  An alternate approach, which may have existed in ancient Greece but is only known for certain to have existed from the 9th century CE, is the “anthyphairetic” definition (Hogendijk 2002 ). The Euclidean algorithm is an iterative process that is used to find the greatest common divisor of a pair of numbers. It may be applied equally well to find the greatest common measure of two geometric magnitudes, but the algorithm will never terminate if the ratio between the two magnitudes is irrational. A sequence of divisions within the algorithm results in a “continued fraction” that corresponds to the ratio between the original two quantities. Khayyam, following several earlier Islamic mathematicians, defines the equality of A/B and C/D according to whether their continued fractions are equal.