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

5.1. Algebraic solubility before LAGRANGE 59
Elementary symmetric relations. A different example of a priori properties of the
roots of an equation was conceived of by men as G. CARDANO (1501–1576), F. VIÈTE
(1540–1603), A. GIRARD (1595–1632), and I. NEWTON (1642–1727) in the sixteenth and
seventeenth centuries. From inspection of equations of low degree they obtained (gen-
erally by analogy and without general proofs) by a tacit theorem on the factorization
of polynomials the dependency of the coefficients of the equation
on the roots x1, . . . , xn given by
x n + an−1x n−1 + an−2x n−2 + · · · + a1x + a0 = 0
an−1 = − (x1 + · · · + xn)
an−2 = x1x2 + · · · + xn−1xn
.
a1 = ± (x1x2 . . . xn−1 + · · · + x2x3 . . . xn)
a0 = ∓x1x2 . . . xn.
(5.1)
These equations established the elementary symmetric relations between the roots and
the coefficients of an equation. When proofs of these relations first emerged, they were
obtained through formal manipulations of the tacitly introduced factors and were,
thus, firmly within the established algebraic style.
The relations (5.1) were to become a central tool in the theory of equations once
NEWTON and E. WARING (∼1736–1798) realized that they were the basic, or ele-
mentary, ones upon which all other symmetric functions of the roots depended rationally.
10
5.1 Algebraic solubility before LAGRANGE
Among the multitude of possible questions concerning the unknown roots, one is
particularly linked to the question of solving equations algebraically. It arose when
mathematicians began investigating the form in which the roots can be written and is
thus a first step in the direction of asking general solubility questions. 11
The general approach taken in solving equations of degrees 2, 3 or 4 had since the
first attempts been to reduce their solution to the solution of equations of lower degree.
The example of the third degree equation solved by S. FERRO (1465–1526) around
1515, by N. TARTAGLIA (1499/1500–1557) in 1539, and by CARDANO, who published
the solution in 1545, might be illustrative 12 . Although CARDANO’S arguments and
style were geometric, its algebraic content is presented in algebraic notation in box 1.
10 See section 5.2.4.
11 This aspect shall be dealt with below (see page 62ff) and section 8.4.
12 In the present form, revised to expose central concepts, CARDANO’S solution closely resembles the
young school-boy’s notes found in the section Ligninger af tredje Grads Opløsning (af Cardan) in
ABEL’S notebook (Abel, MS:829, 139–141).