From: Michael T. Jones (mtj++at++intrinsic.com)
Date: 08/17/2000 11:28:38
| Uh, the jokes continues to go right past my simple mind :-)
Well, perhaps 'joke' was not the right word. The old (and exactly accurate)
reference is what I meant. Most computer documentation lacks references, and
those that have them, usually only go back ten or fewer years. Since
Hamilton invented quaternion algebra in 1843, I thought it would be fun to
mention that in the man page.
| The reference to a paper of july 1844 really is no joke: the math
| behind Quaternions is really that old, or so I've been told
| at this years SIGGRAPH course 'Visualizing Quaternions'.
Absolutely!
Sir William Rowan Hamilton had spent years thinking about what the 3D
version of 2D's complex numbers might be. Amusingly, the answer came to him
in a flash as he was walking with his wife in Dublin. Afraid that he might
forget the key relationship ("i^2 = j^2 = k^2 = ijk = -1"), he *carved* it
in the stone side of a bridge. He wrote a letter to one of his sons about it
20 or so years later. The way he describes the ecstacy of invention speaks
to my soul:
Letter from Sir W. R. Hamilton to Rev. A. H. Hamilton (August 5, 1865)
MY DEAR ARCHIBALD
(1) I had been wishing for an occasion of corresponding a little with you on
QUATERNIONS: and such now presents itself, by your mentioning in your note
of yesterday, received this morning, that you ``have been reflecting on
several points connected with them'' (the quaternions), ``particularly on
the Multiplication of Vectors.''
(2) No more important, or indeed fundamental question, in the whole Theory
of Quaternions, can be proposed than that which thus inquires What is such
MULTIPLICATION? What are its Rules, its Objects, its Results? What Analogies
exist between it and other Operations, which have received the same general
Name? And finally, what is (if any) its Utility?
(3) If I may be allowed to speak of myself in connexion with the subject, I
might do so in a way which would bring you in, by referring to an
ante-quaternionic time, when you were a mere child, but had caught from me
the conception of a Vector, as represented by a Triplet: and indeed I happen
to be able to put the finger of memory upon the year and month - October,
1843 - when having recently returned from visits to Cork and Parsonstown,
connected with a meeting of the British Association, the desire to discover
the laws of the multiplication referred to regained with me a certain
strength and earnestness, which had for years been dormant, but was then on
the point of being gratified, and was occasionally talked of with you. Every
morning in the early part of the above-cited month, on my coming down to
breakfast, your (then) little 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.''
(4) But on the 16th day of the same month - which happened to be a Monday,
and a Council day of the Royal Irish Academy - I was walking in to attend
and preside, and your mother was walking with me, along the Royal Canal, to
which she had perhaps driven; and although she talked with me now and then,
yet an under-current of thought was going on in my mind, which gave at last
a result, whereof it is not too much to say that I felt at once the
importance. An electric circuit seemed to close; and a spark flashed forth,
the herald (as I foresaw, immediately) of many long years to come of
definitely directed thought and work, by myself if spared, and at all events
on the part of others, if I should even be allowed to live long enough
distinctly to communicate the discovery. Nor could I resist the impulse -
unphilosophical as it may have been - to cut with a knife on a stone of
Brougham Bridge, as we passed it, the fundamental formula with the symbols,
i, j, k; namely,
i2 = j2 = k2 = ijk = -1
which contains the Solution of the Problem, but of course, as an
inscription, has long since mouldered away. A more durable notice remains,
however, on the Council Books of the Academy for that day (October 16th,
1843), which records the fact, that I then asked for and obtained leave to
read a Paper on Quaternions, at the First General Meeting of the session:
which reading took place accordingly, on Monday the 13th of the November
following.
With this quaternion of paragraphs I close this letter; but I hope to follow
it up very shortly with another.
Your affectionate father,
WILLIAM ROWAN HAMILTON.
P.S. Hamilton's work on Quaternions was 12 years before Cayley figured out
matrix multiplication, so today's common graphics pipleline notion of
transfoming directions (his "triples") using 3x3 matrices was not yet
forseen.
Michael T. Jones
mtj++at++intrinsic.com <mailto:mtj++at++intrinsic.com> - http://www.intrinsic.com/
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