Re: on quaternions and octonions

From: John Baez (baez_at_galaxy.ucr.edu)
Date: 08/10/04 

Date: Tue, 10 Aug 2004 11:19:46 +0000 (UTC)

In article <40f95438.442470@news.ecn.ab.ca>,
John Savard <jsavard@excxn.aNOSPAMb.cdn.invalid> wrote:

>What I want to know is: when did mathematicians start getting away
>with calling Cayley's octaves "octonions", when that name had already
>been taken (preventing Cayley from using it)?

I may have the answer to your question now - though I still
don't know why you say the name was already taken.

After writing an introductory article on the octonions for the
Bulletin of the AMS, I got a letter from Garry J. Tee of the
Department of Mathematics of the University of Auckland, who
corrected some mistakes and told me some other things as well.

I completely forgot about this until last week, when I was
finishing up a review of Conway and Smith's book _On Quaternions
and Octonions_, again for the Bulletin of the AMS. I want to
publish an erratum listing the mistakes in my article, appearing
in the same issue as this review so people who care about octonions
are more likely to spot it. So, all of a sudden I wanted to reread
Tee's letter! Unfortunately I'm in Cambridge now, and I left the
letter at home, back in California. So, I asked James Dolan to
tell me what it said... which he kindly did.

And here's one thing Tee said in his letter:
 
> |also, i tell that "alexander mcaulay (1863-1931) was the first
> |lecturer in mathematics at the university of tasmania (1893-1896) and
> |then the first professor of mathematics there (1896-1931). he was a
> |prominent advocate of quaternions, and wrote significant books on
> |quaternions and on octonions" (p. 405). our library has copies of
> |both of those books. his smith's prize essay at cambridge university
> |was published as _utility of quaternions in physics_, macmillan,
> |london, 1893. macaulay invented the words "octonion" and "tensor",
> |and his monograph on _octonions: a development of clifford's
> |biquaternions_ was published by _cup_ in 1898 (253 pages).

"_cup_" is Cambridge University Press, so I should be able to find
McAulays' book here and see if he claims to have coined the word
"octonion".

But I've already checked that the book exists and was published in
1898, so the word "octonions" definitely dates back to at least then.

In his book _A Taste of Jordan Algebras_, Kevin McCrimmon writes
about the octonions and says:

 A subsequent flood of (false!!) higher-dimensional algebras
 carried names such as quadrinions, quines, pluquaternions,
 nonions, tettarions, plutonions. Ireland especially seemed a
 factory for such counterfeit division algebras.

.........................................................................

Puzzle #5: When was the Roman empire sold, and who bought it?

For the answer see http://math.ucr.edu/home/baez/puzzles/5.html

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