192 Ordinary Differential Equationshis students, Ludovico Ferrari, who claimed Cardan received the solution
from Ferro through a third party and that Tartaglia, himself, was guilty of
plagiarizing Ferro's solution.
In 1540 , da Coi proposed a problem to Cardan which required the solution
of the quartic equation x^4 + 6x^2 + 36 = 60x. Cardan was unable to so lve this
problem but passed it on to Ferrari. Ferrari solved the problem and in the
process showed how to reduce the solution of all quartic equations of the form
x^4 + px^2 + qx + r = 0 to the solution of a cubic equation. In effect, Ferrari
solved the general quartic polynomial x^4 + ax^3 + bx^2 + ex + d = 0, since it
reduces to x^4 + px^2 + qx + r = 0 by means of a simple linear transformation.
Cardan included Ferrari's solution of the quartic equation in his Ars Magna.
Thus, the Ars Magna contains the first published solution of both the general
cubic equation and the general quartic equation although neither was the work
of the author!
In 1803, in 1805, and again in 1813 the Italian physician Paola Ruffini
published inconclusive proofs that the roots of the fifth and higher order
polynomials cannot be written in terms of the coefficients of the polynomial
by means of radicals. This fact was later successfully proven by the Norwegian
mathematician Niels Henrik Abel in 1824.
Sixteenth century Italian mathematicians assumed every polynomial with
rational coefficients had a root. Late in the century, they were aware that a
quadratic polynomial has two roots, a cubic polynomial h as three roots, and
a quartic polynomial has four roots. Peter Roth seems to be the first writer to
explicitly state the fundamental theorem of algebra in his Arithmetica philo-
sophica published in 1608. The fundamental theorem of algebra states that
an n-th degree polynomial has n roots. D 'Alembert attempted to prove this
theorem in 1746. Euler (1749) and Lagrange also attempted to prove the
theorem. The first rigorous proof is due to Gauss in 1799. A simpler proof
was given in 1849.
The second search for roots of polynomials, and in retrospect the more
fruitful search, consisted of developing techniques for approximating the values
of the roots. As we mentioned earlier , the ancient Babylonians knew in about
2000 B.C. how to solve a quadratic equation algebraically using the quadratic
formula. On a practical level since the quadratic formula involves extracting
a square root, the Babylonians found it necessary to devise a method for
computing a square root. The following scheme for computing the squ are
root of a positive real number is due to the Babylonians.
Let a be a positive real number and let x 1 > 0 be a first a pproximation
(guess) of the value of fa. Either
(1) X1 =Va or (2) X1 < Va or (3) X1 >Va
If (1) X1 =fa, we are done.
If (2) X1 < fa, then fax1 <a and fa< a/x1.