# Domenico Montesano

### Quick Info

Born
22 December 1863
Potenza, Basilicata, Italy
Died
1 October 1930
Naples, Italy

Summary
Domenico Montesano was an Italian mathematician who studied the theory of linear congruences and conic bilinear complexes.

### Biography

Domenico Montesano's parents were Leonardo Antonio Montesano and the duchess Isabella Schiavone of Aragona. Isabella was Leonardo's second wife, his first wife having died at a young age. Leonardo and Isabella had nine children, five boys and four girls. Leonardo practised as a lawyer in Potenza and brought his family up strictly following the ways of the Kingdom of the Two Sicilies which existed until the unification of Italy in 1860. It was a typical intellectual family of the time: all the sons were able to make the most of their talents having keen intelligence as well as integrity and modesty. Two of the boys followed their father to become lawyers, two became medical doctors and Domenico, the subject of this biography, became a professor of mathematics. The four girls, following the customs of the day, were educated only to high school standard and then cultivated the practice of certain home arts such as music, painting and embroidery. One of the two sons who took up medicine was Domenico's younger brother Giuseppe Ferruccio Montesano, born in Potenza on 4 October 1868. He became one of the founders of psychology and child psychiatry in Italy.

After attending schools in Potenza, Domenico entered the University of Rome in 1880 and, after studying under Luigi Cremona and Giuseppe Battaglini, graduated in 1884 with the laureate. Italy was unified three years before Domenico was born and he grew up in a country that was coming to terms with this important event. Only in 1871 did Rome become the capital of Italy and immediately efforts had been made to make the new capital a scientific centre of excellence. Top Italian mathematicians were offered positions there and Montesano's teachers Battaglini and Cremona were two of these. Given that Battaglini and Cremona were distinguished geometers, it is not surprising that Montesano's research was in geometry.

After graduating he remained at the University of Rome for four years as an assistant. During these years he published works such as: Su la corrispondenza reciproca fra due sistemi dello spazio (1885); Su le correlazioni polari dello spazio rispetto alle quali una cubica gobba e polare a se stessa (1886); Su certi gruppi di superficie di secondo grado (1886); Su i complessi di rette di secondo grado generati da due fasci proiettivi di complessi lineari (1886); and Su alcuni complessi di rette-Battaglini (1886). He lectured on projective geometry and his lecture notes were published as Lezioni di geometria proiettiva (1887). This book proved popular and it ran to several editions, for example by 1905 a forth edition was published.

By 1888 Montesano had 13 publications and was in a strong position when he entered the competition for the chair of descriptive and projective geometry at the University of Bologna. He was ranked first by the referees and was appointed to the chair. In 1893 he entered the competition for the chair of projective geometry at the University of Federico II at Naples. The referees appointed to make the decision were Ferdinando Aschieri, Eugenio Bertini, Francesco Chizzoni, Vittorio Martinetti and Giuseppe Veronese. Five candidates were eligible to be considered for the chair, the other four being Luigi Berzolari, Alfonso Del Re, Mario Pieri and Federico Amodeo. It proved a very close contest with Montesano coming out on top just ahead of Berzolari. Two years later, Montesano was promoted to ordinary professor of higher geometry at Naples. For many years he was also responsible for the teaching of higher mathematics at Naples.

Let us note that filling the chair of descriptive and projective geometry at the University of Bologna, which Montesano vacated in 1893 when he went to Naples, remained unfilled until 1896 when a competition was held. Montesano was a referee in this competition which appointed Federigo Enriques with Mario Pieri coming a close second. Montesano was also a referee for the competition for the chair at Turin in 1901 when Gino Fano was appointed. Again in 1905, when competitions for two chairs at Padua were advertised, Montesano acted as a referee for both.

Montesano's research was almost all in geometry, and he followed the lines of research that he had begun under the influence of his two teachers Battaglini and Cremona. In particular he undertook much research on Cremona transformations. In 1898 Montesano published Una estensione del problema della proiettivita a gruppi di complessi e di congrunze lineari di rette . This is discussed by Virgil Snyder in [3] who writes:-
The existence of irrational involutions of points in ordinary space was established by Enriques, and concrete examples of such were later given by Aprile. One of the unsolved problems of modern algebraic geometry is the determination of involutions of minimum order, in particular, to determine whether involutions of order two exist that are certainly irrational. Among the methods used in attacking this problem is the study of those birational involutorial transformations of space contained multiply in a linear line complex. An involution mapped on the general cubic variety of $S_{4}$ is probably an example; two series of such were given at the Bologna Congress of 1928 ... . Here I wish to develop another case, which was first found incidentally by Montesano in connection with another problem.
Let us note that Giorgio Antonio Aprile, mentioned by Snyder in the above quote, was a student of Montesano's at Naples, being awarded his laureate in 1910.

In 1907, in the Rendiconti di Napoli, Montesano published the paper Su nuovi tipi di superficie razionali di 5o ordine showing how to obtain a quintic surface with two consecutive skew double lines by quadratic transformation of a rational sextic. This is discussed in [4].

Montesano was honoured with election to the Royal Academy of Sciences of Naples. In 1921 he became president of the Academy of Science, Physics and Mathematics of Naples. He was described as:-
... an honest, conscientious man who was very well liked.

### References (show)

1. E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).
2. G Scorza, Commemorazione del socio ordinario residente Domenico Montesano, Rend. R. Acc. delle Scienze Fisiche e Mat. di Napoli (3) 36 (1930), 145-154.
3. V Snyder, On an Involutorial Transformation Found by Montesano, Annals of Mathematics (2) 31 (3) (1930), 335-343.
4. A R Williams, The Montesano quintic surface, Bull. Amer. Math. Soc. 34 (1928), 761-770.