In his booklet Geometric Dissections and Transpositions (London: Bell & Sons, 1891) Perigal provided a proof of the Pythagorean theorem based on the idea of dissecting two smaller squares into a larger square.
The five-piece dissection that he found may be generated by overlaying a regular square tiling whose prototile is the larger square with a Pythagorean tiling generated by the two smaller squares.
Perigal had the same dissection printed on his business cards, and it also appears on his tombstone.

In the same book, Perigal expressed the hope that dissection based methods would also solve the 1925 Tarski's problem of circle-squaring by dissection.
That problem had been shown to be impossible to solve in a constructive way in 1963.
Nevertheless, a non-constructive solution has been proposed by Miklós Laczkovich in 1990.

Perigal also proposed the first 6-pieces solution to the square trisection problem.

As well as being interested in mathematics, Perigal was an accomplished lathe worker, and made models of mathematical curves for Augustus De Morgan.
He believed (falsely) that the moon does not rotate with respect to the fixed stars, and used his knowledge of curvilinear motion in an attempt to demonstrate this belief to others.
In his booklet Geometric Dissections and Transpositions (London: Bell & Sons, 1891) Perigal provided a proof of the Pythagorean theorem based on the idea of dissecting two smaller squares into a larger square. The five-piece dissection that he found may be generated by overlaying a regular square tiling whose prototile is the larger square with a Pythagorean tiling generated by the two smaller squares. Perigal had the same dissection printed on his business cards, and it also appears on his tombstone.
In the same book, Perigal expressed the hope that dissection based methods would also solve the 1925 Tarski's problem of circle-squaring by dissection. That problem had been shown to be impossible to solve in a constructive way in 1963. Nevertheless, a non-constructive solution has been proposed by Miklós Laczkovich in 1990.
Perigal also proposed the first 6-pieces solution to the square trisection problem.
As well as being interested in mathematics, Perigal was an accomplished lathe worker, and made models of mathematical curves for Augustus De Morgan. He believed (falsely) that the moon does not rotate with respect to the fixed stars, and used his knowledge of curvilinear motion in an attempt to demonstrate this belief to others.
Henry Perigal