Paolo Ruffini
Encyclopedia
Paolo Ruffini was an Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

and philosopher.

By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics. Among his work was an incomplete proof (Abel–Ruffini theorem
Abel–Ruffini theorem
In algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...

) that quintic (and higher-order) equations
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

(1799), and Ruffini's rule
Ruffini's rule
In mathematics, Ruffini's rule allows the rapid division of any polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in 1809. Ruffini's rule is a special case of synthetic division when the divisor is a linear factor. The Horner scheme is a fast algorithm for dividing...

which is a quick method for polynomial division. Ruffini also made contributions to group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

He practiced as both a professor of mathematics (University of Modena) and a doctor including scientific work on typhus
Typhus
Epidemic typhus is a form of typhus so named because the disease often causes epidemics following wars and natural disasters...

.

## Group Theory

Ruffini’s 1799 work marked a major development for group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

. Ruffini developed Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

's work on permutation theory, following 29 years after Lagrange’s "Réflexions sur la théorie algébrique des equations" (1770-1771) which was largely ignored until Ruffini who established strong connections between permutations and the solvability of algebraic equations. Ruffini was the first to controversially assert the unsolvability by radicals of algebraic equations higher than quartic
Quartic
In mathematics, the term quartic describes something that pertains to the "fourth order", such as the function x^4. It may refer to one of the following:* Quartic function, a polynomial function of degree 4* Quartic curve, an algebraic curve of degree 4...

s. This angered many members of the community such as Malfatti
Gian Francesco Malfatti
Giovanni Francesco Giuseppe, also known as Gian Francesco or Gianfrancesco was an Italian mathematician. He was born in Ala, Trentino, Italy and died in Ferrara....

(1731-1807). Work in this area was later carried on by those such as Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...

and Galois who succeeded in such a proof.

## Publications

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 circulo"

1802: "Della soluzione delle equazioni algebraiche determinata partocolari di grado sup. al 4º"

1804: "Sopra la determinazione delle radici nelle equazioni numeriche di qualunque grado"

1806: "Della immortalità dell’anima"

1807: "Algebra elementare"

1820: "Memoria sul tifo contagioso"

1821: "Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del signor conte Laplace"