Probabilistic (randomized) algorithms before "modern" computer science appeared
This is discussed a bit in my paper with H. C. Williams, "Factoring Integers before Computers"

In a 1917 paper, H. C. Pocklington discussed an algorithm for finding sqrt(a), modulo p, which depended on choosing elements at random to get a nonresidue of a certain form.  In it, he said, "We have to do this [find the nonresidue] by trial, using the Law of Quadratic Reciprocity, which is a defect in the method.  But as for each value of u half the values of t are suitable, there should be no difficulty in finding one."

So this is one of the first explicit mentions of a randomized algorithm.