The search for algorithms to solve algebraic equations has always been one of the important problems of mathematics. Greek mathematics accomplished only the systematic solution of quadratic equations. Despite some progress by Arab mathematicians, most notably Omar Khayyam, nothing resembling a ``formula'' for higher degree equations emerged until the Renaissance. During that time, Greek mathematics was rediscovered and the old problems were attacked by new methods. Further progress for equations of degree three and four became possible through the introduction of algebraic techniques into Europe.
Cardano and several of his contemporaries discovered methods for solving equations such as , published in his Ars Magna (The Great Art) [20, pp. 203--206,]. In the Greek spirit, his arguments are geometric, viewing the cubic term as a volume, although the computation is easily translated into algebra.
The significance of his work (or, at least, of the publication of his book  in 1545) is twofold: it generated widespread interest in the problem of solving algebraic equations, and it raised the specter of imaginary numbers; even equations whose roots are all real may require imaginary numbers in the evaluation of Cardano's formula. (A selection of his work on imaginary roots can be found in [20, pp. 201--202,].) Even by the time Lagrange summarized the state of the art in his lengthy 1770 memoir , no real progress had been made for equations of degree five and higher, despite much effort. Then in the early nineteenth century, Galois completely solved this two millenium old problem, using truly revolutionary methods which paved the way towards the development of abstract algebra.