Pietro Paoli studied first in Livorno then entered the University of Pisa from where he graduated in 1778. Following his graduation, he taught at the University of Mantua from 1778 to 1780, then moved to Pavia where he taught at the University from 1780 to 1784. He was appointed as a lecturer at the University of Pisa on 23 October 1784 and he remained at Pisa for the rest of his career. In 1798 Paoli was promoted to professor at Pisa and, in addition, from 1810 to 1814 he was head of pure mathematics at the university. In 1814 he retired from his chair, being named professor emeritus at this time.
His research was on analytic geometry, calculus, partial derivatives, and differential equations. Among Paoli's publications we mention Liburnensis Opuscula analytica (1780), Ricerche sulle serie (1788) which corrects an error in a 1779 paper by Laplace on series, Della integrazione dell'equazioni a differenze parziali finite ed infinitesime (1800), Sulle oscillazioni di un corpo pendente da un filo estendibile memoria (1815), and Sull'uso del calcolo delle differenze finite nella dottrina degl'integrali definiti memoria (1828).
Paoli was most famed for Elementi di algebra finita ed infinitesimale (1794) which became a classic text used in Italy for many years. The first edition in two volumes was published in Pisa in 1794 and a second edition in three volumes was published in Livorno in 1804. The work was comprehensive treatment of analytical methods in mathematics and, at the time it was written, it incorporated the most modern approach. It was divided into three parts entitled respectively 'The algebra of finite quantity', 'Introduction to infinitesimal analysis', and 'Infinitesimal analysis'. The third part was further divided into two sections, the first containing the differential calculus, the second being devoted to methods related to the integral calculus. Paoli, who corresponded and exchanged books with Lagrange, sent a copy of the first edition of his Elements of algebra to Lagrange who replied with a note of thanks in September 1798.
Although most mathematicians ignored Ruffini's proof of the impossibility of solving equations of degree greater than four by the method of radicals, Paoli read Ruffini's proof and wrote to him in 1799:-
I read with much pleasure your book ... and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four. I rejoice exceedingly with you and with our Italy, which has seen a theory born and perfected and to which other nations have contributed little...
Of course, as well as Ruffini, Paoli was thinking of Lagrange, with whom he corresponded, as an Italian.
In addition to Vincenzo Brunacci who studied with Paoli at Pisa and graduated in 1788, his students included Giovanni Taddeo Farini (1778-1822).
Article by: J J O'Connor and E F Robertson