In Naples, Francesco studied at the Vittorio Emanuele Gymnasium while Enrico, having completed his secondary education, was considering enrolling for a law degree. However, Enrico's uncle, Achille Sannia (born Campobasso in 1823, died in Naples in 1892) had set up a private mathematical school in Naples in 1856 and, under his influence and guidance, Enrico turned from legal studies to mathematics and began to study at Achille Sannia's school preparing for entry to the School of Bridges and Roads in Naples. He studied there for a time and attended lectures by Giuseppe Battaglini, Emanuele Fergola (1830-1915), Remigio Del Grosso (1813-1876) and Fortunato Padula (1816-1881), whose influence made D'Ovidio become interested in an academic career. However, the struggle for Italian independence in 1860-61, which led to the defeat of Ferdinand II, caused considerable disruption to D'Ovidio's studies. Already at this time he was beginning original research in mathematics although he had not taken a standard university course. He wrote some short articles on determinants and conics and these were published in the early volumes of Battaglini's new journal Giornale di Matematiche which was founded in 1863. In Volume 1, he published five papers: Dimostrazione di un teorema del capitano Faure ; Due teoremi di determinanti ; Nota sopra un problema di geometria ; Alcune locali ; and Altra dimostrazione dei teoremi provati a p. 160 . He continued publishing in Giornale di Matematiche with three papers in 1864 in Volume 2 (two of which answered questions posed in the journal), and two papers in the following year in Volume 3 (one answered a question).
D'Ovidio began school teaching at the Umberto High School in Naples and he also taught at the Naval School. To make extra money he also gave private mathematics lessons. He was granted a degree "ad honorem" in mathematics by the University of Naples in 1869 despite never having taken a degree course. He sat no written examination papers for this degree since it was felt that he had already proved his mathematical abilities. In 1869 D'Ovidio, in collaboration with Achille Sannia, published a geometry text for schools. The book Elementi di geometria was very popular and ran to fourteen editions (the third edition was published in 1876, the eighth in 1891, and the fourteenth was published in 1918). In 1872 Eugenio Beltrami persuaded him to enter the competition for the Chair of Algebra and Analytic Geometry at the University of Turin. This was a time of major change for the city of Turin which had been the capital of the unified Italy from 1861 but Rome had become the capital in 1870 after Italian troops entered the city. D'Ovidio was reluctant to leave the Naples area of Italy where he was close to his family but he decided to enter the competition and he was offered the chair on 17 November 1872. D'Ovidio was to work for 46 years in the University of Turin, retiring on 11 August 1918. He was chairman of the Faculty of Science in 1879-80 and rector of the University between 1880 and 1885. Another spell as chairman of the Faculty of Science between 1893 and 1907 ended when he was appointed as a Commissioner of the Polytechnic of Turin. He held this position until 1922.
He married Maria Bonacossa in the summer of 1877; they had two daughters Pia and Laura, and a son who died in 1907 while hiking in the mountains, trying to hold back a friend who was falling. D'Ovidio became a leading figure in Italian mathematics and, as a consequence, often served as a referee for appointing competitions. We give just a few examples. In 1890 D'Ovidio was a referee for a post at the Polytechnic of Turin. There were 24 candidates and Federico Amodeo was chosen for the post (Mario Pieri was placed second). He was a referee, with Eugenio Bertini and Giuseppe Veronese, for appointing the chair of analytic and projective geometry at the University of Rome in 1891. Guido Castelnuovo, who was at that time D'Ovidio's assistant, was appointed to the chair. In 1893 he was a referee for appointing the chair of projective and descriptive geometry at his own University of Turin; Luigi Berzolari was appointed.
Euclidean and non-Euclidean geometry were the areas of special interest to D'Ovidio. He published twenty-two works between 1863 to 1872. These include: Dimostrazione di alcuni teoremi sulle superfici sviluppabili di 5° ordine enunciati dal professor Cremona (1865); Nuova dimostrazione di una formula di Abel (1868); and Nota sui punti, piani e rette in coordinate omogenee (1870). However, his most intense period of research occurred after he was first appointed to Turin. It was a highly significant time for research in geometry with the work by Nikolai Ivanovich Lobachevsky, János Bolyai and Bernhard Riemann on non-Euclidean geometry becoming widely known and Felix Klein had put forward the general view of Euclidean and non-Euclidean geometries as invariants for transformation groups in the 'Erlanger Programm' in 1872. D'Ovidio built on the geometric ideas which these mathematicians had been introduced. His most important work is probably his paper of 1877, Le funzioni metriche fondamentali negli spazi di quante si vogliano dimensioni e di curvatura costante . In this work the he used concepts and methods from projective geometry to derive the metric functions in non-Euclidean n-dimensional spaces, paving the way for subsequent work of Giuseppe Veronese, Corrado Segre and his other students.
D'Ovidio also worked on binary forms, conics and quadrics. The importance of his work in this area is illustrated by the fact that his 1880 paper Studio sulle cubiche gobbe mediante la notazione simbolica delle forme binarie was awarded the gold medal by the National Academy of Sciences of Italy (the "Academy of Forty"). He published a number of important books: La proprietà fondamentale delle curve di secondo ordine studiate sulla equazione generale di secondo grado in coordinate cartesiane (first edition 1876, second edition 1883); Teoria analitica delle forme geometriche fondamentali (1885); Geometria analitica (first edition 1885, fourth edition 1912); Il Io di Euclide esposto da E D'Ovidio (first edition 1887, third edition 1894); and Il libro IIo di Euclide esposto da E D'Ovidio (1889).
He had two very famous assistants, Giuseppe Peano (assistant in 1880-83) and Corrado Segre (assistant in 1883-84). D'Ovidio and Corrado Segre built an important school of geometry at Turin. For example, in 1888, in addition to D'Ovidio and Corrado Segre, the faculty in Turin included Giuseppe Basso, Giuseppe Erba, Angelo Genocchi, Nicodemo Jadanza, Giuseppe Peano, and Francesco Siacci. At this time Mario Pieri, Guido Castelnuovo and Filiberto Castellano were assistants. One of D'Ovidio's greatest achievements was his outstanding support for the students and assistants under his care. In addition to Giuseppe Peano, Corrado Segre, Filiberto Castellano (1860-1919) (who became his assistant in 1881), and Guido Castelnuovo (who was his assistant in 1888-91), we should mention Gino Fano (who was his assistant in 1892-93), Beppo Levi (who was his student 1892-96), and Gino Loria who earned his laurea in 1883 supervised by D'Ovidio and also was his assistant in 1884-86. When D'Ovidio retired in 1918, Italy had just gone through the trauma of World War I. To coincide with his retirement Scritti matematici offerti ad Enrico D'Ovidio was published. The Preface, written by the editors Francesco Gerbaldi (who was also a student of D'Ovidio and then his assistant after 1879) and Gino Loria, begins :-
On the approach of the day on which an inflexible law would retire Senator Enrico D'Ovidio from the university chair, there arose in the minds of many students whom he has had in his long and glorious career as a teacher, the pleasant idea of choosing this occasion - which coincides with his 75th birthday - to manifest to him their sentiments of unalterable affection and, at the same time, to present to him their sincere good wishes ad multos annos. ... And we are certain that to the loved teacher our publication will be doubly gratifying in as much as it serves also to show how Italy, in the tragic hours in which we live - not less than in the more grave and decisive periods of her earlier struggles for redemption - has not ceased to feed the sacred flame of science.The author of  writes:-
It is a pretty Italian custom to mark an epoch in the life of a great scholar, by presenting to him a set of scientific notes written by his friends in his honour. On the occasion of the retirement of Professor d'Ovidio from the chair at Turin, at the age of 75 and after 46 years' service, the present volume is contributed by a distinguished company of nine former assistants, ten old students and one other friend of the veteran.L Wayland Dowling writes :-
Of the 103 persons contributing to the expense of publication, and whose names appear directly following the preface, 47 have been actual students under Professor D'Ovidio, and 20 have been, at one time or another, directly associated with him as Assistenti in the University of Turin. The list contains many names well known to students of mathematics the world over, and bears witness to the great influence Professor D'Ovidio has had upon the growth of mathematics and upon the teaching of mathematics, not alone in Italy, but, mainly perhaps through his disciples, throughout the civilized world. This influence has been exerted from the University of Turin for more than forty years. ... Professor D'Ovidio's great influence has come about rather more through personal contact as a teacher than through published writings, and goes to show that there is such a thing as creative teaching as well as there is creative scholarship - a fact worth noting in a time when so much emphasis is placed upon the latter function and apparently so little is thought about the former. ... [We have] a feeling of profound admiration for the scholars who have made the book possible, and especially at a time when the strain of the war was still in force. ... The total is a wholly worthy epitome of scientific activity even in normal times. It must, indeed, be a source of great satisfaction to Professor D'Ovidio to have so distinct a proof of the esteem with which his many students, associates, and friends regard his long service and his personal qualifications as an inspiring teacher.Many honours came D'Ovidio's way. He was elected to the Accademia pontaniana of Naples on 9 February 1872, the Accademia delle scienze of Turin on 16 January 1879 (he was elected president of the Academy and served in that role during 1902-1910), and the Istituto lombardo di scienze e lettere of Milan on 10 February 1881. He became a corresponding member of the Società reale of Naples on 12 February 1881, becoming a full member on 12 June 1909. He became a corresponding member of the Accademia dei Lincei in Rome on 31 December 1883, becoming a full member on 7 November 1893. He was elected to the National Academy of Sciences of Italy (the "Academy of Forty") in December 1884 and in the same month to the Accademia delle scienze of Naples. In the following year he was elected to the Accademia nazionale di scienze, lettere ed arti of Modena. These were academic awards, but D'Ovidio also received a large number of honours from his country. He was made: Knight of the Order of the Crown of Italy on 28 May 1876; Officer of the Order of the Crown of Italy on 15 January 1882; Commander of the Order of the Crown of Italy on 16 January 1883; Grand Officer of the Order of the Crown of Italy on 11 July 1918; Knight of the Order of Saints Maurice and Lazarus on 2 June 1882; Officer of the Order of Saints Maurice and Lazarus on 5 June 1892; Commander of the Order of Saints Maurice and Lazarus on 3 June 1909; and Grand Officer of the Order of Saints Maurice and Lazarus on 2 July 1922.
D'Ovidio also had a career in parliament. Kennedy writes in :-
He was named a senator in March 1905, but there were rumours that this was due to a mix-up and that the nomination was intended for his brother Francesco, the noted philologist, who was in fact named a senator a few months later.He took his oath on 7 April 1905 and, although he did not make a significant contribution to the parliament, he did make some important speeches on education and was involved in the discussions concerning the law that set up the Royal Polytechnic of Turin in 1906 by combining the already existing Regia Scuola (Royal School) and the Regio Museo Industriale (Royal Industrial Museum).
Article by: J J O'Connor and E F Robertson