After obtaining his laurea, Scorza was appointed as an assistant to Eugenio Bertini, who held the chair of analytical and projective geometry at the University of Pisa. He worked at Pisa for the year 1898-99 but he then spent the year 1899-1900 as an assistant to Corrado Segre at the University of Turin. This was as the result of a somewhat unusual arrangement between Bertini at Pisa and Corrado Segre at Turin who made an agreement to exchange assistants for one year. Scorza spent the year in Turin assisting Corrado Segre, who held the chair of projective and descriptive geometry. While at Turin, he helped in advising Francesco Severi who was undertaking research for his thesis supervised by Corrado Segre. Severi writes about his friendship with Scorza during that year :-
I remember the long walks together in Valentino park or on the small, shady, quiet roads in the Turin hills. Solitary walks on cold winter days or intoxicated by the bright and harmonious awakening in spring when nature was bursting into life: spring in the world around and in our lives. He was reserved, a bit timid, a naive dreamer, as is sometimes the southern temperament.Scorza produced some elegant papers while in Turin and broadened the areas of his research. However, the doubts that had plagued him earlier continued to trouble him. He told Severi that he feared that he was losing his abilities and inventiveness. At the end of the year, the assistants returned to their original positions, so in 1900 Scorza went back to Pisa as Bertini's assistant. He earned the right to lecture in universities and taught courses at the Scuola Normale Superiore in Pisa. However, from 1902 to 1912 he taught in technical institutes in Terni, Bari and Palermo. During these years, he did little research on the geometrical topics that he had worked on earlier and he stopped discussing research with Severi and others :-
But his vast knowledge, and his open and penetrating mind, would not let him be completely idle. He devoted himself to the study of mathematical economics.It was the economist Maffeo Pantaleoni (1857-1924) who encouraged Scorza to take an interest in mathematical economics. Scorza wrote four articles on this topic which were published in the Giornale degli Economisti between April 1902 and February 1903. All except the first of these papers form part of an argument which developed between Scorza and the economist, political scientist and philosopher Vilfredo Pareto (1848-1923). Scorza angered Pareto when he claimed his results were false (see ):-
... given a certain system of prices every individual participating in the exchange tries to regulate their demand and their supply in a way that ophelimity is maximised from this exchange, and that many authors from Walras to Pareto have deduced the conclusion that every participant in the market will achieve the maximum of ophelimity under a regime of free competition. Now, Cassel profoundly observes that there is nothing in that event that is essentially characteristic of free competition, and therefore the reasoning by which this conclusion is drawn from the presence of the equations of maximum satisfaction in the system that determine the equilibrium, is nothing other than gross sophistry.Most historians suggest that Pareto failed to understand fully the points that Scorza was making but eventually sided with him. However, the paper  interprets the argument rather differently. Certainly Scorza's interest in economics was short-lived and he then gave up all scientific research for three years :-
There followed three years of total inactivity in science which would be inexplicable in a young man so well equipped if one were to ignore the crisis that tore at his soul.By the end of 1906, Scorza had begun to have contacts again with Bertini, Severi and other researchers in geometry and, given his outstanding early contributions, they were all keen to persuade him to return to his study of algebraic geometry. Certainly both Bertini and Severi made suggestions of specific questions that would interest Scorza. The depression which had gripped Scorza frequently over the years began to ease around this time. When he had been a student at Pisa he had met Angiola Dragoni, who was studying mathematics there, and they became close friends. She was awarded her laurea in 1902 for her thesis Sulla varietà cubica di S4 dotata di dieci punti doppi Ⓣ but, due to Scorza's doubts and hesitation, he had not been able to propose marriage. However, with the lifting of his depression at the end of 1906, Scorza proposed marriage and the couple were married in 1907. In the following year, on 2 July, their son Giuseppe Scorza-Dragoni was born in Palermo where Scorza was teaching at the technical institute. Giuseppe Scorza-Dragoni (1908-1996) went on to write a laurea advised by Severi (at his father's request) and was professor of mathematical analysis at Padua from 1936 to 1962, and at Rome from 1962 to 1966. After eight years as professor of algebra at Bologna from 1966, he returned to Padua where he worked until he retired.
Scorza's health improved markedly from 1906 and he was able to resume his research in algebraic geometry with a renewed vigour. He published several papers in 1907 including the important Intorno alle corrispondenze (p, p) sulle curve di genere p e ad alcune loro applicazioni Ⓣ and five papers in each of the years of 1908 and 1909. He was awarded his libera docenza in 1907 and began applying for professorships at universities. He was successful in the competition for the chair of projective and descriptive geometry at the University of Cagliari and took up his appointment in 1912. However, he only spent a very short time at Cagliari. Mario Pieri, who had moved to the University of Parma in 1908 so that he might return to his native Tuscany, died on 1 March 1913 and for the remainder of that academic year his teaching was covered by Attilio Vergerio (1877-1937). Vergerio had been Pieri's assistant since 1908 and had delivered Pieri's lectures during his final illness. However, Scorza was appointed to fill Pieri's chair and, from the start of the academic year 1913-14, he taught the courses that Pieri had given.
One of the main ideas for which Scorza is known is that of Riemann matrices. This work is discussed by Guido Zappa in . Zappa writes:-
... the scientific activity of Gaetano Scorza after the year 1915 is studied. In the year 1916 Scorza published a valuable paper in which the concept of Riemann matrix was introduced. In this way a unitary treatment of various geometric theories (abelian integrals, abelian functions, etc.) was given. In the year 1921 Scorza published a new important paper where some problems posed in the previous paper were solved by the theory of algebras.Zappa also looks at this work by Scorza in . He writes:-
... contributions of Italian geometers (during the period (1910-1915) to abelian varieties of higher dimension are illustrated. Particularly, the papers of Gaetano Scorza on singular varieties and on reducible abelian integrals are considered. Scorza was able to achieve a unitary theory, illustrated in a fundamental essay (1916).In 1916 Scorza transferred to the University of Catania, then in 1921 he moved to Naples. In was in that year, while still in Catania, that he published Corpi numerici ed Algebre Ⓣ. The authors of  give the following indication of its content and importance:-
The text is immediately acknowledged for its importance, with awards and influential and encouraging quotations even beyond national boundaries. Scorza exposes a general field theory - for which he uses the term 'corpi numerici' - and an associative algebra theory, pointing also to their applications within the geometrical field. The step towards a wholly abstract presentation of algebra structure is done, at this point.Let us quote from the book Scorza's own justification of his abstract approach:-
Someone who has learnt that different concrete theories, dealing with different entities 'toto coelo', schematically give rise to the same abstract theory, could do this only because, moving out of each of them, they have managed to gain a superior point of view from which to look at them simultaneously. ... The mathematician, who does not possess the general theory of what is called 'corpi numerici' knows, through algebra, the theory of equations; through number theory, the theory of congruencies with respect to a prime modulus; through the treatises on algebraic numbers, the theory of congruencies with respect to a prime ideal; three theories of which, even if he has spotted some analogy, he does not see the innermost bounds.After fourteen years at the University of Naples, in 1935 he moved to Rome, where he remained for the rest of his career. Severi writes about his appointment to the University of Rome :-
His ascension to the university Chair was greeted with applause by all the Italian mathematicians. The Evaluation Commission (of which I was the rapporteur) from the beginning of their deliberations unanimously recognised that he belonged, without question, in the first place.Despite the years of difficulty when he was a young man, Scorza published more than 160 works over his career, most importantly in algebraic geometry and related algebraic fields, but also very many general works and textbooks. We mentioned his main contributions above but we should also note that he wrote eight articles on group theory. Jean Dieudonné wites:-
Two of his eight papers on group theory are concerned with group algebras, and are slightly different presentations of the well-known papers of Frobenius and I Schur. Among the other papers, three develop an idea of M Cipolla on nonabelian groups G, which is to study a partition of the complement of the center, two elements being in the same set if they have the same centralizer; he relates the number of sets of that partition contained in a subgroup H to the number of centralizers containing H. The most original paper is the one in which Scorza apparently was the first to investigate which groups can be the union of a finite number of their proper subgroups. In it, he shows that the number of proper subgroups cannot be two and characterized the groups for which that number is three (the Klein 4-group is the simplest example).He wrote a book on group theory, Gruppi Astratti Ⓣ, published posthumously in 1942. H A Thurston writes in a review:-
This book is concerned with abstract groups, finite or infinite; some results are stated for denumerable groups with the remark that the proof holds in general if the selection axiom is used. ... The book is very clearly written (which makes for easy reading) ...In fact, in addition to his research contributions, he made a major commitment to mathematical education. In 1909, following the death of Giovanni Vailati, Scorza joined Guido Castelnuovo and Federico Enriques as the Italian delegates on the International Commission on Mathematical Instruction. He was the author of two major reports for the Commission, namely L'insegnamento della matematica nelle Scuole e negli Istituti tecnici Ⓣ (1911) which examines mathematics teaching in technical schools and institutes, and Sui libri di testo di geometria per le scuole secondarie superiori (1912) which looked at geometry textbooks written at upper secondary school level. To give some idea of his ideas on teaching we give a quote from La matematica come arte Ⓣ which he published in 1930:-
A mathematics lesson can be a work of art and not a tired repetition of postulates, theorems and corollaries strung one after another with a lifeless mechanical delivery like the disconnected words of a child's nursery rhyme; but provided that the person who gives the lesson appears to the student ... to be a very secure master of what he has to say ... so that he presents each argument to the students as if it sprang ... bright, spontaneous and new, from the living flow of the discussion.Finally, let point out how much Scorza admired Felix Klein and, in particular, Klein's book Elementarmathematik vom höheren Standpunkte aus Ⓣ (1910), which he reviewed in glowing terms :-
Scorza admired Klein for his research method, his mastery of the most disparate branches of mathematics, his capacity for grasping the even most hidden relationships between them, and for having outlined vast programs of work.He was elected to the Reale Accademia dei Lincei in 1926, and he was a member of the Consiglio Superiore della Pubblica Istruzione from 1923 to 1932. He was editor-in-chief of the Rendiconti del seminario matematico della Università di Roma, served as president of the mathematics committee of the Italian National Research Council from 1928 to 1931, and he was vice-president of the International Commission on Mathematical Instruction from 1932 until his death in 1939. Shortly before his death he was appointed a senator of the Kingdom of Italy.
Article by: J J O'Connor and E F Robertson