He went to the University of Bologna in 1863, supported by a scholarship from Forli, with the intention of studying engineering. The University of Bologna was founded in the 11th century and was at the time Bertini entered it (and of course still is today), one of the most famous and oldest universities in Europe. After a period of decline the university had been reorganized in 1860, only three years before Bertini entered, and had resumed its place among Italy's foremost universities. The city of Bologna and the surrounding area had been controlled by the Austrians from 1849 until it became part of the Kingdom of Italy in 1860. When Bertini entered the university in 1863 certain aspects seem from our 21st century perspective, to have been surprisingly modern. Its faculty of science had been developed in the 17th century, and since the 18th century women had been admitted both as students and as teachers. While on his engineering studies, Bertini took a mathematics course given by Cremona and this inspired him to study pure mathematics. Cremona was an ardent Italian nationalist who, after fighting against the Austrians to help achieve an independent Italy, had been appointed as a professor at Bologna three years before Bertini entered the university. Before Bertini could complete his degree he took a break from his studies to take part in the third war for Italian independence, an action which his teacher Cremona strongly approved.
The Kingdom of Italy came into existence in 1860 and was officially proclaimed on 17 March 1861, by a parliament assembled in Turin. They wanted Rome as the capital of their Kingdom but it was held by the Pope supported by the French. In June 1866 war broke out between Austria and Prussia and this diverted attention from Rome to Venice which the Austrians still controlled. The Italian government sent troops to attack the Austrians in an attempt to drive them out of Venice but they were defeated on 24 June at Custozza. Garibaldi led an almost independent army against the Austrians in the Tirol and Bertini joined his force which won some success near Trento. Although losing the main battles for Venice, the success in the Tirol together with French political pressure, led to Italy gaining Venetia at the Treaty of Vienna signed on the 3 October 1866. Bertini returned to his studies at Bologna but was advised by Cremona to transfer to the University of Pisa where he obtained a degree in mathematics in 1867 in the school of Betti and Dini.
In October 1867 Cremona was appointed to the Polytechnic Institute of Milan. Bertini followed his teacher there and, during 1868-69, he studied at Milan attending courses by Cremona, Brioschi and Casorati on Abel's integrals. He began his teaching career in 1870 in a secondary school in Milan, then two years later he went to Rome, again as a secondary school teacher. Cremona recommended him to teach descriptive and projective geometry as a lecturer at the University of Rome. In 1875 he was appointed professor of geometry at the University of Pisa, accepting the offer of a chair for which he had been proposed by Betti. From 1880 to 1892 he held a chair at the University of Pavia where he was part of what Cinquini describes in  as the golden decade of Pavian mathematics. The two most important colleagues of Bertini who contributed to this 'golden decade' were Felice Casorati and Eugenio Beltrami. In 1892 Bertini returned to Pisa where he worked until he retired at the age of 75.
His work in algebraic geometry extended Cremona's work. He studied geometrical properties invariant under Cremona transformations and used the theory to resolve the singularities of a curve. The paper  by Kleiman studies what the authors calls the two fundamental theorems of Bertini. These two fundamental theorems are among the ones most used in algebraic geometry. The first theorem is a statement about singular points of members of a pencil of hypersurfaces in an algebraic variety. The second theorem is about the irreducibility of a general member of a linear system of hypersurfaces.
Carruccio writes in  that:-
His treatises are noteworthy for their order and clarity.We should note that Bertini had a number of outstanding students and their work continued the Italian tradition of outstanding contributions to geometry. We mention L Berzolari, C Rosati, G Scorza, G Fubini, G Albanese and L Campedelli. At Pisa, Enriques was his assistant.
Those who knew Bertini wrote that he kept a youthful enthusiasm for science to the end of his life.
Article by: J J O'Connor and E F Robertson