 A. L. Reynaud in 1811, who showed that the number of division steps on input ("u", "v") is bounded by "v"; later he improved this to "v"/2  + 2.Later, in 1841, P. J. E. Finck showed that the number of division steps is at most 2 log2 "v" + 1, and hence Euclid's algorithm runs in time polynomial in the size of the input.Émile Léger, in 1837, studied the worst case, which is when the inputs are consecutive Fibonacci numbers.