Here you will find the explanation of who Paolo Ruffini was: his biography, his main mathematical contributions, all his works, what he invented, anecdotes, etc.

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## Who was Paolo Ruffini?

**Paolo Ruffini was a famous mathematician, philosopher and physician who was born on September 22, 1765 in Valentano, Italy, and died on May 10, 1822 in Modena, now Italy.**

The figure of Paolo Ruffini stands out for his relevant contributions to science, especially in the field of mathematics. In fact, a very important mathematical rule was named after Paolo Ruffini: the Ruffini’s rule (or synthetic division of polynomials).

This is a biographical summary of Paolo Ruffini, but he evidently made many more notable contributions to the mathematical field. Next, we will explain in detail the biography of Paolo Ruffini and what these contributions were.

## Biography of Paolo Ruffini

Paolo Ruffini was the son of Maria Francesca Ippoliti and Basilio Ruffini, a doctor by profession. And, as we have explained, the birth of Paolo Ruffini took place in the town of Valentano, then belonging to the Papal States. However, his family had to move to Reggio, Duchy of Modena, in northern Italy, and it was there that Paolo spent most of his life.

Despite the fact that as a child he seemed destined for a religious career, in 1783 Paolo entered the University of Modena to study **mathematics, medicine, philosophy and literature**. And surprisingly, he first graduated in philosophy, medicine and surgery, specifically on June 9, 1788. A little later he graduated in mathematics.

When he was still a student at the university, Paolo Ruffini already began to teach Fundamentals of Analysis during the 1787-88 course, since his previous instructor, Paolo Cassiano, was elected councilor and, therefore, had to leave University.

Later, **in 1791, Paolo Ruffini managed to become a professor in Elements of Mathematics**, thus replacing Fantini, who had been his professor in geometry. However, that same year he showed that he was not just a mathematician, because also in 1791 he obtained permission to practice medicine in Modena and the license to teach clinical medical classes at the same university where he studied.

To understand the context of the time in which Paolo Ruffini lived, it was a time of war as France was expanding across the European continent after the famous French Revolution. In 1796, Napoleon Bonaparte, general of the French troops, occupied the territory of Modena and this situation directly affected the mathematician Ruffini.

Napoleon then founded the Cisalpine Republic, made up of the regions of Lombardy, Emilia, Modena and Bologna, and proposed to Ruffini to occupy a position on its council. But Paolo rejected the proposal because to do so he had to take an oath of allegiance, which was contrary to his political and religious principles. **Because of this, Ruffini is fired from his position at the university and is prohibited from teaching**, but as a calm man that he was, he took this new paradigm in a positive way, taking advantage of this time as an opportunity to dedicate himself to the practice of medicine and his research on the resolution of the fifth degree equation by radicals, a type of algebraic equation that is very difficult to solve.

Because equations of second degree had been solved since the time of the Babylonians, the equation of third degree (or cubic equation) had been deciphered by Ferro and Tartaglia and the quartic equation had been solved by Ferrari in 1540, but 250 years had passed without anyone being able to find the solution to the quintic (fifth degree equation). Despite the fact that illustrious mathematicians such as Tschirnhaus, Euler, Bézout, Vandermonde, Waring and Lagrange tried it throughout their mathematical careers.

But everything seemed to indicate that the resolution of the quintic equation could be done in some way through radicals, even the Italian scientist Lagrange defended this theory in one of his works. However, **in the year 1799 Paolo Ruffini was readmitted to the University of Modena and then he published his book called Theory of equations** in which he proved the opposite, that is, there is no formula for equations of degree 5 or higher. Although he was correct, in his publication he made some errors that were later corrected (1824) by the Norwegian mathematician Niels Henrik Abel, thus calling it the Abel-Ruffini theorem.

Still, it seems that Ruffini was ahead of his time as the mathematical world ignored his findings. That is why in 1802 he wrote a second proof: *Riflessioni intorno alla rettificazione ed alla quadratura del circolo* and the memory *Della soluzione delle equazioni algebraiche determinata partocolari di Grado sup. al 4º*. And in 1804 he edited the memory called *Sopra la determinzione delle radici nelle equazioni numeriche di qualunque grado*, in which Ruffini describes a method with which the roots of an equation could be approximated, this procedure was later known as Horner’s method because he popularized it.

In 1806, he accepted a chair of Applied Mathematics at the military school in Modena. And that same year he also dedicated a work, *Dell’ immortalità dell’ anima* to Pius VII, at that time pope of the Catholic Church. Facts like this prove his strong religious belief.

The following year, in 1807, *Algebra elementare* (*Algebra e suo apendice)*, another of his famous works, was printed.

Around the year 1809, he discovered Ruffini’s rule, which is the most important contribution that Paolo Ruffini made to mathematics, without a doubt.

A few years later, in 1813, his *Riflessioni intorno alla soluzione delle equazioni algebraiche generali* were published. Although the mathematical community still did not recognize the prestige of Paolo Ruffini, at a later stage the qualified French mathematician Augustin Louis Cauchy did admit the importance of Ruffini writing works like this.

Then, **in 1814, Paolo Ruffini was appointed rector of the University of Modena**, where he not only held the chair of mathematics but also that of medicine. This has to make us reflect on how genius Ruffini was, since he was able to master two totally different disciplines and, in addition, he reached excellence in both.

In 1816 he was appointed president of the Italian society “Dei Quaranta”, of which he had been a member since the year of the turn of the century (1800). He also managed to be the president of the Italian Institute of Sciences.

Although Paolo Ruffini’s mathematical career is full of successes, in 1817 the bad news began. During that year he fell ill with typhus, an epidemic with a high mortality at that time. And, although he was able to partially recover, he had to leave the university chair in 1819. In 1820 he published an article (*Memoria sul tifo contagioso*) based on his experience with this disease.

Even before his death, Paolo Ruffini wrote about probability in his publication *Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del Sig. Conte de la Place* (1821).

Finally, Paolo Ruffini died on May 9, 1822, in the Italian town of Modena, where he spent most of his time, trained mainly as a mathematician, doctor and philosopher, and became a genius who will be remembered forever in the history of mathematics.

## Paolo Ruffini’s main contributions to mathematics

The most notable mathematical contributions of Paolo Ruffini are the following:

- His most notable contribution is Ruffini’s rule, a very important mathematical rule that can be used to do many operations: quickly divide a polynomial by a binomial of the form
*(x-r)*, find the roots of a polynomial, factor a polynomial,… - Another very important contribution was the demonstration that polynomial equations of degree greater than 4 are unsolvable by radicals. This may seem very obvious now, but it was a problem that had remained open since the 16th century.
- He found a method for roughly calculating the roots of an equation.
- As we have seen in his biographical explanation, Paolo Ruffini participated in the elaboration of the Abel-Ruffini theorem to a considerable extent.
- Likewise, it established the bases of the theory of the transformations of equations.

## Works of Paolo Ruffini

Although all his works have already been commented above, below you can see biographically all the publications of Paolo Ruffini:

- 1799:
*Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto*. - 1802:
*Riflessioni intorno alla rettificazione ed alla quadratura del circolo*y la memoria*Della soluzione delle equazioni algebraiche determinate partocolari di grado superiore al quarto*. - 1804:
*Sopra la determinazione delle radici nelle equazioni numeriche di qualunque grado*. - 1806:
*Dell’ inmortalità dell’ anima*. - 1807:
*Algebra elementare*(*Algebra e suo apendice)*. - 1813:
*Riflessioni intorno alla soluzione delle equazioni algebraiche generali*. - 1820:
*Memoria sul tifo contagioso*. - 1821:
*Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del Sig. Conte de la Place*.