Self-published, 1830s; engraved on Perigal's tombstone

Perigal's Quartering

Henry Perigal · 1830 · DissectionDifficultyPen & paper

The square on the hypotenuse can be assembled, by sliding alone, from four congruent pieces of the larger leg-square and one whole copy of the smaller leg-square.

Of the two squares on the legs, take the larger one — call it $B$ — and mark its center.
Draw two lines through that center: one parallel to the hypotenuse, one perpendicular to it. Together they cut $B$ into four congruent pieces.
The smaller square, $A$ , is left whole. It becomes the fifth piece of the puzzle.
Henry Perigal's discovery: these five pieces — the four quarters of $B$ and the whole of $A$ — slide into place, with no rotation at all, to exactly cover the square on the hypotenuse.
Every scrap of area from the two leg-squares is accounted for inside the hypotenuse's square, and nothing is left over. So $a^{2} + b^{2} = c^{2}$ .

Therefore c² = a² + b². ∎

Henry Perigal was a London stockbroker and amateur mathematician who spent decades on dissection proofs; this one was important enough to him that he had it engraved on his own tombstone in East Finchley. The four-piece cut works for any right triangle — the two lines through the center always land the pieces correctly, whatever the ratio of the legs.