph Bertrand's Algebra elementare Ⓣ(Elementary algebra).
- 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.
- 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 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.
- 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 Ⓣ(Treatise on substitutions and on algebraic equations) (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.
- 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.
- 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.
- In his work on capillarity, 'Memoria sopra la teoria della capillarità' Ⓣ(Memoir on the theory of capillaries) 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.
- When Betti wrote 'Teoria della elasticità', the theory of elasticity was already mature with known principles, though not completely shared.
- 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.
- Betti published a memoir on the theory of elliptic functions La teorica delle funzioni ellitiche Ⓣ(The theory of elliptic functions) (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".
- The Betti numbers were so named by Henri Poincaré who was inspired to study topology through Betti's work on the subject.
- 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.
- 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.

Born 21 October 1823, Pistoia, Tuscany (now Italy). Died 11 August 1892, Soiana, Pisa, Italy.

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Algebra, Group Theory, Knot Theory, Origin Italy, Physics, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive