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

5.2. LAGRANGE’s theory of equations 71
Figure 5.2: EDWARD WARING (1734–1798)
theorem stated that the coefficients, here A0, . . . , A ¯ω, were rational functions of the co-
efficients of the given equation. By Waring’s formulae, any such rational function of the
coefficients was a symmetric function of the roots.
5.2.4 Waring’s formulae
The elementary symmetric functions of the roots of an equation, which since the times
of VIÈTE and NEWTON had been known to agree with the coefficients (see section 5),
was seen by the little known British mathematician WARING to provide a basis for the
study of all symmetric functions of the equation’s roots. In his Miscellanea analytica of
1762, WARING demonstrated that all rational symmetric functions of the roots could
be expressed rationally in the elementary symmetric functions. 39 In his other more
influential work Meditationes algebraicae, 40 to which LAGRANGE referred, 41 the result
was contained in the first chapter. There, WARING dealt with the determination of the
power sums of the roots x1, . . . , xµ (modern notation)
39 (Waerden, 1985, 76–77).
40 (Waring, 1770).
41 (Lagrange, 1770–1771, 369–370).
µ
∑ x
k=1
m k
for integer m