Marcolongo enrolled in the degree course in mathematics at the University of Rome where he studied with Guiseppe Battaglini, Luigi Cremona and Valentino Cerruti (1850-1909), who lectured to him in mechanics. He graduated in mathematics in 1886 and in 1888 he became an ordinary mathematics assistant to Valentino Cerruti at the University of Rome. He published his first papers, consisting of some brief notes of algebra, in the Journal of Mathematics directed by Guiseppe Battaglini, in 1887. These are the three papers Generalizzazione di un teorema sui determinanti Ⓣ, Su di un teorema di algebra elementare Ⓣ, and Sull' analisi indetermina di 2° grado. Nota I° Ⓣ.
For a list of papers and books by Marcolongo, see THIS LINK.
In 1892 he was declared eligible in competitions for the position of extraordinary professor of rational mechanics at the University of Naples and of mathematical physics at Pavia but he was not ranked as the top candidate. In November 1895 he was ranked first in a competition for an extraordinary professorship of rational mechanics at the University of Messina. Marcolongo then began applying for chairs so, in 1897, entered competitions for the chair of rational mechanics at the University of Padua and the chair of higher mathematics at the University of Turin. In both of these competitions his main opponent was Tullio Levi-Civita who was eleven years younger than Marcolongo. These competitions proved somewhat awkward with Marcolongo being declared the winner of both of these despite having received two votes less that Levi-Civita. The Ministry of Education, which had to make the final decisions based on the advice from the judging panels, offered Marcolongo the Padua chair but he turned it down. It was then offered to Levi-Civita, who accepted. The Ministry offered to confirm Marcolongo as extraordinary professor of rational mechanics at Messina, so he remained there. They also decided to leave the chair at the University of Turin vacant.
In 1899, after receiving the opinion of a commission composed of Vito Volterra, Giacinto Morera and Carlo Somigliana (1860-1955), Marcolongo was promoted to the chair of rational mechanics at Messina. In December 1907 he moved to Naples, where he succeeded Francesco Siacci when appointed to the chair of rational mechanics. Siacci had died in May 1907 leaving the chair vacant. Marcolongo would have liked to return to his home city of Rome and in 1909 he applied for a position there but he was unsuccessful. He continued to hold the chair of rational mechanics at Naples until his retirement in October 1935. Marcolongo married Maria Santi and their son Fernando Marcolongo was born in Messina on 28 August 1905. Fernando Marcolongo became a medical researcher with an impressive list of over 70 top quality publications. Another son, Aldo Marcolongo, was born in Messina on 14 January 1904 and he became a chemist.
Roberto Marcolongo became a corresponding member of the Accademia Nazionale dei Lincei in 1912 and a full member in 1921. In 1924, at the suggestion of Giovanni Gentile (1875-1944), he was awarded the Ordine della Corona d'Italia.
For the rest of this biography we follow reasonably closely a translation of  adding extra material.
Marcolongo was among the Italian mathematical physicists who accepted the theory of relativity. His mathematical research concerns the theory of elasticity, the kinematics of rigid bodies and, most of all, the vector calculus, given that :-
... his attitude led him especially towards the analytical algorithmic part.At the Fourth International Congress of Mathematicians held in Rome in April 1908 he delivered a paper on L'unificazione delle teorie vettoriali Ⓣ, a research program derived from the geometric calculus of Giuseppe Peano and developed in the following years in a series of articles and monographs written in collaboration with Cesare Burali-Forti (of whom he later published the obituary  in the Bollettino dell'Unione matematica italiana Ⓣ).
For a list of Marcolongo's publications, including those referred to on the vector calculus, see THIS LINK.
For reviews of some books by Marcolongo, in particular two works written in collaboration with Cesare Burali-Forti on vector calculus, see THIS LINK.
In fact Marcolongo established a lengthy and fruitful collaboration with Cesare Burali-Forti and they were jokingly baptised by their colleagues as the "vector binomial." Their Elementi di calcolo vettoriale con numerose applicazioni alla geometria, alla meccanica e alla fisica matematica Ⓣ (Bologna 1909, French translation, Paris 1910, 2nd ed., Bologna 1921, 3rd ed., Milan 1925) presented the fundamentals and notation of a so-called minimum system and gave applications to the mechanics of continuous geometry and differential geometry on a surface. The Analysis vectorielle générale Ⓣ (two volumes published in Pavia, 1912-13), on which Marcolongo collaborated with Matteo Bottasso and Cesare Burali-Forti, introduces and shows the possibilities of applying vector homographies, a type of linear transformation between vectors, which constitutes an alternative to the vector calculus of other authors, such as J Willard Gibbs. A second edition of the first volume of this work, entitled Trasformazioni lineari Ⓣ (Bologna 1929), was published in a encyclopaedia of vector analysis written in collaboration with Pietro Burgatti and Tommaso Boggio, two other members of the so-called school of Italian vectorists. The methods of vector calculus gradually won the favour of the scholars, and their use spread in the universities in Italy and abroad. There was however no lack of discussions and criticism, above all on the part of the British and American quaternionists and the disciples of J Willard Gibbs.
Marcolongo also had a strong interest in the history of science and technology. Already in Elementi di calcolo vettoriale con numerose applicazioni alla geometria, alla meccanica ed alla fisica matematica Ⓣ, published by Zanichelli in Bologna in 1909, the mathematics was accompanied by in-depth notes on the history of the vector calculus; later he specialised in the history of mechanics, with particular attention to the Italian author, Galileo Galilei and, above all, to the drawings and manuscripts of Leonardo da Vinci. Among his monographs in this field: Il problema dei tre corpi da Newton (1686) al nostri giorni Ⓣ (Pisa 1915, other editions Naples 1915, Milan 1918, Bologna 1919), and Leonardo da Vinci, artista-scienziato Ⓣ (Ulrico Hoepli, Milan, 1939; 2nd ed., Ulrico Hoepli, Milan, 1943) which is the synthesis of many previous works. In fact in Marcolongo's publication list there are 24 papers relating to Leonardo da Vinci which he published before the book appeared in 1939.
Marcolongo also distinguished himself for his contribution to the dissemination of scientific culture in Italy, which over the years became increasingly central to his work. First of all, he made a considerable effort to maintain the rich Italian tradition in classical mechanics and at the same time worked hard to promote knowledge of the new physical theories. Examples of these efforts are his inaugural lecture at the University of Messina on "Modern theories of mathematical physics" (1903), a contribution to the Fifth International Congress of Philosophy (Naples 1925) entitled "Between relativists and anti-relativists" and especially his book Relatività Ⓣ (G Principato, Messina, 1921). Giulio Maltese writes :-
In 1906 Marcolongo demonstrated the existence of scalar and vector potentials and defined Lorentz's transformation as the one that left Maxwell's equations unaltered, and in 1913 he studied the transformation laws of various quantities occurring in special relativity. Lorentz, Minkowski, and Poincaré were the authors he cited most frequently; Einstein's name never appeared. ... Marcolongo's main purpose was therefore to demonstrate the power of his mathematical methods. He reported his advances in vectorial formalism to Levi-Civita, who used them in his papers on the asymptotic of electric radiations.The above quote mentions that Marcolongo informed Levi-Civita about his advances and we note that he did this in letters he wrote to Levi-Civita on 1 August 1904, 13 March 1905, 12 November 1905, 22 March 1906, and 25 April 1912.
Two texts on university teaching in Marcolongo's publication list stand out as being particularly influential. These, like the two historical monographs mentioned above, were published in the particularly important series of Hoepli Manuals. They are: Teoria matematica dell'equilibrio dei corpi elastici Ⓣ (Ulrico Hoepli, Milan, 1904) and Meccanica razionale Ⓣ (2 volumes, Ulrico Hoepli, Milan, 1905), which was translated into German as Theoretische Mechanik Ⓣ (B G Teubner, Leipzig, 1911).
Interest by Marcolongo in secondary education, especially for technical and vocational schools, is already evident in his Relazione sull'ordinamento dei programmi della Scuola d'arti e mestieri di Messina Ⓣ (Messina, 1902). From the 1920s he also published highly regarded mathematical textbooks for secondary schools: (with Cesare Burali-Forti) Corso di matematica pel 2° biennio degli istituti tecnici Ⓣ (Naples, 1920); Algebra pei licei scientifici Ⓣ (Naples, 1923); Elementi di trigonometria ad uso degli istituti medi superiori e degli istituti industriali Ⓣ (Rome, 1929); Complementi di algebra e di analisi per i licei scientifici Ⓣ (Rome, 1930); and (with L Marzella) Corso di matematica ad uso degli istituti tecnici commerciali e per geometri Ⓣ (Rome, 1938). Since the 1920s he also carried out a large body of work, widely used in mathematics and in the history of science, as a collaborator on the Italian Encyclopaedia. In addition he had numerous articles published in the journals Sapere Ⓣ and Scientia Ⓣ. For example in Sapere he published articles such as Le curve di Lissajous e le curve giroscopiche Ⓣ in 1936, Leonardo nel "paradiso delle scienze matematiche", Fisico, matematico, astronomo Ⓣ in 1938, and Le nuove scienze, la meccanica del secol novo Ⓣ in 1942. In Scientia, Rivista di Scienza Ⓣ he published numerous reviews of books such as A D'Abro's book The evolution of Scientific thought from Newton to Einstein which he reviewed in 1931 and Tullio Levi-Civita's book Fondamenti di Meccanica Relativistica Ⓣ which he reviewed in 1932. Also in Scientia he published La relatività ristretta Ⓣ in two parts, both published in 1924.
Marcolongo found himself involved in bitter controversies derived from scientific and academic rivalries that featured Italian vectorialists. His position, however, was, in his words, "less rigid and more eclectic" and therefore more open to connection with the trends of international scientific culture. Regarding the purist criteria, radically hostile to the use of the Burali-Forti coordinates, he had a more conciliatory attitude; he did not share the position decidedly contrary to the theory of relativity of Burali-Forti and Boggio, and not even the criticisms that these two mathematicians put forward against the absolute differential calculus of Levi-Civita and G Ricci-Curbastro. In fact the close cooperation between Marcolongo and Burali-Forti only ran into difficulties over Burali-Forti's attack on Einstein's theory of relativity in his paper Espaces courbes. Critique de la relativité Ⓣ (1924). Marcolongo wrote that this was :-
... the only time when the peace and solidity of the vector binomial [Burali-Forti and Marcolongo] seemed to be compromised... . It was not possible for them to agree, first on the new range to be given to vector methods; then, even more deeply, on the essence of the whole theory. Despite the interest he [Burali-Forti] felt for all the modern physical questions and above all for those of high and absorbing philosophical interest, he remained firmly obedient to the classic systems and in attack as in defence he could not remain calm and objective.In October 1932, Marcolongo enrolled in the National Fascist Party. This party had been founded by Benito Mussolini in 1921 and by the time Marcolongo joined it was the only political party allowed in Italy. On 27 May 1933, membership of the National Fascist Party was declared a basic requirement for public office. In the following years, which saw the devastation of the Italian scientific community following the promulgation of the racial laws, he participated in two important cultural initiatives. As a member of the Royal Vinciana Commission, he participated in the organization of the Mostra di Leonardo e delle invenzioni italiane Ⓣ held in the Palazzo dell'Arte (now the Triennale) in Milan from 9 May 1939 to 1 October 1939. A quote from the exhibition catalogue shows the political nature of this event:-
The purpose of the exhibition is to celebrate the universal and unequalled genius of Leonardo da Vinci, assumed as practically the symbol of all Latin and Christian, and therefore Roman, civilization, and to highlight the spiritual connections uniting this great man of accomplishments and creator with the realizations of Mussolinian and Imperial Italy. The combining of the Vincian celebrations with the exhibition of Italian inventions tends to demonstrate the continuity of the creative genius of the race and the great possibilities opening up to those within the climate of Fascist will.In December 1939 he became a member of the sub-committee for mathematics of the Science Exhibition in the context of the works for the Esposizione universale Ⓣ of Rome in 1942. The committee undertook to identify the so-called Italian contribution to the mathematical sciences, thus removing the names of Italian Jewish mathematicians, including Levi-Civita himself. The exhibition never took place because of World War II.
In April 1942 Marcolongo was appointed to the Reale Accademia d'Italia. This academy, which only operated between 1929 and 1944, was founded by Mussolini with the aim of:-
... promoting and coordinating the Italian intellectual movement in the field of science, literature and the arts, to keep the national character pure, according to the genius and traditions of the race and to favour its expansion and influence beyond boundaries of the State.After being appointed professor emeritus in 1937, Marcolongo returned to Rome, the city of his birth, where he died on 16 May 1943. He therefore died before the Allies invaded Sicily in July 1943 and mainland Italy in September 1943. We note that Marcolongo's son, Fernando, was an extraordinary professor of medical pathology at the University of Catania when the Allies invaded Sicily, but he survived the war.
Article by: J J O'Connor and E F Robertson