From 1879 to 1883 he was a student at the University of Turin and he published his first paper Intorno alla geometria su un complesso tetraedrale in 1883. He was awarded his doctorate in July 1883 for his dissertation on spherical geometry with the same title as his first publication. His thesis supervisor at Turin was Enrico D'Ovidio. He spent 1883-84 at the University of Pavia and during 1884-86 he was an assistant to D'Ovidio in algebra and analytic geometry at the University of Turin. He published Sur les différentes espèces de complexes du 2e degré des droites qui coupent harmoniquement deux surfaces du second ordre in Mathematische Annalen in June 1884, with Corrado Segre as his coauthor. During his time as D'Ovidio's assistant, Loria published a total of sixteen papers on :-
... the geometry of straight lines and spheres, hyperspatial projective geometry, entities generated by algebraic correspondences between fundamental forms, and Cremona transformations in space.Loria was appointed as a Privatdocent at the University of Turin in July 1886. Then, in November 1886 after a competition, he was appointed as an extraordinary professor of higher geometry in the University of Genoa, where he was to spend the rest of his academic life. He was promoted to ordinary professor of higher geometry in November 1891, and continued to hold this chair until he retired on 1 August 1935 at the age of 73. He was also "charge de cours" of higher analysis 1892-97, and "charge de cours" of descriptive geometry 1897-1935. He served as dean of the faculty of sciences at Genoa from 1903 to 1906. In 1903 he married Ida Levi Gattinara and they made their home at 41 Manin Square, Genoa. Following his retirement he was made professor emeritus in July 1936.
Although Loria continued to undertake research into geometry throughout his life, his main research activity was in the history of mathematics. His first contributions to this topic were Sur une démonstration du théorème fondamental de la théorie des équation algébriques (1887) and Notizie storiche sulla geometria numerativa (1888). The authors of  explain Loria's aim when writing works on the history of mathematics:-
In his day, Loria was arguably the pre-eminent historian of mathematics in Italy. A full professor of higher geometry at the University of Genoa beginning in 1891, Loria wrote the history of mathematics as a mathematician writing for other mathematicians. He emphasised this approach repeatedly in his works. For instance, in the introduction to his 'Storia delle matematiche dall'alba della civilità al tramonto del secolo XIX' , he stated that general history of mathematics was written "by a mathematician for mathematicians". In his 'Le scienze esatte nell'antica Grecia' (1914), he explained that the book had been written "by a mathematician for persons who, although with modest scientific background, are interested in mathematics". Concerned only with exact sciences, Loria gelt himself compelled to justify his incursions into "foreign fields" like philosophy, geodesy, and astronomy, forced as he said by "the indissoluble links" among the various branches of his subject matter.His first paper to create a real impact among historians was written around the time he was made a full professor at Genoa :-
Loria's main works are devoted to the history of geometry, his area of research and teaching as a mathematician. His first important historical work was an essay ['Nicola Fergola e la scuola di matematici che lo ebbe a duce' (1892)] on Nicolò Fergola (1753-1824), which marked the rediscovery of a virtually forgotten school of geometry that flourished in Naples during the early decades of the 19th century. Loria himself first heard of the existence of Fergola's school when he read Chasles's 'Aperçu historique sur l'origine et le developpement des méthods en geométrie' (1875) that "we owe several important works that re-establish ancient geometrical analysis in its original purity to the celebrated Fergola and his students.Loria wrote a number of texts which became classics. For example: Le scienze esatte nell'antica grecia (1893, 2nd edition 1914); Spezielle Algebraische und Transcendente Ebene Kurven. Theorie und Geschichte (1902, 2nd edition 1910, Italian edition 1930-31); Vorlesungen über darstellende Geometrie (2 volumes) (1907, 1913); Metodi di Geometria Descrittiva (1909, 2nd edition 1919, third edition 1925); Storia della geometria descrittiva dalle origini sino ai giorni nostri (1921); Curve sghembe speciali algebriche e trascendenti; Teoria e Storia (2 volumes) (1925); Histoire des Mathématiques dans l'Antiquité Hellènique (3 volumes) (1929, 1931, 1933); and Storia delle matematiche (1929). Reviewing the first volume of Storia delle matematiche , McClenon writes :-
This book is one that in an English translation would be very useful and welcome to teachers of mathematics, both in universities and in secondary schools. Its value to mathematicians will depend largely on their personal interest in the history and development of the subject; and the second volume, which will begin with the 16th century, will doubtless offer considerably more of interest to them. The present volume is in any case not just "another history of mathematics," for it is written in an original, individual, and vivid style which, combined with its general accuracy in matters of fact, assures it a place of its own in the literature of the subject.Another important contribution by Loria was the founding of Bollettino di bibliografia e storia delle scienze matematiche in 1898. This journal for the history of mathematics was associated with the Giornale di Matematiche but published independently of it until 1922, after which it was issued as a section with separate page numbers in Il Bollettino di Matematica :-
Following Boncompagni's example, Loria founded the 'Bollettino di bibliografia e storia delle scienze matematiche' in 1898. Loria, however, adapted editorial policies different from those of Boncompagni. In Loria's view, subsequent historians "slightly modified (Boncompagni's) principles". On the one hand they tried to pay attention to influential mathematicians; on the other hand, they presented quotations in a clear and exact form, though reduced to their essentials in order to save space and spare the time of scholars by not "inflicting them with such boredom that drives readers away." Loria's 'Bollettino' thus sought to avoid the overly rigorous, erudite style of Boncompagni. It included generally short though well-documented papers, and gave ample space to reviews.We should also comment on Loria's contributions to the teaching of mathematics since these are highly significant. In 1893 he published Della varia fortuna d'Euclide in relazione con i problemi dell'insegnamento geometrico elementare in which he established:-
... a target which the modern teaching of elementary geometry should reasonably aim at.Geometry had to be taught in a rigorous but interesting way:-
... enemies as we are of any concession made at the sacrifice of geometric rigour, we are favourable on the other hand to any legitimate means of enlivening and keeping awake the interest of young people in geometry. Interest which, as a rule, belongs to a few select people.Geometry, he argued, was not a 'dead language', but a 'living language'. In the paper La storia della matematica come anello di congiunzione fra l'insegnamento secondario e l'insegnamento universitario , which he published in 1899, Loria argued that teachers of mathematics should attend lectures on the history of mathematics before they began to teach in classrooms. Soon Loria was playing a large role in attempting to improve school teaching in mathematics. He gave the main plenary lecture at the Mathesis Congress in Padua in September 1909, choosing as his title La scuola media e la sua attuale crisi di sviluppo . The International Commission on Mathematical Instruction, meeting in Paris in 1914, asked him to compile a report Rapporto generale sulla preparazione teorica e pratica dei professori di matematica dell'insegnamento secondario to be delivered at the planned international conference in Munich in 1915. As part of his preparation he visited Göttingen where he had discussions with Felix Klein but, following the outbreak of World War I in 1914, the 1915 conference failed to take place and Loria's report was put on hold. In 1928 the International Commission on Mathematical Instruction meeting in Bologna asked Loria to prepare a report for the International Congress of Mathematicians to take place in Zürich in 1932. The Congress was held from 5 to 12 September 1932 and he delivered his report La préparation théorique et pratique des professeurs de mathématiques de L'Enseignement secondaire dans les divers pays which was published in full in 1933 in L'Enseignement Mathématique. In recognition of his efforts, the Commission conferred the title of Honorary Member on Loria on 15 July 1936.
Loria retired from the Chair of Higher Geometry at the University of Genoa in 1935. He remained very active with four historical papers appearing in 1936, six in 1937 and five in 1938. However, World War II proved an extremely difficult time for Loria. Even before Italy joined the war, the Italian government imposed anti-Semitic laws in 1938. Jews were labelled unpatriotic and prevented from holding government positions or teaching posts. Loria, in common with all Italian Jews, had to register with the authorities. In 1939 a law was passed requiring all books by Jewish authors to be removed from shops. All this was hard to bear, but much worse was to come. The Allies invaded Sicily in July 1943 and German troops took over control of Italy which surrendered in September 1943. Jews were faced with extermination, but the Waldenses saved Loria and his brother. This was a religious movement which had begun in the Middle Ages and had eventually became a Reformed branch of the Protestant Church. The centre of the Waldenses church was the mountain valley of Torre Pellice about 45 km southwest of Turin. The Waldenses saved many Jews from extermination by the Nazis, hiding them in the valley of Torre Pellice; it was there that Loria and his brother were given a safe refuse. However, sadly, Loria's brother Achille died at Luserna San Giovanni (Torre Pellice) on 6 November 1943. Loria returned to Genoa in 1945. He wrote no further works on geometry, but did continue to publish historical works up to his final contribution published in 1953 La matematica nel suo millenario sviluppo, ha seguito una direzione costante? .
For his outstanding mathematical achievements, Loria received many honours and awards. He was elected to: the Accademia dei Lincei; the Accademia Virgiliana di Scienze, Lettere ed Arti, Mantua; the Accademia di Scienze, Lettere ed Arti, Modena; the Accademia Pontaniana, Naples; the Accademia dei Scienze, Turin; the Societa Italiana per il Progresso di Scienze, Rome; and the Societa di Scienze e Lettere, Genoa. In 1926 he was awarded the Silver Medal of the Association Francaise pour l'Avancement des Sciences. He was also elected to the Deutsche Akademie der Wissenschaften, Halle; to the Czechoslovakian Society for Mathematics and Physics, Prague; to the Königliche Abhandlungen Gesellschaft der Wissenschaften, Prague; to the Masarik Academy of Work, Prague; to the Mathematical Association, London; to the Amsterdam Mathematical Society; to the Kharkov Mathematical Society; and to the Berlin Mathematical Society. In 1907 he received Binoux prize from the French Institute, then in 1922 he was awarded a second Binoux prize for Storia della geometria descrittiva dalle origini sino ai giorni nostri (History of Descriptive Geometry from Origins up to our Times). He explained how he came to write this work (see for example ):-
It is my conviction that whoever aspires to bring any contribution to our scientific knowledge cannot avoid taking notice of what his predecessors have done; this is especially true of a teacher since his course should be a true mirror of the science of his time. With this concept in mind I forced myself to gain an exact knowledge of the vast scientific literature relating to descriptive geometry, without neglecting in the least the precursors of Monge. The notes thus collected were of great use to me in my university courses; but, up to a certain point, I lost no time in noticing that, as a whole, they constituted the skeleton of a complete history of this material which was lacking, but only on this side of the Alps. Therefore, since in 1918 was approaching the first centenary of the death of Monge, I decided to celebrate it by publishing a historical work of this sort. The events of the recent war did not permit me to complete this task which I had undertaken. But the delay was short. And the French Institute in conferring the Binoux Prize  for my 'History of Descriptive Geometry from Origins up to our Times' has accorded to my labours the most coveted and most significant reward to which I could aspire.In addition to the historical work we have mentioned above, Loria also wrote around fifty biographies of mathematicians, many of which we have used in writing our biographies and are referenced in this archive. He explained why he was so interested in writing biographies (see for example ):-
Ever since I was in my teens the reading of the biographies of the most eminent thinkers exercised on me an irresistible fascination; more even than the fantastic adventures conceived in the whimsy of the old-time writers of romances, I was especially interested in knowing the circumstances through which 'man makes himself eternal.' Was it perhaps to draw from them a norm for the government of my own life? I do not know. I was then ignorant of the maxim, formed by Cicero in the most high sounding togaed Latin, according to which 'history is the master of life,' just as I was ignorant of the disrespectful ironic reply of Hegel, according to which instead 'history teaches only that men have never learned anything about it.' But I felt, however dimly, that, although, just as there do not exist two individuals perfectly equal physically or morally, so there never unfold two existences identical in everything, still the spectacle offered by the evidence of the life of one person might suggest, by way of generalization, some directives of universal character by which I might launch my own career in the world.Finally, we note that Pepe, in , describes Loria as:-
... the most notable historian of Italian mathematics.
Article by: J J O'Connor and E F Robertson