RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

5.5. RUFFINI’s proofs of the insolubility of the quintic 85
Figure 5.4: PAOLO RUFFINI (1765–1822)
on numerous occasions, and his final proof published 1813. 74
5.5.2 RUFFINI’s first proof
The writings of RUFFINI were deeply inspired by LAGRANGE’S analysis of the solu-
bility of equations described in section 5.2. 75 LAGRANGE’S ideas, concepts, and nota-
tion permeate RUFFINI’S works; and on numerous occasions RUFFINI openly acknowl-
edged his debt to LAGRANGE. 76 As LAGRANGE had done, RUFFINI studied equations
of low degrees in order to establish patterns subjectable of generalization. Prior to
applying his analysis to the fifth degree equation, RUFFINI propounded the corner
stone of his investigation. Central to his line of argument was his classification of per-
mutations. Founded in LAGRANGE’S studies of the behavior of functions when their
arguments were permuted, RUFFINI set out to classify all such permutations of argu-
ments which left the function (formally) unaltered. RUFFINI’S concept of permutation
(Italian: “permutazione”) differed from the modern one, and can most easily be un-
derstood if translated into the modern concept introduced by CAUCHY in the 1840s of
74 (Ruffini, 1813). The presentation of RUFFINI’S proofs will largely rely on secondary sources, primarily
(Burkhardt, 1892), (Wussing, 1969, 56–59), and (Kiernan, 1971, 56–60). In (R. G. Ayoub, 1980), his
proofs are interpreted using concepts from GALOIS theory.
75 (Lagrange, 1770–1771).
76 See for instance his preliminary discourse in (Ruffini, 1799, 3–4).