I was reading a book about computational complexity theory and the author made a claim that the study of time complexity of algorithms started with a result on the upper bound on the number of operations needed by euclid's algorithm on two numbers by Gabriel Lamé in the early/mid 1800s.
I haven't read his original paper but I suspect that Lame's result may have been inspired by previous results on convergence of approximation algorithms on polynomials, specifically Lagrange's.