Even as he taught mathematics, he continued to study medicine and in 1791, he received his license to practice medicine.
1796 was the time of Napoleon Bonaparte who took over Modena. Napoleon set up what became the Cisalpine Republic that included Modena. All faculty of the university were required to swear an oath of allegiance to the new republic.
Ruffini refused to swear this oath and so lost his job and was no longer allowed to teach. This change enabled him to focus on medicine and to actively study his own projects in mathematics.
One of his math projects was an effort to solve the quintic equation in terms of radicals. The quadratic equation was long known to be solvable. The cubic equation had been solved by Giralomo Cardano and the quartic equation had been solved by Lodovico Ferrari. No one had yet solved the quintic.
It was at this point, that Paolo Ruffini approached the problem differently than others had before him. He was the first person to propose that the problem was not solvable in terms of radicals. In 1799, Ruffini published his math treatise which he titled: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
In the introduction, he writes:
In writing this work, he made arguments in terms of group theory. Since group theory had not yet been invented, Ruffini had to invent concepts which made his argument even hard to follow. In this work, we find such concepts as order, cycles, and the notion of primitive elements. Joseph-Louis Lagrange had previously written about permutations but Ruffini applied these points to his argument in ways that Lagrange had not. In this work, he successfully proved some very important theorems of group theory.
The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof.
Unfortunately, he did not succeed in proving his claim about the quintic equation. He came very close. So close, in fact, that it was later accepted by Augustin Louis Cauchy as valid. In retrospect, it is clear that his argument had a significant gap which would later be resolved by Niels Abel. Ruffini assumed but failed to demonstrate that:
"...if an expression by radicals is a root of the general equation of some degree, then every function of it is composed is a rational expression of the roots." (Jean-Pierre Tignol, p211).In fairness to Ruffini, he received no response from the mathematical community on this point. He sent a copy to Lagrange in 1801. He got no response. He sent out a second copy of his book with the following message:
Because of the uncertainty that you may have received my book, I send you another copy. If I have erred in any proof, or if I have said something which I believed new, and which is in reality not new, finally if I have written a useless book, I pray you point it out to me sincerely.He still got no response. In 1802, he wrote a follow up letter. Again, no response.
A mathematician named Gian Malfatti did respond with objections that today are viewed as a misunderstanding of Ruffini's ideas. Ruffini responded with additional proofs that are today accepted as answering Malfatti's objections.
Why didn't anyone respond? The answer seems to be that Ruffini was tackling a question in a way that no one wanted to be right. The mathematical community was not yet ready to concede that quintic equations were not solvable by radicals. In addition, Ruffini's proof was over 516 pages and the mathematical argument was difficult to follow.
In 1810, Ruffini asked the Institute of Science in Paris to make a formal statement on the validity of his proof. After over a year, Adrien-Marie Legendre, Lagrange, and Sylvestre Francois de Lacroix concluded that there was nothing of importance here and that it was not "worthy of attention."
Ruffini made the same request of the Royal Society. The response was polite but again reflected disinterest in the topic area.
In 1814, after the fall of Napoleon, Ruffini was reinstated back at the University of Modena. All this time, Ruffini had a reputation for honesty and integrity. As a doctor, he treated patients from the richest to the poorest backgrounds. In 1817, there was a typhus epidemic. Ruffini proceeded to continue to treat patients and caught the disease himself. Although, he made a recovery, he never regained his health and had to resign from his university position in 1819.
The year before his death, he received the following correspondence from Cauchy which would represent the only recognition he received while he was alive from a major mathematician:
... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.Even so, Ruffini does not get credit for proof that the quintic equation is not solvable by radicals. That honor goes to Niels Abel. On this point, R. G. Ayoub writes:
... the mathematical community was not ready to accept so revolutionary an idea: that a polynomial could not be solved in radicals. Then, too, the method of permutations was too exotic and, it must be conceeded, Ruffini's early account is not easy to follow. ... between 1800 and 1820 say, the mood of the mathematical community ... changed from one attempting to solve the quintic to one proving its impossibility...Paolo Ruffini died on May 10, 1822 in Modena.