Guido attended the Foscarini grammar school in Venice, where his mathematics teacher was Aureliano Faifofer. After he had completed his secondary education he studied mathematics at the University of Padua. There Castelnuovo was taught by Veronese who gave him an interest in geometry. It was to become the main research interest of his career. Castelnuovo graduated from Padua in 1886 and spent the following year in Rome on a postgraduate scholarship. Veronese had advised Castelnuovo about the best places in which to continue his research but pointed out the problems of going to Rome:-
Rome should be the best place, but unfortunately Cremona is too busy in politics, I don't know how successfully, and he has not been interested in science for a long time. But I think he is still giving the Higher Geometry course.Soon after graduating Castelnuovo sent a copy of one of his papers to Corrado Segre, the professor in Turin. Castelnuovo expressed his surprise when he received a copy of the paper back with carefully thought out notes and suggestions on it and wrote to Segre saying how unusually helpful his comments had been. Segre replied:-
You are right. It is a bad habit, even if widespread, to praise papers without reading them. On the contrary I have the habit of reading all the papers I receive and of sending my remarks to the authors: I do with others what I would like the others to do with me.These letters mark the beginning of a long correspondence and collaboration between the two mathematicians.
After his year in Rome Castelnuovo obtained his first appointment as an assistant to D'Ovidio at the University of Turin. While at Turin he was strongly influenced by Corrado Segre. This was a period when Castelnuovo produced research of the highest quality on the theory of algebraic curves. In 1873 Alexander von Brill and Max Noether had published a joint work on properties of linear series. Castelnuovo made a major step forward reinterpreting the results of this paper in projective terms.
In 1891 Castelnuovo was appointed to the Chair of Analytic and Projective Geometry at the University of Rome. In Rome Castelnuovo was a colleague of Cremona but although he had given up active research he was still teaching the Higher Geometry course despite the fact that he had "not been interested in science for a long time", as Veronese had commented five years earlier.
After Cremona's death in 1903, Castelnuovo began to teach the advanced geometry courses. He divided his course into two parts, the first part being a general overview of mathematics while the second part was on the theory of plane algebraic curves. This structure was quite deliberate as it formed part of his philosophy on how mathematics should be taught. He wrote:-
... the reason for the division is that on the one hand it is necessary to have general culture, on the other hand it is necessary to have deep knowledge of a particular field.Another influence on the way he taught was his belief that students should:-
... see how problems arise from applications, but to look at the theoretical interest they have, beyond the interest in their applications. ... theoretical results could find unexpected applications in the future.Later in his career at Rome he taught a course on algebraic functions and abelian integrals in which he treated the theory of Riemann surfaces,and courses on non-euclidean geometry, differential geometry, interpolation and approximation, and probability theory. He explained why he found probability an interesting topic to teach:-
Probability is a science of recent formation; hence in it, better than in other branches of mathematics, one can see the relationship between the empirical contribution and the one given by reasoning, and between the process of inductive and deductive logic used in it. The fact that it is a science in the making explains why it is appropriate to give frequent examples to show the applications of known methods or to introduce new ones.Castelnuovo also wrote a book on probability, publishing Calcolo della probabilità Ⓣ in 1919 and a text on the theory of relativity in 1923. A summary of his important probability text is given in . An interest in the history of mathematics is evident from the interesting history book Le origini del calcolo infinitesimale nell'era moderna Ⓣ (1938) which he wrote on the calculus up to the time of Newton and Leibniz. His interest in the history of mathematics was also evident in all the courses he taught. Another of his interests was natural philosophy, in particular he was interested in determinism and chance, causality and indeterminacy.
Castelnuovo's most important work, however, was done in algebraic geometry, publishing Geometria analitica e proiettiva Ⓣ in 1903. His areas of interest in geometry included the geometry of algebraic curves, linear systems of plane curves from the point of view of birational invariants, and the theory of surfaces. In the area of algebraic curves we should mention the Castelnuovo-Severi inequality and a related criterion which Castelnuovo found for the linearity of an algebraic system on a curve. He published three famous papers on linear systems dating from the early 1890s. In the second of these, which appeared in 1891, he gave the first systematic use of the characteristic series and of the adjoint system. His study of the fundamental curves of the system led him to investigate the theory of surfaces, a topic on which he collaborated with Enriques.
Castelnuovo produced a series of papers over a period of 20 years which, together with Enriques, finally produced a classification of algebraic surfaces. Their collaboration began in 1892, shortly after Castelnuovo had taken up his chair in Rome. At first the collaboration was between the established mathematician Castelnuovo and the young twenty year old Enriques so the relationship was that of teacher and student. Enriques struggled with the concepts as he tried to understand the theory of algebraic surfaces, making mistakes in his attempts. He showed his appreciation of Castelnuovo's help and patience writing:-
If there is a thing which moves me and encourages me to correct my attitude, apart from the feeling of necessity of dong so, more than the rigour, even if benevolent, of Segre, it is your magnanimity. You, who are witness of my errors more than anyone else, never criticised me in a harsh way and you never showed impatience with me.They pursued the idea that a Riemann-Roch theorem for surfaces would be a powerful tool. Their classification of algebraic surfaces was published in 1914 but their collaboration had led to many joint papers during the course of the work. Details of their collaboration is given in . In 1901 Castelnuovo and Enriques had submitted their joint work for the Royal Prize in Mathematics awarded by the Accademia dei Lincei. Veronese, Cerruti, Bianchi, Dini and D'Ovidio formed the committee which had to decide whether to award the prize. After lengthy discussions it was decided not to make an award because Castelnuovo and Enriques had made a joint submission. They argued that it was joint work done in close collaboration so it was impossible to disentangle their individual contributions, but no award was made. Both received the prize in subsequent years; Castelnuovo in 1905 and Enriques in 1907. See  for details.
Castelnuovo is also remembered for the Kronecker-Castelnuovo theorem which states:-
If the sections of an irreducible algebraic surface with a doubly infinite system of planes turn out to be ruler curves, then the above surface is either ruled or the Roman surface of Steiner.Kronecker had first stated a version of this theorem in a lecture which he gave to the Accademia dei Lincei in 1886. Castelnuovo had only recently graduated when he was informed by Cremona of Kronecker's lecture and he found his own proof of the result. Kronecker never published the theorem and it was Castelnuovo's version which appeared in print.
In all Castelnuovo produced over 100 publications if one counts his books, articles and scientific memoirs. Further information on him has become available recently since his daughter Emma Castelnuovo, the author of , preserved the archive of Castelnuovo's papers and various historians of mathematics such as Gario and Conte have begun to study this material, see for example , and .
Castelnuovo retired from teaching at the University of Rome in 1935. This was a period of great political difficulty in Italy as in the rest of Europe. Italy's increasingly close alliance with Hitler and his policies was resented and feared by many in Italy and most certainly by Castelnuovo. There was a sudden decision to impose anti-Semitic laws in 1938 which condemned Jews as unpatriotic, excluded them from government posts and from state universities. Gario writes :-
In 1938, like all Italian Jews, he suffered the humiliation of the racial laws which banned thousands of Jews from Italian society.Castelnuovo, being a Jew, was forced into hiding during the years the Nazis were in power in Italy but he organised special courses to instruct Jewish students disadvantaged by anti-Semitic government policies.
Castelnuovo was appointed as a special commissioner of the Consiglio Nazionale delle Ricerche in June 1944, after Rome was liberated, being given the task of reviving the scientific institutions in Italy which had suffered badly under Mussolini during his 20 years in power. Then he became president of the Accademia dei Lincei, remaining in this position until his death. On 5 December 1949 he had the honour of being named senator of the Italian Republic. Among his many foreign honours was election to the Académie des Sciences of Paris.
Castelnuovo was also deeply involved in the debate on mathematical instruction at all levels and had a major influence in the development of the ministerial curricula for the scientific lyceum and technical institutes. Calculus was not brought into the school syllabus in Italy as early as in many other countries and the concept of a function only brought in around 1910 after Castelnuovo's efforts.
Article by: J J O'Connor and E F Robertson