Paolo Ruffini

Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher.

By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics. Among his work was an incomplete proof (Abel–Ruffini theorem) that quintic (and higher-order) equations cannot be solved by radicals (1799), and Ruffini's rule which is a quick method for polynomial division. Ruffini also made contributions to group theory in addition to probability and quadrature of the circle.

He practiced as both a professor of mathematics (University of Modena) and a doctor including scientific work on typhus.

Ruffini’s 1799 work marked a major development for group theory. Ruffini developed Joseph Louis Lagrange's work on permutation theory, following 29 years after Lagrange’s "Réflexions sur la théorie algébrique des equations" (1770–1771) which was largely ignored until Ruffini who established strong connections between permutations and the solvability of algebraic equations. Ruffini was the first to controversially assert the unsolvability by radicals of algebraic equations higher than quartics. This angered many members of the community such as Malfatti (1731–1807). Work in this area was later carried on by those such as Abel and Galois who succeeded in such a proof.

• 1799: "Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto" (General Theory of equations, in which the algebraic solution of general equations of degree higher than four is proven impossible).
• 1802: "Riflessioni intorno alla rettificazione ed alla quadratura del circulo" (Reflections on the rectification and the squaring of the circle)
• 1802: "Della soluzione delle equazioni algebraiche determinate particolari di grado superiore al quarto" (On the solution of certain determined algebraic equations of degree higher than four)
• 1804: "Sopra la determinazione delle radici nelle equazioni numeriche di qualunque grado" (Above the determination of the roots in the numerical equations of any degree)
• 1806: "Della immortalità dell’anima" (On the immortality of the soul)
• 1807: "Algebra elementare" (Elementary algebra)
• 1820: "Memoria sul tifo contagioso" (Essay on contagious typhoid)
• 1821: "Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del signor conte Laplace" (Critical reflections on the philosophical essay about probability by Count Laplace)