Physician and mathematician, b. at Valentano in the Duchy of Castro, 3 Sept., 1765; d. at Modena, 10 May, 1822. At first he intended to enter Holy orders and went so far as to receive the tonsure, but changing his mind, he began the study of mathematics and medicine in the University of Modena, where he received the degree of doctor. At the age of twenty-three he was appointed professor of analysis after having substituted for a year for his teacher Cassiani. In 1791, the chair of elementary mathematics was entrusted to him. In the meantime, he did not neglect the study and practice of medicine. At the time of the French invasion of Italy (1796), he was unexpectedly appointed a member of the Juniori in the legislative body at Milan. It was not without difficulty that he succeeded in returning to his lectures at Modena. Because he refused to take the republican oath without the conditional declaration dictated by his conscience, he was dismissed from his position as a public lecturer; but with the return of the Austrians in 1799 he was restored to his former post and maintained therein by succeeding governments. A call to the chair of higher mathematics in Pavia he declined, because he did not wish to give up his medical practice among his dear Modenese. The university having been degraded to the rank of lyceum, he accepted (1806) the chair of applied mathematics at the newly established military school. In 1814 Franceso IV re-established the university and appointed Ruffini rector for life, and at the same time professor of practical medicine and applied mathematics. By his lectures with the patients actually present he revived the clinical studies which had been neglected for several years. During the typhus epidemic of 1817 he sacrificed himself for his fellow citizens, and finally succumbed. Although he recovered, he never regained his strength. He was buried in the Church of Santa Maria di Pomposa, between the tombs of Sigonio and Muratori.

Ruffini's sole medical treatise is a "Memoria sul tifo contagioso". As a mathematician his name is inseparably associated with the proof of the impossibility of solving algebraically the quintic equation, on which subject he wrote several treatises ("Teoria generale delle equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al 4°", 2 vols., Bologna, 1798; "Della soluzione delle equazioni alg. determinate particolari di grado sup. al 4°" in "Mem. Soc. Ital.", IX, 1802, which was awarded a prize by the National Institute of Milan; "Della insolubilità etc. qualunque metodo si adoperi, algebraico esso sia o trascendente" in "Mem. Inst. Naz. Ital.", I, 1806). He also proved the impossibility of the quadrature of the circle ("Riflessioni intorno alla rettificazione ed alla quadratura del circolo" in "Mem. Soc. Ital.", IX, 1802). Less known, however, is the fact that Ruffini published the now familiar "Horner's method" of approximation to the roots of numerical equations fifteen years before Horner's first paper on it appeared in the "Philosophical Transactions" of 1819 (pt. I, pp. 308-35). In 1802 the Italian Society of Forty offered a gold medal for the best method of determining the root of a numerical equation of any degree. In 1804 the medal was awarded to Ruffini, and the dissertation was published under the title "Sopra la determinazione delle radice nelle equazioni numeriche di qualunque grado". In a paper read before the Southwestern Section of the American Math. Soc. (26 Nov., 1910), Professor Florian Cajori pointed out that the computation demanded by Ruffini is identical with that in "Horner's method", and that this method is elaborated by Ruffini with a clearness and thoroughness not surpassed in Horner's own exposition of 1819. In view of this fact, Professor Cajori insists that the name of Ruffini should be associated with that of Horner in the designation of the method. Ruffini again wrote on this subject in 1807 (Algebra elementare, cap. iv, v), and in 1813 (Memorie Soc. It., XVI, XVII). Ruffini was during his whole life a zealous Catholic. His convictions find expression in his apologetic works: "Dell' immortalità dell' anima" (Modena, 1806), dedicated to Pius VII, who sent him a gold medal; "Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del Sig. Conte de la Place" (Modena, 1821), in which he proves himself to be as familiar with metaphysics as with questions of religion.

FANTONETTI, Note storiche sopra i socj defunti: Paolo Ruffini in Mem. Imp. Reg. Ist. del Regno Lomb. Ven., V (1838), 40-41; LOMBARDI, Notizie sulla vita di Paolo Ruffini (Florence, 1824); POGGENDORF, Biogr.-Litt. Handwörterb. zur Gesch. der Exact. Wiss. (1858-63); CAJORI, Horner's Method of Approximation Anticipated by Ruffini in Bull. of American Math. Soc. (May, 1911).

J. Stein.