17 - Solution of Equations by Radicals

Summary

Preliminary Remarks

The problem of solving algebraic equations has been of interest to mathematicians for about four thousand years. Clay tablets from 1700 BC Babylon contain the essence of the quadratic formula for solutions of second-degree equations. Unfortunately, the tablets do not indicate how the Babylonians arrived at the methods they described. Greek mathematicians later considered the problem from a geometric point of view. Indian mathematicians from the second century AD made significant advances in algebra, especially in the development of the kind of notational system by which the symbolic algebra of the seventeenth century was made possible. These Indian mathematicians also described a method for solving the quadratic using factorization by completing the square. Medieval Islamic mathematicians continued this algebraic tradition with several original contributions of their own. For example, they considered the cubic equation and gave algebraic and geometric methods for solving special cubics, although a general method for solving the cubic was not found until the sixteenth century.

Artis Magnae, sive de Regulis Algebraicis by Girolamo Cardano (1501–1576) presented the first known general method for solving a cubic, as well as a method for solving a quartic. Published in 1545, this book contained the work of Scipione del Ferro, Niccolò Tartaglia, Lodovico Ferrari, and Cardano himself.