Pieri left Turin in 1900 and Boggio continued to teach projective and descriptive geometry. While he tutored geometry at the university in his assistant position, Boggio was undertaking research in applied mathematics. He published four papers in 1900, for example Sull'equilibrio delle membrane elastiche piane and Un teorema di reciprocità sulle funzioni di Green d'ordine qualunque . In the first of these, Boggio obtained a solution for the problem of an elastic membrane, displaced in its own plane with known displacements on the boundary. In 1901 he published seven papers including Sopra alcune funzioni armoniche o bi-armoniche in un campo ellittico od ellissoidico and Sull'equilibrio delle piastre elastiche incastrate . In 1903 he was appointed to teach mathematical physics at the University of Pavia and as an assistant to Giuseppe Peano to teach calculus at the University of Turin. Boggio remained at Turin and Pavia, teaching a variety of different courses, until 1905 when, after a competition, he was appointed Professor of Mathematics of Finance at the Royal Higher School of Commerce of Genoa, later part of the Faculty of Economics and Commerce of the University of Genoa. In 1906 the Paris Academy of Sciences proposed 'The theory of the equilibrium of supported elastic plates' as the topic for their competition for the Vaillant Prize. Twelve mathematicians, from different countries, submitted entries for the prize which was judged by Henri Poincaré. Four entries were deemed worthy of a share of the 4000 franc prize, namely those by Boggio, Jacques Hadamard, Arthur Korn (1870-1945) and Giuseppe Lauricella (1867-1913).
In 1908 Boggio moved again, this time to the position of Professor of Rational Mechanics at Messina in northeast Sicily. However disaster struck Messina on 28 December 1908 when an earthquake almost totally destroyed the city. Boggio was extremely fortunate to escape with his life as 78000 people were killed by the earthquake. Messina was no longer a viable place for Boggio to work and, following a unanimous vote by the Faculty of Rational Mechanics and Mathematical Physics at the University of Florence, he was appointed to teach there. This only lasted a short time for, following the death of Giacinto Morera in 1907, a competition was announced to fill his chair in Turin. Boggio was successful and, in November 1909, he was appointed Professor of Higher Mechanics at the University of Turin. In addition to teaching at the University, he also taught courses on Higher Mechanics and on Mathematical Analysis at the Military Academy in Turin. He also gave courses in various disciplines at the University of Modena.
In 1918 Enrico D'Ovidio retired from his chair in Turin and Boggio took over teaching algebraic analysis and analytic geometry. A text which Boggio wrote on the differential calculus with geometrical applications, published in 1921, was reviewed by his colleague Peano who says the books use of vector methods:-
... constitutes that royal road sought in vain since the time of Euclid.Mathematics was reorganised at Turin in 1922. Boggio was director of the School of Algebra and Analytic Geometry in 1921-22. In 1923, as required by the Ministry of Public Instruction, the Chair of Complementary Mathematics was established. Topics in this were taught by Boggio in session 1924-25 before a new professor, Francesco Tricomi, was appointed in 1925. One of Boggio's most unfortunate publications was the book Espaces courbes. Critique de la Relativité written in collaboration with Cesare Burali-Forti and published in 1924. G Y Rainich writes in the review :-
This book by two known Italian mathematicians makes one feel sad. It is an example of how intolerance can mislead even powerful minds in a field where we would least expect it. ... The authors of the book under review make it their purpose to get rid of all extraneous features in the theory of curved spaces and this purpose is very commendable but it must be said at once that in spite of some good ideas (they recognize, for instance, the importance for the theory of what they call homographies, i.e. linear and multilinear vector functions) their attempt results in a failure. The situation may be best characterized by stating that the authors have not succeeded in introducing the most fundamental concept in the theory of curved space - the curvature, or the Riemann, tensor-in an intrinsic or absolute way, i.e. without the use of extraneous or arbitrary things. In their attempt to eliminate extraneous things they stopped half way: they got rid of coordinates but instead of studying curved space directly they use a representation of it on a Euclidean space, a representation which, as the authors themselves recognize, involves a certain degree of arbitrariness. But the really strange thing is that because in their treatment this tensor is introduced with the aid of notions which have no intrinsic significance the authors conclude that the tensor itself is of no or little importance. On this point (which is the central point in their criticism of the application of geometry of curved space to physics) Burali-Forti and Boggio are behind those geometers who while using coordinates succeed in discriminating as to which expressions have a meaning independent of them. And it must be remarked that, of course, it is possible to introduce the Riemann tensor intrinsically and that, in fact, the authors themselves were not so very far from it when they introduced the Riemann curvature. ... Outside of this main line of attack on the relativity theory the authors bring forth against this theory all possible arguments without finding anything to say in its favour. Most of these arguments cannot be taken seriously ...We must not allow this rather unfortunate publication in any way dim our view of the quality of Boggio's other contributions which were very substantial. Examples of his work which has proved important is Sulle funzioni di Green d'ordine m (1905), which contains what is known today as 'Boggio's Principle', and Sull'equazione del moto vibratorio delle membrane elastiche which contains his lower-bound lemma of certain elliptic operators. Several papers have been written during the last five years which generalise these and other results by Boggio. Also his various generalizations and applications of the Lebesgue integral are still of interest today. The famous Boggio-Hadamard conjecture about the sign-definiteness of the Green function of the clamped plate in smooth and convex domains was disproved by Duffin in 1948. The conjecture essentially claimed that the biharmonic Green functions with clamped boundary condition are always positive or, in physical terms, a clamped thin elastic plate is always bent to the direction of a point load placed at any position on the plate.
Boggio taught Higher Geometry from 1938 to 1940, then both Higher geometry, and Analytic and Projective geometry in 1940-41. The years of World War II were extremely difficult ones for Boggio. Kennedy writes in :-
In addition to his professorship, he also taught many courses at the Military Academy and he gave private lessons, even to his own university students, a fact which lowered him in the estimation of many. Boggio suffered many family difficulties. His wife is said to have been of little support to him, a daughter died during World War II and his second son died at the age of 46 (the first son emigrated to Argentina), leaving him to care for his daughter-in-law and two grandchildren.His first son, Mario, was an engineer and emigrated with his family to Argentina. The second son, who died at the age of 46, had graduated in philosophy. His daughter died in a sanatorium. These tragedies shook Boggio greatly but he bore the pain with great resignation.
After the war ended he taught Higher Geometry from 1945 to 1947, then Numerical Mathematics and Graph Theory in 1947-48. In session 1949-50 he taught Infinitesimal Calculus but by this time he was officially retired and taught as an assistant to the chair. He continued to publish after he retired, publishing Sur un théorème de Darboux in 1960 and Sopra alcune questioni di meccanica razionale in the following year. Following his death, he was buried in the small cemetery at Axams, near Innsbruck, next to the grave of his second son.
Cataldo Agostinelli gives an indication of his character in :-
He was a modest man, with simple ways and needs, yet he was strong and decent, friendly towards his colleagues and kind to his students. Generous and open-hearted, he worked willingly for his colleagues and friends, and was always generous to students with help and advice. He scrupulously fulfilled his academic duties. He was not free from flaws and shortcomings, like any human being, frequently causing opposition, so he did not receive the awards for his academic contributions that he deserved.He did, however, receive many honours. He was elected to the Academy of Sciences of Turin in 1924 and was a member of the National Committee for Mathematics Research. In January 1926 became a knight of the Order of the Crown of Italy, in 1931 he was appointed Grand Officer and, in 1953, Commander of the Order of Merit of the Italian Republic. When he retired he was awarded the gold medal for merit from the Academy of Culture and Art. Shortly before his death he became the president of the Academy of Sciences of Modena. He had been made an honorary member of that academy in recognition of his work at the University of Modena carried out in extremely difficult circumstances during and shortly after World War II.
Article by: J J O'Connor and E F Robertson