Paolo Ruffini

Paolo Ruffini was an Italian mathematician and philosopher. He lived from 22 September 1765 to 10 May 1822.

He is remembered chiefly for what is now known as the Abel–Ruffini theorem. This is one of his best-known ideas in mathematics.

Ruffini also made a major contribution to the theory of equations. He developed what is called the theory of substitutions, which came before modern group theory.

If you are looking for the comedian, see Paolo Ruffini (actor).

Biography

Paolo Ruffini was born in 1765 in Valentano in the Papal States, into a middle-class family. His father, Basilio, was a physician. When Paolo was a teenager, the family moved to Reggio Emilia, near Modena. As a boy he planned to join the Holy orders and even received the tonsure, but later he changed his mind. He studied at the University of Modena, where his mathematics teachers included Luigi Fantini, Paolo Cassiani and Giovanni Battista Venturi. When only 23, he was elected Professor of Mathematics there. He had also been elected a member of the National Academy of Sciences, the so-called "Society of the Forty", in 1780. A conservative Catholic, he lost his university post during the French invasion of Italy because he would not take the oath to the new government. In 1799 he published a bold and controversial theory claiming that quintics could not be solved by radicals, which leading mathematicians had believed.

Ruffini's ideas were ahead of their time, and many mathematicians did not accept them at first. Niels Henrik Abel is sometimes wrongly credited with Ruffini's ideas, and Lagrange found the work impressive but hesitated to accept it. The first mathematician to recognise its importance was Augustin Cauchy, who built on some of his results from 1813 to 1815. The work became widely recognised at the end of the nineteenth century thanks to Burkhardt and Pierpont. In 1804 Ruffini won a gold medal for the best method of finding the roots of a numerical equation of any degree, the famous Ruffini's rule. After Napoleon's defeat in 1814, he returned to the University of Modena as rector, and in 1816 he succeeded Antonio Cagnoli as president of the National Academy of Sciences. While helping victims of the 1817–18 typhus epidemic, he caught severe typhoid fever and never fully recovered. He died in Modena on 10 May 1822 and was buried in the Church of Santa Maria della Pomposa, between the tombs of Sigonius and Muratori. He also wrote scientific works on typhus in 1820. His mathematical works were edited by Ettore Bortolotti and published in 1915 by the Italian Mathematical Union.

Contributions to mathematics

Ruffini's works include notable developments in algebra. His most important achievement was proving that quintic (and higher-order) equations cannot be solved by radicals (Abel–Ruffini theorem). His first proof, in 1799, was incomplete, but it laid the foundation for the later work of Niels Henrik Abel, who gave a complete proof in 1824. Ruffini is also well known for Ruffini's rule, a quick method for dividing polynomials by linear factors. He described this rule in a paper published in 1804, and it remains a valuable tool. In 1813 he applied his method to finding the roots of numbers.

Ruffini also greatly improved group theory, building on the work of Joseph-Louis Lagrange on permutation theory. Lagrange's work was mostly ignored until Ruffini showed strong connections between permutations and solving algebraic equations. He introduced ideas such as the order of an element, conjugacy, and cycle decomposition in permutation groups, making him a pioneer in this field. He was also the first to tell apart transitive and intransitive permutation groups, and among the transitive ones, primitive and imprimitive groups. Ruffini also wrote about probability and the quadrature of the circle. In later years he wrote about philosophy from a Catholic perspective. In his essay The immateriality of the soul (1806), he opposed the philosophical system of Erasmus Darwin, and in Riflessioni critiche (1821), he refuted the work of Pierre-Simon Laplace on probability.

Legacy

At first, many mathematicians doubted Ruffini's proof. Strong objections came from Gian Francesco Malfatti and Gregorio Fontana. Between 1802 and 1806, Ruffini improved and completed his proof. He answered the objections of Malfatti and Fontana and used some suggestions from Pietro Abbati Marescotti. In his 1813 paper Riflessione intorno alla soluzione delle equazioni algebraiche generali, Ruffini gave a different demonstration of his theorem. It mostly matches what is now called Wantzel's version of the Ruffini–Abel theorem.

Only later in Ruffini's life was his proof recognized as basically correct. Mathematicians such as Cauchy, Abel and Galois built on his work. His ideas about the structure of permutation groups were important for understanding the limits of algebraic solutions.

Publications

Ruffini wrote many works between 1799 and 1821. His first, in 1799, was a General Theory of Equations, which argued that general equations of higher degree than four cannot be solved by algebra. Other works covered the squaring of the circle, the roots of numerical equations, and the immortality of the soul. He also wrote an Elementary Algebra in 1807.

His later works included a 1820 essay on contagious typhoid and a 1821 critical response to a probability essay by Count Laplace.