Cubic Equation: From Babylon to the Renaissance – 3/24/2021 – Marcelo Viana

Clay documents excavated in Mesopotamia show that the solution of the quadratic equation ax2 + bx + c = 0 as early as 2000 BC. Was known and must be even older. The story of the cubic equation ax3 + bx2 + cx + d = 0 starts at this point, but is longer and more interesting.

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

The problem of multiplying the cube, which corresponds to the equation x3 = 2, was 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 by finding two proportional averages between one line segment and another twice as long. In this way he came very close to solving the problem with intersections of cone curves.

Methods for solving various cubic equations can be found in the manuscript “The nine chapters of mathematical art”, which was written between the 10th and 2nd centuries BC. Was compiled in China. In the 3rd century, Diophantus found whole and rational roots of certain cubic equations in Greece. Four centuries later, the Chinese mathematician and astronomer Wang Xiaotong numerically solved two dozen cubic equations.

The great Persian poet and mathematician Omar Khayyam (1048–1131) was the first to discover that cubic equations can have more than one solution, and also stated that in general they cannot be solved with just a ruler and compass. His work was resumed in the following century by the Indian mathematician Bhaskara (1114–1185), but without great success.

As recently as the 12th century, the Persian Sharaf al-Din al-Tusi dealt with 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, the sign of which determines how many real roots the equation has.

Leonardo de Pisa (1170–1250), better known as Fibonacci, the greatest mathematician of medieval Europe, also contributed to the problem by introducing an approximate method of solving it and applying it to the equation x3 + 2×2 + 10x = 20.

All of these important advances affected specific cases, but that changed in the early 16th century when the Italian mathematician Scipione del Ferro (1465–1526) finally discovered a general method of solving the cubic equation.

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