Sir William Rowan Hamilton
Hamilton was born in Dublin in 1805. That is, 4 years before Nelson’s Pillar was completed in the
centre of Dublin. As he grew up
Hamilton lived in Trim, Co. Meath, but he returned to lecture in Trinity
College, Dublin and lived in the Observatory House at Dunsink, near Finglas,
just a few miles from the centre of the city. He died in 1865, aged 60.
He liked to take a stroll with his wife. He also liked to think about mathematical problems. One day – the 16th of
October, 1843 – he combined both.
As the couple were walking along the canal and passing under Broom
Bridge ((or Brougham Bridge) near Glasnevin and which links Finglas and Cabra) he had a flash of
insight into the problem of quaternions.
Not having a quill or ink on his person he quickly thought of his
pocket-knife and carved the relevant mathematical formula into the stone of the
bridge in case he forgot what it was.
Mr. Hamilton was quite a famous person and Dublin people came to know
the ideas of quaternions even though they referred to them as ‘quart ‘er
onions’ – a quart being a standard measure of beer or porter and they could
never quite make out why onions had to be measured like this. (James Joyce also referred to quarts
but called them quarks and quarks have become famous as elementary particles of
atoms in the theories of physics).
Dev (Eamon deValera) also knew his onions and whiled away his time (or
some of it anyway) while imprisoned following the 1916 uprising by carving the
formula for quaternions on his prison wall. Later, in 1958, Taoiseach Eamon deValera dedicated a plaque
to Hamilton and that plaque is still visible on the side of Broom Bridge.
No wonder he would have had trouble remembering it!
The formula is famous because it contains the rules for a new
mathematics which does not allow commuting – Hamilton would have loved the
irony of the building of a commuter train-station at Broom Bridge in recent
years. Commuting in mathematics
means that calculations work the same either backwards or forwards. For example, 3 times 2 results in
6; 2 times 3 also results in
6. But Quaternions do not
commute!
The genius of Hamilton was to realise that i times j would not produce the same result as j
times i. Furthermore, in order to
get his system of letter-mathematics to work he required the third letter k –
kool, eh? and just beside the kanal!
What he discovered was that i times j produced the same result as –j
times i. ij = -ji.
The formula iČ = -1 was within the Complex Number System, a system
containing two parts - the ordinary part for familiar real numbers and
the extraordinary or imaginary part for numbers which obeyed the
different law that two identical numbers multiplied together could produce a
negative result.
This system was well known before Hamilton got his mind to work on
it. Hamilton wanted to know if he
could extend this system to develop a system with three parts (thus involving real
parts, i-parts and j-parts). His discovery was that four parts were necessary to
construct a viable mathematical system and that even then the new system had to
break the long-held mathematical law of commutativity. However, the new system of 4 parts
(quaternions) obeyed all other mathematical laws and Hamilton was delighted.
For those who wonder if Hamilton just dreamt it up and that these things
have no earthly relevance here is a point to ponder. The mathematical system for calculations which are required
to produce the gyrations of the computer-game-girl Lara Croft in such games as
Tomb Raider is nothing less than our friends the quaternions translated into
computer language.
In 2001, at the Young Scientist of Ireland Exhibition, a young girl,
Sarah Madden, attending St. Mary’s Secondary School in Glasnevin, received the
second top prize for her inventive exploration and attempts to create a number
system involving Panjic and Noppic numbers intended to replace the Complex
Number System.
For those who wish to get started thinking about such things a good
place to start is with the Problem of Cardano (Girolamo Cardano of Milan,
Italy) which he solved in 1545 (with some help from a former friend Nicolo
Fontana (also called Tartaglia) of Venice, Italy). The problem is:
Divide 10 into two parts such that the product of the two parts is 40
while the sum of the two parts is 10.
Hint: He had to invent a new number system to solve this problem.
On the anniversary of Hamilton’s invention, the 16th
October, each year there is an excursion from the Observatory in Dunsink along
the bank of the Royal Canal to Broom Bridge. This excursion has been organised by Dr. O Cairbre of the
Mathematics Department of NUI Maynooth.
This year (2001) he was joined by groups organised in St. Patrick’s
Training College, Drumcondra, Dublin, as well as students, teachers and
professors from surrounding schools and colleges in Dublin and Kildare.
Check out this web-site for more information on Hamilton or details of Hamilton’s life or a picture of the plaque or an engraving of Hamilton carving his formula in the bridge. (Pictures from the St. Andrew’s, Scotland, web-site).
Page created by Neil Hallinan
