
IRELAND'S GREATEST MATHEMATICIAN
By
Professor William Reville, University College, Cork.
We are fast approaching the 200th anniversary of the birth of William
Rowan Hamilton, the greatest ever Irish mathematician. A child prodigy,
he achieved high academic position at a very young age and made mathematical
discoveries of the first order. Unfortunately his personal life was unhappy,
mainly because circumstances prevented him from marrying his one true
love.
William was born in Dublin on the 4th August 1805 to
Archibald Hamilton, a solicitor, and Sarah Hutton. William’s father
was away a lot on legal business and William lived with his uncle, the
Rev. James Hamilton, at Trim from 1808 to 1823. James was a gifted teacher
and undertook to educate William. He used some unusual methods. It is
said that he bored a hole in the wall between William’s bedroom
and his, through which he passed a length of string. This was attached
to one of William’s toes when he retired for the night and James
would tug the string early in the morning to remind his charge to begin
his studies.
William showed prodigious learning abilities and, by the age of 5, had
already learned Latin, Greek and Hebrew. His interest in mathematics was
probably sparked when, aged 12, he was introduced to the American boy
Zerah Colburn, whose prodigious feats of mental arithmetic were being
exhibited in Dublin. Hamilton engaged him in tests of arithmetical ability,
but usually came off second best.
At the age of 15, Hamilton began to study the works
of Newton and Laplace. He discovered an error in Laplace’s work
Mechanique Celeste [italics]. This brought him to the attention of John
Brinkley, Astronomer Royal of Ireland, who declared Hamilton to be ‘the
first mathematician of his age’. Hamilton entered TCD at the age
of 18 in 1823 and achieved an ‘optime’ in Classics in his
first year, an honour awarded only once in 20 years.
Uncle James took William to visit the Disney family
in Summerhill in August 1824, where William fell in love with the Disney
daughter Catherine. He longed to marry her but was not in a position to
propose since he had 3 years of study left at TCD.
In February
1825 it was announced that Catherine was to marry Rev. William Barlow,
15 years her senior but in a much better position than William Rowan
Hamilton to offer Catherine material comforts. Catherine married
Barlow in May 1825. William was distraught. He fell ill, contemplated
suicide, and started writing poetry, a habit that he continued throughout
his life in times of despair.
Hamilton’s mathematical studies were going
very well. In 1824 he submitted his first paper on mathematics to
The Royal Irish Academy. His finals examiner persuaded him to apply
for the vacant post of Astronomer Royal at Dunsink Observatory and
in 1827 he was appointed Professor of Astronomy at TCD although
still a 21 year old undergraduate. Hamilton had little experience
in practical astronomical observing. His interest in astronomy waned
and he spent almost all of his time doing mathematics.
Hamilton made major scientific contributions
in mechanics, optics, geometry and algebra. 
Sir
William Rowan Hamilton and one of his sons (circa 1845) 
In 1832 he made the daring mathematical prediction
that a ray of light passed through a biaxial crystal would be refracted
into the shape of a cone. Humphrey Lloyd, a TCD physicist, experimentally
confirmed Hamilton’s prediction and the news astounded the scientific
world. Hamilton was knighted for his discovery in 1835 and awarded a
royal pension of £200.
In 1835 Hamilton made his greatest contribution  his
general theory of dynamics. He rewrote Newton’s Laws of Motion
in a powerful general way by expressing the energy of mechanical
systems as special variables. The energy when written in this way
is called the Hamiltonian. Hamiltonians were crucial to the 20th
century development of quantum mechanics.
In 1843 Hamilton invented quaternions, a method of describing
rotations in three dimensions. There is an arithmetic principle called
the commutative law which says that A multiplied by B is equal to B multiplied
by A. But in 3 dimensions the order of actions is important. Think of
the three spatial axes x, y and z. If you first rotate something around
the x axis and next around the y axis, you end up in a different position
to where you would be if you rotated first around the y axis and next
around the x axis. Hamilton expressed his invention in the equation i2=j2=k2=ijk=1
[2s are superscripts]. i, j and k are unit imaginaries corresponding to
the x, y, z spatial axes and the equation also employs the imaginary number
v1 [square root of minus one]. Because there are 4 parts i, j, k and
v–1 [square root of minus one], he called them quaternions. This
leap of creativity paved the way for the introduction of vectors.
Hamilton married Helen Bayley but it was not a match
based on love and was illfated from the beginning. They had 2 sons and
a daughter. William suffered from bouts of depression and started drinking
heavily. In 1847 his uncle James died and his TCD mathematician colleague
James McCullagh committed suicide. Catherine Barlow began a confused correspondence
with him, but, feeling guilty, confessed to her husband and attempted
suicide. William became very depressed and his alcohol problem accelerated.
Catherine died, plunging Hamilton into deep grief. Nevertheless he continued
to work and wrote a new book Elements of Quaternions. He died on 2 September
1865 after a severe attack of gout precipitated by excessive drinking
and overeating.
(This article first appeared in The Irish Times,
February 26, 2004.)
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