In 1543 Turkish forces overran its capital, Constantinople. Byzantine scholars found refuge in Italy, where rich and powerful families like the Medici added scholars to their entourage and manuscripts to their libraries. The appearance of Gutenberg's printing press made mathematical ideas far more widely available. Over 200 new books on mathematics appeared in Italy before 1500.
In 1545 a book entitled Ars Magna, or The Great Art, by the Italian mathematician Girolamo Cardano (1501-1576), appeared. This work incorporated significant new results—the solution of the cubic and quartic equations. In modern treatments of algebra, a quadratic equation is any equation of the form
where A is a number other than zero, and B and C are constants that can be positive, negative, or zero. The letter x, of course, is the unknown to be found. Until quite recently, however, mathematicians did not have the tools available to deal with the case in which A, B, and C are all positive numbers. Further, there was a tendency to avoid the appearance of negative numbers. A method of solution for some quadratic equations had been developed by the ancient Babylonians based on a process now taught as "completing the square." The al-jabr discusses the six possible variations of the quadratic equation that can be written without negative numbers or a zero, for example, Ax2 = Bx, or Ax2 + Bx = C.
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