**Pietro Paoli**studied first in Livorno with the Jesuits then entered the University of Pisa in 1774. His original intention was to study law but he became interested in the mathematical and physical sciences and he graduated in 1778. Following his graduation, he taught at the Gymnasium in Mantua from 1780 to 1782, then moved to Pavia where he was appointed to the chair of Elementary Mathematics at the University. He held this chair in Pavia for two years then, on 23 October 1784, he was appointed by Leopold II, Grand Duke of Tuscany, to the chair of algebra at the University of Pisa.

The University of Pisa, being in the Duchy of Tuscany, had been under Lorraine rule from 1737. It had prospered during this period with the opening of an observatory and the creating of chairs in experimental physics and chemistry. Paoli continued to hold the chair of algebra at Pisa until 1798 when he was appointed to the chair of higher mathematics. However, in the following year, Lorraine rule ended when Tuscany was invaded by French troops. The University of Pisa was closed by Ferdinand III, the Grand Duke of Tuscany, in the year 1799 but it reopened under French control in 1800 and Paoli continued to hold the chair of higher mathematics. The University became an Imperial Academy in 1803 and at this time Paoli was named as an Honorary Inspector. In 1805 he was appointed "Royal hydraulic consultant". 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.

The University of Pisa became a Normal School in 1813 and Paoli was one of the commissioners appointed to draft new regulations for the University. In 1816 he was appointed as Superintendent of Education of the Grand Duchy of Tuscany and he moved to Florence. In addition to this role, in 1817 he was also appointed as President of the Commission in charge of drafting the new General Land Register of Tuscany. This was a long and important task, and it was completed in 1834.

His research was on analytic geometry, calculus, partial derivatives, and differential equations. His contributions show that he possessed a deep knowledge of the works of Lagrange, Laplace and Monge. 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). The first of these, the book *Opuscula analytica* Ⓣ, was first published in 1780 and dedicated to Leopold II, the Grand Duke of Tuscany. A second edition of this book was published in 1783. He also published the book *Memorie sul Calcolo Integrale e sopra alcuni Problemi Meccanici* Ⓣ in 1793. This text was based on ideas of Leonhard Euler, Alexis Fontaine des Bertins and the Marquis de Condorcet.

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 consisted of a 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. We see from the Preface that Paoli wanted to write the book because he was unhappy with the standard of mathematics teaching in Italy at this time. He writes:-

John Cuzzocrea and Shlomo Sawilowsky write [2]:-Among all those who in Italy are given to the study of mathematics, if we except a few sublime geniuses, who, with their strength of spirit have triumphed over all obstacles and reached a place at the highest level of geometry, there are few others that come to mediocrity. Not, we must repeat, because of the lack of talent that abounds in Italy, as in other places, but because of badly designed methods of teaching mathematics: since young people are not properly taught the very basic elements, which appear easy because they are inaccurate and do not treat, in each branch of science, a particular case. The first problem that arises is that young people will be satisfied with such an approach, rather than a rigorous demonstration that has not be given when it is considered in its full generality. In addition, it is certain that no one will be able to read the works of the great geometers, who assume their reader's science has been brought to that rigorous level, when they write. Now who has not found that having the elements of knowledge that you had only a century ago, the first reading of the books of Euler, d'Alembert, and Lagrange present insuperable difficulties. Hence, most of the time, it happens that everyone abandons the career they are undertaking, or is content to remain in the restricted sphere of knowledge passing their lives with the most basic elements, and sometimes in this position comforted by the inexperience of the teachers who, not being in a state to explain the difficulties that they encounter as well, advise the student to refrain from certain researches, on which they pour contempt and characterise as useless.

Although most mathematicians ignored Paolo 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:-Cajori(1929)cited the first edition of 'Elementi di algebra'Ⓣas helping to establish various1718^{th}and^{th}century mathematics symbols in Italy. Some examples include trigonometric, inverse, and powers of trigonometric functions; and Euler's e as the base of natural logarithms.

Of course, as well as Ruffini, Paoli was thinking of Lagrange, with whom he corresponded, as an Italian.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...

Perhaps Paoli's most famous student was Vincenzo Brunacci (1768-1818) who studied with him at Pisa, graduated in 1788, and published the two volume *Elementi di algebra e di Geometria* Ⓣ in 1809. In addition to Brunacci, Paoli's students included Giovanni Taddeo Farini (1778-1822) and Antonio Bordoni.

Paoli received many honours including election to the academies of Bologna, Naples, and Mantua, and to the Mathematical Circle of Palermo. He was a founding member of the National Academy of Sciences of Italy (the "Academy of Forty") in 1782 and elected to the Accademia delle Scienze di Torino on 13 July 1811.

**Article by:** *J J O'Connor* and *E F Robertson*

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