**Giuseppe Battaglini** was an Italian mathematician who was important for his work in non-Euclidean geometry.

- If the University of Naples had been a quality institution at this time, Battaglini would have studied there but it was a place with low standards, sadly neglected by the government.
- So, in 1844, Battaglini entered the Scuola di Ponti e Strade from which he graduated in 1848.
- Politics was not the only obstacle to Battaglini.
- Battaglini, however, began to publish papers such as: Sugli assi principali Ⓣ(On the main axes) (1850); Inscrivere in una superficie di 2 o grado un poligono in modo che i lati passino per punti dati Ⓣ(Inscribe a polygon in a surface of degree 2 so that the sides pass through given points) (1851); Soluzione di un problema di geometria a tre coordinate.
- Battaglini was named professor of higher geometry at the University of Naples in 1860.
- In 1863, Battaglini was one of the three founders of the Giornale di matematiche, the others being Nicola Trudi (1811-1894) and Vincenzo Janni (1819-1891).
- Battaglini edited the journal, aimed at university students, which became the main outlet for papers in non-Euclidean geometry in Italy.
- Many articles by Battaglini appear in the journal from 1863 onwards, but his first memoir on non-Euclidean geometry Sulla geometria immaginaria di Lobachevsky Ⓣ(On the imaginary geometry of Lobachevsky) was published in 1867.
- the part G Battaglini, often mentioned only for the foundation of his 'Giornale di Matematiche', has had in the elaboration and in the divulgation of non-Euclidean geometry.
- Battaglini was one of these and, appointed by Giuseppe Garibaldi, he took up his chair in Rome in 1872.
- Battaglini, whose research focused mainly on geometric topics was, nevertheless, deeply interested in new developments in other mathematical disciplines, particularly algebra.
- Immediately after the publication of Camille Jordan's 'Traité des substitutions et des équations algébriques', Battaglini realized the importance of such a ground-breaking treatise.
- There Battaglini published a work on the theory of substitutions and Galois theory written by the young Vincenzo Janni (1871, 1872, 1873).
- Battaglini's interest in group theory also manifested itself in the course he gave on the theory of groups of substitutions at the University of Rome during the academic year 1875-1876.
- In fact, at the end of the introduction of his 1878 memoir "Sopra l'isomorfismo dei gruppi di sostituzioni" Ⓣ(On the isomorphism of groups of substitutions), the paper he extracted from his 1877 thesis written under Battaglini's supervision, Capelli credited his supervisor's lectures for his introduction to the subject.
- As well as Capelli and Frattini, the two students of Battaglini mentioned in this quote, we should also mention his students Enrico D'Ovidio, Riccardo De Paolis (1854-1892), Ettore Caporali (1855-1886), and Domenico Montesano (1863-1930) all of whom became well-known algebraic geometers.
- Battaglini retired for health reasons in 1885 and returned to Naples where he remained until his death.
- Some of Battaglini's results have proved significant.
- For example, in his doctoral dissertation of 1868, Klein introduced a classification scheme for second-degree line complexes based on Battaglini's earlier work.
- Battaglini and Hoüel also exchanged ideas relating to mathematical education in various European countries.

Born 11 January 1826, Naples, Kingdom of the Two Sicilies (now Italy). Died 29, Apr 1894, Naples, Italy.

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Origin Italy

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