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6.AR. QUADRISECTED SQUARE PUZZLE
This is usually done by two perpendicular cuts through the centre. A dissection proof of the Theorem of Pythagoras described by Henry Perigal (Messenger of Mathematics 2 (1873) 104) uses the same shapes -- cf 6.AS.2.

The pieces make a number of other different shapes.

Crambrook. 1843. P. 4, no. 17: Four pieces to form a Square. This might be the dissection being considered here??

A. Héraud. Jeux et Récréations Scientifiques -- Chimie, Histoire Naturelle, Mathématiques. (1884); Baillière, Paris, 1903. Pp. 303 304: Casse tête. Uses two cuts which are perpendicular but are not through the centre. He claims there are 120 ways to try to assemble it, but his mathematics is shaky -- he adds the numbers of ways at each stage rather than multiplying! Also, as Strens notes in the margin of his copy (now at Calgary), if the crossing is off-centre, then many of the edges have different lengths and the number of ways to try is really only one. Actually, I'm not at all sure what the number of ways to try is -- Héraud seems to assume one tries each orientation of each piece, but some intelligence sees that a piece can only fit one way beside another.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. P. 14: The divided square puzzle. Crossing is off-centre.

Tom Tit, vol 3. 1893. Carré casse-tête, pp. 179-180. = K, no. 26: Puzzle squares, pp. 68 69. = R&A, Puzzling squares, p. 99. Not illustrated, but described: cut a square into four parts by two perpendicular cuts, not necessarily through the centre.

A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 -- BMC]. No. 77: Pattern making, pp. 69-70 & 109. Make five other shapes.

M. Adams. Puzzle Book. 1939. Prob. C.12: The broken square, pp. 125 & 173. As above, but notes that the pieces also make a square with a square hole.