8.5 William Rowan Hamilton (1805–1865)
branches of mathematics and its applications, he considered the task of finding a
similar algebra of triplets to be of vital importance.
Addition and subtraction of triplets were to be defined componentwise, in the
obvious way. As for multiplication, he imposed several conditions it would have
to satisfy: It had to be associative, commutative, and distributive (over addition),
division had to be possible, the “law of the moduli” had to hold (the modulus of the
product is equal the product of the moduli, where the modulus of the triplet (a, b,c)
is a
2
+b
2
+c
2
), and finally, the product of triplets had to have geometric significance,
just as the product of vectors in the plane does [2].
We have pointed out that Hamilton rejected the formal definition of complex
numbers as vectors in the plane, but he was happy to use the vector representation as
an aid to intuition. With that in mind he represented triplets (a, b,c) also as vectors
in 3-space, in the form a + bi + cj, where the properties of j were to be determined.
In attempts at defining the product of triplets his task reduced to determining the
products ij and j
2
.
Hamilton worked for fifteen years trying to find a multiplication for triplets which
would satisfy the conditions stated above.As we know,he did not succeed, and turned
to quadruples, (a, b, c, d), which he also denoted by a +bi+cj+dk.Ablow-by-blow
account of his attempts to define products of triplets is given in a nice article by van
derWaerden [12]. Hamilton’s own account of these events can be found in [3]. Below
he describes his “flash of insight” on the invention of the quaternions, in 1843. The
account appears in a letter of 1865 to his son Archibald:
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 anti-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 brotherWilliamEdwin, 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.”
Butonthe16thdayofthesamemonth—whichhappenedtobeaMonday,
and a Council day of the Royal Irish Academy—I was walking in to attend
andpreside, and your motherwas walking withme, along theRoyal Canal, to
whichshehadperhapsdriven;andalthoughshetalkedwithmenowandthen,
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
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