We should mention the political events which occurred in Europe around this time and had a large influence on the lives of many of the mathematicians we study in this archive. In April 1796 the French army under the command of Napoleon had invaded Italy. There followed a period when many intellectuals moved to a new way of thinking and enthusiastically took part in the democratic movement. Plana, although a young man at the time, was certainly one of those who embraced this new way. This aspect of his life between 1796 and 1799 is studied in .
In 1800 Plana entered the École Polytechnique in Paris. There he was greatly influenced by Lagrange who was one of his teachers and, of course, also someone of Italian birth. Fourier, like Lagrange, was greatly impressed by Plana's abilities and he tried to arrange for him to be appointed to the chair of mathematics at the school of artillery at Grenoble. However, Fourier failed in his attempt on behalf of Plana, so he tried again, this time to have Plana appointed to the chair of mathematics at the school of artillery in the Piedmont which was located part in Turin and part in Alessandria. This attempt was successful and Plana returned to Italy in 1803 to take up this post.
After further victories by Napoleon, the Treaty of Pressburg was signed by Austria and France on 26 December 1805. Austria was forced to give up much territory to France, and in particular Napoleon wanted Austrian influence completely removed from Italy. As a consequence Piedmont was given to France so Plana found himself in France again without making a move! In 1811 Lagrange recommended Plana for the chair of astronomy at the University of Turin, and Plana was appointed to the position. He would teach in Turin for the rest of his life, teaching both astronomy and mathematics, and teaching both at the university and at the school of artillery there.
Topics Plana worked on, in addition to astronomy, were integrals, elliptic functions, heat, electrostatics and geodesy. In astronomy his most famous work relates to the motion of the moon. Plana had already worked with Francesco Carlini on geodesy, and the director of the observatory in Milan suggested to Plana that he might collaborate with Carlini on problems relating to the motion of the moon. This episode is described in detail in  and we quote here part of the summary of that paper:-
In 1818 Laplace proposed that the Académie des Sciences in Paris set up a prize to be awarded to whoever succeeded in constructing lunar tables based solely on the law of universal gravity. In 1820 the prize was awarded to Carlini and Plana and to Damoiseau by a committee of which Laplace was a member. But Laplace strongly criticised the Carlini-Plana approach to lunar theory. A dispute ensued ... letters [were] exchanged between Carlini-Plana and Laplace, and ... papers published in Connaissance des temps and in Zach's Correspondance. After the exchanges, public and private, between Carlini-Plana and Laplace, the latter concluded that the results of the Italian astronomers and those arrived at by Damoiseau following the method of Laplace's 'Mecanique céleste' were fairly close, and that the purpose of the Académie in establishing the prize had been reasonably fulfilled.In fact Plana fell out with Carlini and Carlini withdrew from the collaboration. Plana continued with the work on his own and published Théorie du mouvement de la lune in Turin in 1832. By this time Plana was astronomer royal, and he went on to become a hereditary baron in 1844 and a senator in 1848. When he was nearly 80 years old, in 1860, he was elected a member of the Académie des Sciences in Paris.
Finally let us return to talk a little of Plana's association with the Turin Academy of Sciences. The article  describing the Academy of Sciences states that Plana's studies on lunar motion were the most important presented to the Academy at the beginning of the nineteenth century. An important event for mathematics in Turin occurred when Cauchy lived in Turin during the year 1832-33. He held the chair of mathematical physics at the University of Turin during a period when he was highly active in research, yet political events forced him to leave Paris. Cauchy and Plana certainly interacted during this time and their relationship is discussed in .
Another famous scientist who interacted with Plana was Babbage. He was elected to the Turin Academy of Sciences in 1841 and the article  examines his relations with the Academy and with some of its members, including in particular Plana.
Perhaps the most important aspect of Plana's contribution is summed up by Tricomi in  as follows:-
Plana is generally considered one of the major Italian scientists of his age because, at a time when the quality of instruction at Italian universities had greatly deteriorated, his teaching was of the highest quality, quite comparable with that of the grandes écoles of Paris, at which he had studied.
Article by: J J O'Connor and E F Robertson