This worrying result was first published by German mathematician Oskar Schlömilch in 1868. (The discrepancy is explained by minute gaps in the diagonals, as explained here.)

Charles Dodgson (Lewis Carroll) seems to have been taken with the paradox — his papers show that between 1890 and 1893 he was working to determine all the squares that might similarly be converted into rectangles with a “gain” of one unit of area, apparently unaware that V. Schlegel had carried out the same task much earlier.

(Warren Weaver, “Lewis Carroll and a Geometrical Paradox,” American Mathematical Monthly 45:4 [April 1938], 234-236.)