# Mathematics at TCD 1592-1992

# 400

years of

MATHEMATICS

*by * T. D. Spearman

# QUATERNIONS

In the early 1830s Hamilton had taken an important step in
de-mythologising complex numbers by treating them as ordered pairs
of real numbers. He thus avoided any specific reference to the
square
root of a negative number. It seemed natural to extend this concept
to triplets (*a*, *b*, *c*) but every attempt to
define
an acceptable multiplication law for triplets ended in failure.
This problem of triplets became an obsession. Hamilton believed that
triplets should exist, not just as abstract mathematical objects but
as entities with their own inherent meaning and significance in
relation
to our three-dimensional world, and as such that they should be
subject to
the basic algebraic operations of addition and multiplication.
For more than a decade he wrestled with the problem, to no avail,
and then one afternoon as he walked with his wife from their home
at Dunsink Observatory towards town, on his way to an Academy
meeting,
it was as though the veil which had for so long obscured his vision
was suddenly lifted and the solution was plain to see. Years later,
recalling his own sensations on that day which had remained vividly
engrained in his memory, hamilton wrote this remarkable description
of the moment of inspiration:

`I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations betweeni,j,k; exactly as I have used them ever since. I pulled out on the spot a pocket-book, which still exists, and made an entry, on which, at the very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be fifteen) years to come. But then it is fair to say that this was because I felt a problem to have been at that moment solved - an intellectual want relieved - which had haunted me for at least fifteen years before.'

What Hamilton had realised was that one had to go beyond triplets
to quaternions, sets of four real numbers, to construct a suitable
algebra, and what he say in his mind and recorded not only in his
notebook but also, using his pocket knife, on a stone of
Brougham Bridge,
was the law of multiplication which these quaternions must satisfy.
The most radical feature of this discovery was that the algebra of
quaternions was not commutative, *a* × *b* was not
equal to *b* × *a*, a possibility which had not
hiterto been contemplated and which opened up new horizons
in mathematics.