A lot of people counted Latin squares, going back to Euler (1782) and Cayley and Frolov (independently, 1890). Many of those who tried got the wrong answer. A summary is in this paper:

B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups and loops, J. Combin. Designs, 15 (2007) 98-119. A copy with a correction to Theorem 2 is here.

My favourite from the pre-computer age is:

P. N. Saxena, A simplified method of enumerating Latin squares by MacMahon’s differential operators; II. The 7 × 7 Latin squares, J. Indian Soc. Agric. Statistics, 3 (1951) 24–79.

Saxena devoted 55 pages to the most intricate case-by-case calculations but amazingly got the right answer.