24/03/2021

Cubic equation: from Babylon to the Renaissance

Foto: Freepik

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

Clay documents excavated in Mesopotamia show that the solution to the quadratic equation ^ax² + bx + c = 0 was already known around 2000 BC and is likely even older. The history of the cubic equation ^ax³ + ^bx² + cx + d = 0 begins around that time, but is longer and more interesting.

There are tables of cube roots on clay tablets from Babylon (20th to 16th centuries BC), but we don't know if they were used to solve equations.

The problem of doubling the cube, which corresponds to the equation ^x³ = 2, began to be studied in ancient Egypt. In the 5th century BC, the Greek Hippocrates of Chios (not to be confused with his contemporary Hippocrates of Kos, the father of medicine) reduced this problem to finding two proportional means between a line segment and another with twice the length. In this way, he came very close to solving the problem through intersections of conic sections.

Methods for solving various cubic equations appear in the manuscript "The Nine Chapters on the Mathematical Art," compiled in China between the 10th and 2nd centuries BC. In the 3rd century, in Greece, Diophantus found integer and rational roots of certain cubic equations. Four centuries later, the Chinese mathematician and astronomer Wang Xiaotong numerically solved two dozen cubic equations.

Back in the 12th century, the Persian Sharaf al-Din al-Tusi discussed 13 types of cubic equations in his "Treatise on Equations," including some that have no positive solutions. He also pointed out the importance of the discriminant ^b²c² ^– ^4ac³ – ^4b³d – ^27a²d² + ^18abcd , whose sign determines how many real roots the equation has.

Leonardo of Pisa (1170–1250), better known as Fibonacci, the greatest mathematician of the European Middle Ages, also contributed to the problem, presenting a method of approximate resolution and applying it to the equation ^x³ + ^2x² + 10x = 20.

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