cubic equation . Extracting the square root of the bi-quadratic, he got . Solving the cubic for y and substituting, it remained only to determine x from the resulting quadratic. Ferrari pursued a similar method with other numerical bi-quadratic equations.[7] Cardan had the pleasure of publishing this discovery in his Ars Magna in 1545. Ferrari's solution is sometimes ascribed to Bombelli, but he is no more the discoverer of it than Cardan is of the solution called by his name.
To Cardan algebra is much indebted. In his Ars Magna he takes notice of negative roots of an equation, calling them fictitious, while the positive roots are called real. Imaginary roots he does not consider; cases where they appear he calls impossible. Cardan also observed the difficulty in the irreducible case in the cubics, which, like the quadrature of the circle, has since "so much tormented the perverse ingenuity of mathematicians." But he did not understand its nature. It remained for Raphael Bombelli of Bologna, who published in 1572 an algebra of great merit, to point out the reality of the apparently imaginary expression which the root assumes, and thus to lay the foundation of a more intimate knowledge of imaginary quantities.
After this brilliant success in solving equations of the third and fourth degrees, there was probably no one who doubted, that with aid of irrationals of higher degrees, the solution of equations of any degree whatever could be found. But all attempts at the algebraic solution of the quintic were fruitless, and, finally, Abel demonstrated that all hopes of finding algebraic solutions to equations of higher than the fourth degree were purely Utopian.
Since no solution by radicals of equations of higher degrees
