Napoleon had led the French army into the north part of Italy in 1796 and won military victories against the Austrian army. In the following year, he had reorganized the northern Italian republics and, in particular, founded the Military School in Pavia. During his time at the military school Bordoni published a number of mathematical works including: Memorie matematiche: 1, sopra le trasformazioni delle equazioni: Sopra l'integrazione delle equazioni a differenze finite e parziali delle funzioni a piu variabili Ⓣ (1808); Nuovo rapporto tra la teoria del centro di gravita e quella della composizione delle forze Ⓣ (1810); Memoria matematica dedicata al sig. dottor Vincenzo Brunacci ... Ⓣ (1810); Sopra l'equilibrio di un poligono qualunque Ⓣ (1814); Nuovi teoremi di meccanica elementare Ⓣ (1815); Dell'ombra del toro, dell'ovolo dello spaccato di una volta emisferica e della scozia Ⓣ (1815); and De' contorni delle ombre ordinarie Ⓣ (1816).
In 1815, following the defeat of Napoleon, Francis I founded of the Kingdom of Lombardy-Venetia, a strange region consisting of two separated areas of land. Control was taken away from Milan and given to Venice and the administrative system set up by Napoleon was dismantled. There was widespread discontent, and severe measures were imposed to keep control. These political events led to the closing of the military school in Pavia in 1816 and Bordoni was out of a job. However, this situation did not last long for, on 1 November 1817, he was appointed Professor of Elementary Pure Mathematics at the University of Pavia. In the following year he was appointed to the Chair of Higher Calculus, Geodesy and Hydrometrics. We note that Higher Calculus (actually called Sublime Calculus) essentially meant Mathematical Analysis. At Pavia, Bordoni joined the other Chairs in Mathematics, namely Marchesi in Architecture, Lotteri in Preliminary Calculus, and Gratognini in Mechanics, Statics, Hydrodynamics, Hydraulics (Applied Mathematics). Four new mathematics courses were established in 1819, namely Descriptive Geometry, Static and Hydraulic Architecture, and Hydrometry and Drawing.
Bordoni began to study differential geometry as early as the 1820s. When he became acquainted with the work of Liouville and the ideas of Gauss, he encouraged his colleagues and students at the University of Pavia to develop them. His works published during the 1820s include: Degli argini di terra Ⓣ (1820); Sull'equilibrio delle curve a doppia curvatura rigide ovvero completamente o solo in parte elastiche Ⓣ (1820); Sull'equilibrio astratto delle volte Ⓣ (1821); Annotazioni agli elementi di meccanica e d'idraulica del prof. Giuseppe Venturoli Ⓣ (1821); Sui sistemi di due forze equivalenti fra loro e ad un qualsivoglia Ⓣ (1821); De' contorni delle penombre ordinarie Ⓣ (1822); Della distanza delle linee e delle superficie cha hanno le normali comuni Ⓣ (1822); Sopra de' momenti ordinarj Ⓣ (1822); Proposizioni di geodesia elementare Ⓣ (1823); Sulla stereometria Ⓣ (1824); Trattato di geodesia elementare Ⓣ (1825); Nota di stereotomia sopra i cunei dei ponti in isbieco Ⓣ (1826); and Sul teorema guldiniano Ⓣ (1827). Not only was he working hard within Mathematics but he also took a major role in running the University and, in 1827 and 1828, he served as rector of the University.
When Bordoni was appointed to the University of Pavia, Mathematics was part of the Faculty of Philosophy. Major changes in the Mathematical Sciences course structure were introduced in 1827 and further changes made in 1838. The article  describes these in detail. In 1847 the Faculty of Mathematics was granted autonomy and, two years later, Bordoni put forward proposals to reorganise Mathematical Studies :-
... in which he hoped that practical courses in engineering would be organized and specializations in architecture as well as in rural, mechanical, mining, hydraulic and civil, military engineering would be provided for.His proposals were, however, largely ignored. In 1854 Bordoni was elected Director of Mathematical Studies and, in 1859, he tried again to make major changes to the structure of Mathematics, in particular emphasising the further development of advanced courses in Civil Engineering. He put forward proposals for a four-year course leading to the degree of Doctor in the Studies of Civil Engineering and Architecture. In addition he made proposals as to how the material should be examined. However, his proposals went further :-
In order to be qualified to practice the profession a three-year apprenticeship would be compulsorily required, the first year to be spent at the same University under the direction of two professors, while the others would be under the direction of an Engineer practicing the profession. After this training, a very severe examination should be taken "from any board of Public Constructions".Again his proposals were not adopted. In February 1860, Bordoni was appointed as a senator but died one month later. Among the publications from the latter part of his career we mention the texts Lezioni di calcolo sublime Ⓣ (1831), Sopra gli scolastici Ⓣ (1837) and Trattato di geodesia elementare Ⓣ (1843). He was honoured with election to various academies, including the Accademia dei XL (now the National Academy of Sciences of Italy). He was also honoured more than twenty years after his death when the Istituto Tecnico Bordoni, established in Pavia in 1861, was named after Bordoni in 1882.
Among Bordoni's students at Pavia were Felice Casorati, who became his assistant after graduating in 1856. Earlier Francesco Brioschi had been Bordini's student graduating 1845 having written a thesis under Bordini's supervision. Brioschi was appointed professor of applied mathematics at the University of Pavia in 1852, becoming Bordoni's colleague. Luigi Cremona was also taught by Bordoni when a student of Civil Engineering at Pavia beginning in 1849. He was awarded his doctorate from Pavia in 1853.
Article by: J J O'Connor and E F Robertson