Paolo Ruffini | Biography

Paolo Ruffini made significant contributions in the areas of medicine and philosophy, as well as mathematics, where he developed the theory that a quintic equation cannot be solved by radicals. This theory later came to be known as the Abel-Ruffini theorem, named after Ruffini and the Norwegian mathematician Niels Abel, who published the first accepted version of this theorem. Ruffini also played a crucial role in the development of group theory.

Ruffini was born September 22, 1765, in Valentano, Papal States (also known as Valentano, Viterbo and now part of Italy), to Basilio Ruffini, a physician, and his wife, Maria Francesca Ippoliti. He moved with his parents to Modena, Duchy of Modena (now also part of Italy) when he was a teenager and attended the University of Modena, where he studied medicine, philosophy, literature, and mathematics. He practiced mathematics with many well-known instructors and mathematicians, including Luigi Fantini, who taught geometry, and Paolo Cassiani, who taught infinitesimal calculus. The well-rounded Ruffini excelled in all his classes, so much so that when Cassiani took a leave from his foundations of analysis course in the fall of 1787 to accept a position as councillor of the Este domains, Ruffini, at that time still a student, instructed the class himself.

Ruffini graduated from the university with a degree in philosophy and medicine the following summer, and soon after that obtained a degree in mathematics as well. That fall he was appointed professor of the foundations of analysis. He remained in that position until 1791, when he replaced Fantini as professor of the elements of mathematics. That same year Ruffini obtained his license to practice medicine by the Collegiate Medical Court of Modena. In addition to work in the academic field, he also began to practice medicine.

Ruffini's professional situation changed drastically in 1796, however, when Napoleon's troops occupied Modena and he was appointed representative from the department of Panaro to the Junior Council of the Cisalpine Republic, against his wishes. Ruffini was relieved of his post in 1798 and returned to scientific research, but was banned from teaching or holding a public office after he refused to swear an oath of allegiance to the republic, citing religious reasons. Ruffini continued to practice medicine and conduct mathematical research, and during this time published what later became known as the Abel-Ruffini theorem, which stated that a general algebraic equation of greater than the fourth degree could not be solved using only radicals, such as square roots, cube roots, etc.

The theorem was first published in Teoria generale delle equazioni in 1799, and later in revised form in Riflessioni intorno alla soluzione delle equazioni algebriche generalie in 1813. Upon publication, the theorem was met with skepticism among other mathematicians, with the exception of Augustin-Louis Cauchy, who told Ruffini in an 1821 letter that his work deserved the attention of the mathematics community and that he wholeheartedly agreed with Ruffini's findings. The theorem gained general acceptance when it was proven by Abel in 1824.

Ruffini also developed the theory of substitutions, considered a significant contribution to the theory of equations. This theory laid the foundation for Èvariste Galois' general theory of the solubility of algebraic equations and served as an important forerunner of group theory.

Ruffini later developed the basic rule for determining the quotient and the remainder resulting from the division of a polynomial in the variable x by a binomial of the form x-a, using approximation by means of infinite algorithms. Despite a growing trend toward acknowledging the uncertainty of the foundations of infinitesimal analysis, Ruffini, with the support of Cauchy and Abel, demanded rigor of himself and others. During his tenure as president of the Societa Italiana dei Quaranta, he refused to accept to papers by Giuliano Frullani because they relied on series for which the convergence had not been demonstrated.

Ruffini moved on to apply his manner of thinking to philosophical and biological matters, determining that the faculties of the soul could not be measured because they do not correspond to magnitudes. Ruffini returned to academia with the fall of Napoleon in Modena in 1814. At this time, Ruffini was appointed rector of the university, as well as chair of the mathematics and practical medicine departments.

Ruffini continued as a practicing physician as well, treating impoverished and royal citizens of Modena alike. He contracted typhus while aiding victims of the 1817-18 typhus epidemic, but recovered enough to write about the experience from a medical and personal perspective. In 1822, however, he contracted chronic pericarditis, accompanied by a serious fever, and he died on May 10 of that year.