could not produce a formula which was generally applicable to equations
involving x^5 , the ‘quintic’ equations. What was so special about the power of five?
In 1826, the short-lived Niels Abel came up with a remarkable answer to this
quintic equation conumdrum. He actually proved a negative concept, nearly
always a more difficult task than proving that something can be done. Abel
proved there could not be a formula for solving all quintic equations, and
concluded that any further search for this particular holy grail would be futile.
Abel convinced the top rung of mathematicians, but news took a long time to
filter through to the wider mathematical world. Some mathematicians refused to
accept the result, and well into the 19th century people were still publishing work
which claimed to have found the non-existent formula.
The modern world
For 500 years algebra meant ‘the theory of equations’ but developments took
a new turn in the 19th century. People realized that symbols in algebra could
represent more than just numbers – they could represent ‘propositions’ and so
algebra could be related to the study of logic. They could even represent higher-
dimensional objects such as those found in matrix algebra (see page 156). And,
as many non-mathematicians have long suspected, they could even represent
nothing at all and just be symbols moved about according to certain (formal)
rules.
A significant event in modern algebra occurred in 1843 when the Irishman
William Rowan Hamilton discovered the quaternions. Hamilton was seeking a
system of symbols that would extend two-dimensional complex numbers to
higher dimensions. For many years he tried three-dimensional symbols, but no
satisfactory system resulted. When he came down for breakfast each morning his
sons would ask him, ‘Well, Papa, can you multiply triplets?’ and he was bound to
answer that he could only add and subtract them.
Success came rather unexpectedly. The three-dimensional quest was a dead
end – he should have gone for four-dimensional symbols. This flash of
inspiration came to him as he walked with his wife along the Royal Canal to
Dublin. He was ecstatic about the sensation of discovery. Without hesitation, the
38-year-old vandal, Astronomer Royal of Ireland and Knight of the Realm, carved
the defining relations into the stone on Brougham Bridge – a spot that is