Origins of the analysis of the Euclidean algorithm

The Euclidean algorithm for computing the greatest common divisor of two integers is, as D. E. Knuth has remarked, “the oldest nontrivial algorithm that has survived to the present day.” Credit for the first analysis of the running time of the algorithm is traditionally assigned to Gabriel Lamé, for his 1844 paper. This article explores the historical origins of the analysis of the Euclidean algorithm. A weak bound on the running time of this algorithm was given as early as 1811 by Antoine-André-Louis Reynaud. Furthermore, Lamé's basic result was known to Émile Léger in 1837, and a complete, valid proof along different lines was given by Pierre-Joseph-Étienne Finck in 1841.