Alain,

Thanks for this inc.  Not only should it prove useful in itself, but I'll be
talking about quaternions (briefly) in my multivariable Calculus class, and
I'll share this with my students (if you don't mind) as a good
demonstration of some of their uses.  Sir William Rowan Hamilton would be
proud.

BTW, Hamilton provides me with my favorite quote I use in college algebra
(the uniquely American way of terming what should be remedial algebra --
worse yet, the prerequisite for "college" algebra at this school is called
"higher algebra;" higher than what, I'm not sure.)

Hamilton, from his "Theory of Conjugate Functions" (1837):

"[I]t requires no particular scepticism to doubt, or even to disbelieve, the
doctrine of Negatives and Imaginaries, when set forth (as it has commonly
been) with principles like these:  that a greater magnitude may be
subtracted from a less, and that the remainder is less than nothing; that
two negative numbers, or numbers denoting magnitudes each less than
nothing, may be multiplied, the one by the other, and that the product will
be a positive number, or a number denoting a magnitude greater than
nothing; and that although the square of a number, or the product obtained
by multiplying that number by itself, is therefore always positive, whether
the number be positive or negative, yet that numbers, called Imaginary, can
be found or conceived or determined, and operated on by all the rules of
positive and negative numbers, as if they were subject to those rules,
although they have negative squares, and mut therefore be supposed to be
themselves neither positive or negative, nor yet null numbers, so that the
magnitudes which they are supposed to denote can neither be greater than
nothing, nor less than nothing, nor even equal to nothing.  It must be hard
to found a science on such grounds as these."

Dave Matthews

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