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

5.6. CAUCHY’ theory of permutations and a new proof of RUFFINI’s theorem 91
Figure 5.5: AUGUSTIN-LOUIS CAUCHY (1789–1857)
that no function of five quantities could have three or four different values when its ar-
guments were permuted (see above). Before going into this particular result, however,
CAUCHY devised the terminology and notation which he was going to use. Precisely
in formulating exact and useful notation and terminology, CAUCHY advanced well
beyond his predecessors and laid the foundations upon which the nineteenth-century
theory of permutations would later build.
Notational advances. With CAUCHY, the term “permutation” came to mean an ar-
rangement of indices, thereby replacing the “arrangements” of which RUFFINI spoke.
A “substitution” was subsequently defined to be a transition from one permutation
to another (which is the modern meaning of “permutation”), and CAUCHY devised
writing it as, for instance,
� �
1.2.4.3
. (5.12)
2.4.3.1
CAUCHY’S convention was that in the expression K, to which the substitution (5.12)
was to be applied, the index 2 was to replace the index 1, the index 4 to replace 2, 3
should replace 4, and 1 should replace 3. More generally, CAUCHY wrote
� A1
A2
�