**Paolo Ruffini** was an Italian mathematician who gave a proof that the quintic equation could not be solved earlier than Abel.

- Among his teachers of mathematics at Modena were Luigi Fantini, who taught Ruffini geometry, and Paolo Cassiani, who taught him calculus.
- Cassiani's course at Modena on the foundations of analysis was taken over by Ruffini in 1787-88 although he was still a student at this time.
- On 9 June 1788 Ruffini graduated with a degree in philosophy, medicine and surgery.
- Ruffini must have made a good job of the foundations of analysis course he took over from Cassiani for, on 15 October 1788, he was appointed professor of the foundations of analysis.
- Fantini, who had taught Ruffini geometry when he was an undergraduate, found his eyesight deteriorating and in 1791 he had to resign his post at Modena.
- Ruffini was appointed to fill the position of Professor of the Elements of Mathematics in 1791.
- However, Ruffini was not only a mathematician.
- Although not wishing to get involved, Ruffini found himself appointed as a representative to the Junior Council of the Cisalpine Republic.
- He was required to swear an oath of allegiance to the republic and this Ruffini found he could not bring himself to do on religious grounds.
- Ruffini did not seem greatly disturbed by the loss of his chair, in fact he was a very calm man who took all the dramatic events around him in his stride.
- It appears that nobody before Ruffini really believed that the quintic could not be solved by radicals.
- In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
- Ruffini used group theory in his work but he had to invent the subject for himself.
- Ruffini is the first to introduce the notion of the order of an element, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive.
- It is remarkable work and, except for one gap, proves the theorem as Ruffini claimed.
- However there was a strange lack of response to Ruffini's work from mathematicians.
- In 1801 Ruffini sent a copy of his book to Lagrange.
- This patriotic reaction apart, the world of mathematics seemed to almost ignore Ruffini's great result.
- So how did Ruffini react?
- At least Ruffini received comments from Malfatti concerning this paper, but unfortunately Malfatti had not understood Ruffini's arguments and raised a fallacious objection.
- Ruffini published further proofs in 1808 and 1813.
- It is no surprise that it should resemble Ruffini's proof, since Wantzel says in his paper ..."using works of Abel and Ruffini...".
- Ruffini did not stop trying to have his work recognised by the mathematical community.
- One has to feel desperately sorry for Ruffini.
- If some mathematician had written to him showing him there was an error or even a gap in the proof, then at least Ruffini would have had the chance to correct it.
- Ruffini asked the Institute in Paris to pronounce on the correctness of his proof and Lagrange, Legendre and Lacroix were appointed to examine it.
- The Royal Society were also asked to pronounce on the correctness and Ruffini received a somewhat kinder reply which said that although they did not give approval of particular pieces of work they were quite sure that it proved what was claimed.
- Cauchy had written a major work on permutation groups between 1813 and 1815 and in it he generalised some of Ruffini's results.
- He had certainly been greatly influenced by Ruffini's ideas.
- This influence through Cauchy is perhaps the only way in which Ruffini's work was to make an impact on the development of mathematics.
- We left the story of Ruffini's career around 1799 when he began his publications on the quintic.
- After the fall of Napoleon, Ruffini became rector of the University of Modena in 1814.
- As well as the rectorship, Ruffini held a chair of applied mathematics, a chair of practical medicine and a chair of clinical medicine in the University of Modena.
- In 1817 there was a typhus epidemic and Ruffini continued to treat his patients until he caught the disease himself.
- There are further aspects of Ruffini's work which should be mentioned.
- Given the information in this article about the insolubility of the quintic, it is reasonable to ask why Abel has been credited with proving the theorem while Ruffini has not.
- Then, too, the method of permutations was too exotic and, it must be conceeded, Ruffini's early account is not easy to follow.

Born 22 September 1765, Valentano, Papal States (now Italy). Died 10 May 1822, Modena, Duchy of Modena (now Italy).

View full biography at MacTutor

Algebra, Ancient Arab, Ancient Chinese, Chinese, Group Theory, Origin Italy

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive