Einstein set up
x' = x-vt, the transformation from the moving frame to the
stationary frame but
Einstein OMITTED
-x' = -x+vt for the return journey.

Sagnac is isomorphic to Einstein's thought experiment, but Einstein
BLUNDERED the simple algebra at schoolboy level, right here:

  http://www.fourmilab.ch/etexts/einstein/specrel/www/figures/img22.gif

He claimed the return distance is x-vt with time t = (x-vt)/(c+v)
when it is in fact t = (vt-x)/(c+v).

From this BLUNDER he calculated the cuckoo malformations that
he blamed on Lorentz and called "transformations".

The rigorous solution is NOT difficult.