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

82 Chapter 5. Towards unsolvable equations
upon combinations, i.e. permutations — inside which the solution could be investi-
gated. However, for equations of the fifth and higher degrees the required number of
calculations and combinations would be exceeding practical possibilities.
“These are, if I am not mistaken, the true principles of the solution of equations,
and the most appropriate analysis leading to it. As one can see, it all comes
down to a sort of calculus of combinations, by which one finds à priori the results
for which one should be prepared. It should, by the way, be applicable to
equations of the fifth degree and higher degrees, of which the solution is until
now unknown. But this application demands a too great number of researches
and combinations, of which the success is still in serious doubt, for us to follow
this path in the present work. We hope, though, to be able to follow it at another
time, and we content ourselves by having laid the foundations of a theory which
appears to us to be new and general.” 69
LAGRANGE never wrote the definitive work which he had reserved the right to
do. By the time GALOIS had substantiated LAGRANGE’S claim for generality and ap-
plicability of his theory of combinations (see chapter 8.5), LAGRANGE was no longer
around to celebrate the ultimate vindication of his research in the field of algebraic
solubility.
Both WARING and LAGRANGE believed by 1770 that their theories were the nec-
essary stepping stones towards the study of solutions to general equations. However,
they both acknowledged that the amount of work required to apply these theories to
the quintic equation was beyond their own limitations. Before the end of the century,
even more radical opinions were voiced in print.
5.4.2 Outright impossibility
In the introduction to his first proof (published 1799 but constructed two years earlier)
of the Fundamental Theorem of Algebra, GAUSS gave detailed discussions and criticisms
of previously attempted proofs. In EULER’S attempt dating back to 1749, GAUSS found
the implicit assumption that any polynomial equation could be solved by radicals.
“In a few words: It is without sufficient reason assumed that the solution of
any equation can be reduced to the resolution of pure equations. Perhaps it would
not be too difficult to prove the impossibility for the fifth degree with all rigor; I
will communicate my investigations on this subject on another occasion. At this
place, it suffices to emphasize that the general solution of equations, in this sense,
69 “Voilà, si je me ne trompe, les vrais principes de la résolution des équations et l’analyse la plus
propre à y conduire; tout se réduit, comme on voit, à une espèce de calcul des combinaisons, par
lequel on trouve à priori les résultats auxquels on doit s’attendre. Il serait à propos d’en faire
l’application aux équations du cinquième degré et des degrés supérieurs, dont la résolution est
jusqu’à présent inconnue; mais cette application demande un trop grand nombre de recherches et
de combinaisons, dont le succés est encore d’ailleurs fort douteux, pour que nous puissions quant
à présent nous livrer à ce travail; nous espérons cependant pouvoir y revenir dans un autre temps,
et nous nous contenterons ici d’avoir posé les fondements d’une théorie qui nous paraît nouvelle et
générale.” (Lagrange, 1770–1771, 403).