- THEORY OF EQUATIONS 439
Tschirnhaus claimed that the the middle terms (the á& above) would be elimi-
nated by a polynomial of the sort just discussed, provided that the 6jt are suitably
chosen. Such a change of variable is now called a Tschirnhaus transformation. If
a Tschirnhaus transformation could be found for the general equation of degree n,
and a formula existed for solving the general equation of degree ç — 1, the two could
be combined to generate a formula for solving the general equation of degree n. At
the time, there was not even a Tschirnhaus transformation for the cubic equation.
Tschirnhaus was to provide one.
He illustrated his transformation using the example x^3 - qx - r = 0. Taking
y = x^2 — ax — 6, he noted that y satisfied the equation
y^3 + (36 - 2q)y^2 + (3b^2 + 3ar - 4qb + q^2 - a^2 q)y
+ (b^2 - 2q6^2 + 36ar + q^2 b - aqr - a^2 qb + a^3 r - r^2 ) = 0.He eliminated the square term by choosing 6 = 2q/3, then removed the linear term
by solving for a in the quadratic equation
qa^2 - 3ra + 4q^2 /3 = 0.In this way, he had found at the very least a second solution of the general cubic
equation, independent of the solution given by Cardano. And, what is more im-
portant, he had indicated a plausible way by which any equation might be solved.
If it worked, it would prove that every polynomial equation could be solved using
rational operations and root extractions, thereby proving at the same time that the
complex numbers are algebraically closed. Unfortunately, detailed examination of
the problem revealed difficulties that Tschirnhaus had apparently not noticed at
the time of his letter to Leibniz.
The main difficulty is that when the variable ÷ is eliminated between two
polynomial equations pn(x) = 0 and y = pn-i(x), where pn is of degree ç and p„_i
of degree ç — 1, the degrees of the equations needed to eliminate the successive
coefficients in the equation for y increase to (ç - 1)!, not ç - l.^2 It is only in the
case of a cubic, where (n — 1)! = ç - 1, that the program can be made to work in
general. It may, however, work for a particular equation of higher degree. Leibniz,
at any rate, was not convinced. He wrote to Tschirnhaus,
I do not believe that [your method] will be successful for equations
of higher degree, except in special cases. I believe that I have a
proof for this. [Kracht and Kreyszig, 1990, p. 27]Tschirnhaus' method had intuitive plausibility: If there existed an algorithm
for solving all equations, that algorithm should be a procedure like the Tschirnhaus
transformation. Because the method does not work, the thought suggests itself that
there may be equations that cannot be solved algebraically. The work of Tschirn-
haus and Girard had produced two important insights into the general problem of
polynomial equations: (1) the coefficients are symmetric functions of the roots; (2)
solving the equation should be a matter of finding a sequence of operations that
would eliminate coefficients until a pure equation yn = C was obtained. Since the
problem was still unresolved, still more new insights were needed.
(^2) Seki Kowa knew the rational procedures (what he called folding, as discussed in Section 3 of
Chapter 14) for eliminating x. It does seem a pity that the contemporaries Tschirnhaus and Seki
Kowa lived so far apart. They would have had much to talk about if they could have met.