He matriculated at the University of Turin in 1897 and he showed his quality as a mathematics student by winning several awards and scholarships. At the same time as he was studying for his degree, he was also working for a mathematics teaching qualification at the School of Education. He graduated with honours in mathematics from the University of Turin on 5 July 1901, having gained the highest possible grade, and gained his teaching certificate four days later. After graduating, he was appointed assistant professor in projective geometry at the University of Turin where he taught for three years. In fact at Turin he was an assistant to Gino Fano who had been appointed as a professor at the university in 1901. He published his first paper Sopra le coniche bitangenti alle superficie algebriche Ⓣ in 1903 then, in 1904, he was awarded a scholarship from the Collegio Carlo Alberto to allow him to improve his knowledge of mathematics by attending courses by Henri Poincaré and Émile Picard at the higher education institutes in Paris.
Arriving back in Italy Bottasso was appointed as an assistant professor of projective geometry at the University of Bologna. After three years in this post he had a number of different positions to which he was appointed after winning competitions: he was a professor in the Technical School of Racconigi (Racconigi is a town in the province of Cuneo in Piedmont, 40 km south of Turin), a professor in the Technical School of Susa (a town which is also in Piedmont, 50 km west of Turin) and then at the Technical School of Turin. Following this series of appointments, he became a lecturer in algebra and analytic geometry at the University of Parvia (the city is 35 km south of Milan). It was only during his time in Parvia that he resumed publishing mathematical papers with his second and third papers I caratteri d'un piano multiplo ciclico la cui curva di diramazione è irriducibile e generale nel suo ordine Ⓣ (25 pages) and Alcune singolarità elementari d'un piano multiplo ciclico la cui curva di diramazione è irriducibile Ⓣ (26 pages) both appearing in 1909 in the Proceedings of the Accademia delle Scienze di Torino. In 1910 he was appointed as a professor at the Turin Military Academy, a post he held until 1916 when courses for officers were abolished. From October 1915 to January 1917 he taught mathematics in the Royal Technical Institute "M Buniva" in Pinerolo (a town about 35 km southwest of Turin), being an extraordinary professor for the first year. Finally, not long before his death, he was appointed professor of rational mechanics and mathematical-physics at the University of Messina.
Bottasso studied differential geometry and mechanics but also made contributions to actuarial and financial mathematics. In his paper Alcune applicazioni delle formule di Fernet Ⓣ (Atti Acc. Sci. Torino, 1911) he pointed out:-
... the simplicity and the quickness of vector calculus in the approach to different problems for which Cartesian methods are too difficult.He used the vector calculus in studying problems in geometry, mechanics and physics. He worked with Cesare Burali-Forti and Roberto Marcolongo on the Analyse vectorielle générale Ⓣ, by writing the volume Astatique Ⓣ. Giuseppe Peano writes in a review:-
The perspicuity of exposition, based on the ingenious use of a single homography, has allowed the author to bring together a wealth of material in a short space. The meticulous care used in discussing and characterizing, in a subtle geometric sense, the different special cases, to cover all the issues, and finally, the many new and elegant properties, make this little volume, which also fills a gap in our mathematical literature, very interesting.In the paper Il teorema di Rouché-Capelli per i sistemi di equazioni integrali Ⓣ (Atti Acc. Sci. Torino, 1912) Bottasso underlined the analogy between vector homography and integral equations, and used vector homography to solve integral equations. In 1913, for his excellent contributions, he was awarded a mathematics prize, the Ministerial Prize, by the Accademia Nazionale dei Lincei. He was also honoured with election to the Academia pro Interlingua in 1915.
He was a lecturer at the 'Conferenze matematiche', designed to update secondary school teachers of mathematics, organised by Giuseppe Peano and Tommaso Boggio at the University of Turin between 1915 and 1916. Bottasso gave lectures to the high school teachers on numerical calculus in March 1915.
An interesting episode regarding Bottasso reported in the paper La Valle Pesio relating to the time when he was teaching at the Turin Military Academy. This is quoted in :-
One night - reports the columnist D.G. - I was with many of my friends, students of the Military Academy under the arcades of Via Po in Turin. We were talking animatedly and cheerfully. All of a sudden everyone was silent and stiffened in a greeting worthy of a general. I turned and saw that professor Bottasso was passing. Seeing me with his pupils, he greeted me as we greet people from Chiusa away from their valley. The students realized that I knew him and were very generous with their praise and admiration for their teacher.Lionello Bottasso, writing about his grandfather Matteo Bottasso in , describes his character:-
[He] is remembered as a warm person, even if a little shy. He loved the life of his country ...
Article by: J J O'Connor and E F Robertson