RUFFINI, Paolo. Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica dell equazioni generali di grado superiore al quarto.
Bologna: Stamperia di S. Tommaso d' Aquino, 1799. First edition.

The first statement and proof of the Abel-Ruffini theorem, that the general equation of degree higher than four cannot be solved algebraically. Although Ruffini's proof was not in general accpeted he developed, in this lengthy treatise, many new fundamental methods and the concept of permutation groups, which was essential to the later work of Abel and Galois. It thus marks the transition from classical to abstract algebra. "The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini's work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls permutazione and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely permutazione semplice which are cyclic groups in modern notation, and permutazione composta which are non-cyclic groups. The permutazione composta Ruffini divides into three types which in today's notation are intransitive groups, transitive imprimitive groups and transitive primitive groups." (MacTutor History of Mathematics).

Two volumes bound in one. Small 4to: 207 x 147 mm. Contemporary half-calf with gilt lettering on spine. Old inoffensive repair to the first and second pages of the introduction. A very good copy. viii, 206, (4:errata); (2:title), 207-509, (7:errata) and two large folding tables.