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 between i, 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.