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History

Cardan also observed the difficulty in the irre ducible 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 re mained for Raphael Bombelli of Bologna, who published in 1572 an algebra of great merit, to point out the reality of the appar ently imaginary expression which the root assumes, and thus to lay the foundation of a more intimate knowledge of imagi nary 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 alge braic solutions to equations of higher than the fourth degree were purely Utopian.