Early Attempts at FTA

A glance into early attempts even to formulate the Fundamental Theorem of Algebra correctly, demonstrates how much development of mathematics has been impeded by the absence of a proper language to describe mathematical concepts.

The first formulation of the theorem was due to Albert Girard (1595-1632), a native of Lorraine who worked in Holland and was the editor of the works of Simon Stevin. His L'invention nouvelle en l'algèbre (Amsterdam, 1629) bases its fame on its formulation of the fundamental theorem of algebra, which also shows that he already took complex number seriously. It must be noted that Struik's is the only book that mentions Girard. The rest in my library that mention the Fundamental Theorem of Algebra, start with d'Alembert. The theorem has been for a long time known as the d'Alembert's Theorem. Here how Girard put it:

Theorem II. All equations of algebra receive as many solutions as the denomination of the highest term shows, except the incomplete, and the first faction of the solutions is equal to the number of the first mixed, their second faction is equal to the number of the second mixed; their third to the third mixed, and so on, so that the last faction is equal to the closure, and this according to the signs that can be observed in the alternate order.

("Incomplete polynomials" are the ones in which some coefficients vanish. "Mixed" are, in my belief, the monomials akxk.) He later wrote,

As to the incomplete equations, they have not always so many solutions, nevertheless we can well explain the solutions whose existence is impossible, and show wherein lies the impossibility because of the defectiveness and incompleteness of the equation.

René Descartes (1596-1650), the father of Analytic Geometry, who made an attempt to formalize the whole of science by reducing it to the simplest and most obvious maxim "Cogito, ergo sum", still had troubles accepting negative numbers. In the following piece from his Géométrie the "false roots" he struggles with were unfortunate enough to need a minus sign in front of them.

Know then that in every equation there are as many distinct roots, that is, values of the unknown quantity, as is the number of dimensions of the unknown quantity.

Suppose, for example, x equal to 2, or x - 2 equal to nothing, and again, x equal to 3, or x - 3 equal to nothing. Multiplying together the two equations we have xx - 5x + 6 equal to nothing, or xx equal to 5x - 6. This is an equation in which x has the value 2 and at the same time has the value 3...

It often happens, however, that some roots are false, or less than nothing. Thus, if we suppose x to represent the defect of a quantity 5, we have x + 5 equal to nothing which, multiplied by x³ - 9x² + 26x - 24, gives x4 - 4x3 - 19x2 + 106x - 120 equal to nothing, as an equation having four roots, namely three true roots, 2, 3, 4, and one false root, 5.

The French mathematician Jean d'Alembert (1717-1783) tried but failed to prove the theorem in 1748. In the words of W.Dunham, d'Alembert recognized the importance of such a statement and made a stab at a proof. His stab, unfortunately, was wide of the mark. Perhaps to accord him the honor of trying, the result was long known as "d'Alembert's theorem", in spite of the fact that he came nowhere near proving it. This seems somewhat akin to renaming Moscow after Napoleon simply because he tried to reach it.

Euler made an attempt to prove the theorem in 1749 in a paper Recherches sur les racines imaginaires des équations. Gauss who opened his 1799 dissertation with a critical review of previous attempts, observed that Euler's proof lacks "the clarity which is required in mathematics." Euler proved a series of theorems for various polynomial degrees. First for n = 4, then n = 8, with special corollaries filling in the gaps. Then he stated the theorem for n = 2k. The proof leads to an equation of degree N = 32870 which he calls "oddly even" (impairement pair) meaning that N/2 is odd.

Euler believed that the proof is solid ("je crois qu'on n'y trouvera rien à redire"), but to strengthen the argument he gives extra proofs for degrees 6, 4k + 2, 8k + 4,..., 2kp, p an odd number. If Euler mixed a matter of belief with a matter of having proven something rigorously - who may not?

References

1. W.Dunham, Journey through Genius, Penguin Books, 1991
2. D.J.Struik, A Source Book in Mathematics, 1200-1800, Princeton University Press, Third Printing, 1990  