History of Quaternions - Octonions

Octonions

Octonions were developed independently by Arthur Cayley in 1845 and John T. Graves, a friend of Hamilton's. Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold, why should you stop there?"

Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843 presenting a kind of double quaternion that is now days often called an octonion, and showing that they were what we now call normed division algebra; Graves called them octaves. Hamilton needed a way to distinguish between two different types of double quaternions, the associative bi-quaternions and the octaves. He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion. observed in reply that they were not associative, which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it; Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845 – as an appendix to a paper on a different subject. Hamilton was stung into protesting Graves' priority in discovery, if not publication; nevertheless, octonions are known by the name Cayley gave them – or as Cayley numbers.

The major deduction from the existence of octonions was the eight squares theorem, which follows directly from the product rule from octonions, had also been previously discovered as a purely algebraic identity, by Ferdinand Degen in 1818.