In 1770, Lagrange presented a lengthy paper, which would be read during 1771: Réflexions sur la résolution algébrique des équations (Reflections on the Algebraic Solution of Equations). Here is what he wrote in the introduction:
"As for solving algebraic equations, we have made little progress since Cardano's time; he was the first to publish solutions for cubic and quartic equations. […] In this paper, I propose to examine the various methods devised thus far for the algebraic solution of equations, to reduce them to general principles, and to show a priori why these methods succeed for the third and fourth degrees but fail for higher degrees. This examination will have two benefits: on the one hand, it will shed greater light on the known solutions for the third and fourth degrees; on the other, it will help those wishing to tackle higher degrees by sparing them many fruitless steps and attempts."
The aim, then, is to understand the general principles governing the solution of quadratic, cubic and quartic equations, and to see whether they can be applied to equations of arbitrary degree. Lagrange's idea is to find simple polynomial functions of the roots that satisfy an equation of lower degree than the original equation and whose coefficients are invariant under permutations of the roots. This would become known as Lagrange's method of solution.
Quartic equations
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Consider a quartic equation, following Lagrange but simplifying matters slightly. Let ax4 + bx3 + cx2 + dx + e = 0 be the polynomial equation. First, it is easy to reduce it to y4 + py2 + qy + r = 0 by making the substitution x = y − b / (4a). This new equation has no cubic term.
