History of quaternions

Subject: Re: Q: Geometric Algebra instead of Vector Algebra??
From: Doug B Sweetser <sweetser@alum.mit.edu>
Date: 1996/09/30
Message-Id: <52pnk4$113k@pulp.ucs.ualberta.ca>
Newsgroups: sci.physics.research
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This is my own (biased by my current research) view of the history
surrounding vectors - with some facts thrown in.

Hamilton was THE big mathematician of his day. Complex numbers were a
hot subject for research. An obvious question was that if a rule for
multiplying two numbers together was known, what about multiplying three
numbers? This simple question had bothered Hamilton for over a decade.
And the pressure was not merely from within. Hamilton wrote to his son:

"Every morning in the early part of the above-cited month [Oct. 1843] on
my coming down to breakfast, your brother William Edwin and yourself used
to ask me, 'Well, Papa, can you multiply triplets?' Whereto I was always
obliged to reply, with a sad shake of the head, 'No, I can only add and
subtract them.'"

We can guess how Hollywood would handle the Brougham Bridge scene. He
had found a long sought-after solution, but it was weird, very weird, it
was 4D. One of the first things Hamilton did was get rid of the fourth
dimension, setting it equal to zero, and calling the result a "proper
quaternion." He spent a good fraction of the rest of his life trying to
find a use for quaternions. Quaternions were viewed by the end of the
nineteenth century as an oversold novelty.

In the early years of this century, Prof. Gibbs of Yale found a use for
proper quaternions by reducing the extra fluid surrounding Hamilton's
work and adding key ingredients from Rodrigues concerning the application
to the rotation of spheres. He ended up with the vector dot product and
cross product we know today. This was a useful and potent brew. Our
investment in vectors is enormous, eclipsing their originators (Harvard
had >1000 references under "vector", about 20 under "quaternions", most
of those written before the turn of the century).

In the early years of this century, Albert Einstein found a use for four
dimensions in order to make the speed of light constant for all inertial
observers. Here was a topic tailor-made for a 4D tool, but as we all
know, Albert was not a big math buff, and just built a machine that
worked from locally available parts (much like Michelangelo with that
rock the other guy didn't want). We can say now that Einstein discovered
the Lorentz transformation and how it acts on spacetime four vectors.

By the sheer volume of its success, I believe vector analysis properly
holds its place above quaternions or their generalization, the geometric
algebras. I personally think that there may be 4D roads in relativity
that can be efficiently traveled only by quaternions, and that is the
path I am now exploring.

Doug Sweetser