This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...