Sir William Rowan Hamilton:
The History of Ireland’s Greatest Mathematician
And His Accomplishments
Paper 5
History of Mathematics
December 11, 2002
Problem13.24
Abstract
Sir William Rowan Hamilton was one of the greatest
mathematicians from the nineteenth century. As a child,
Hamilton showed promise in becoming the second Sir Isaac
Newton, and contributing much to mathematics. His
contributions were to the fields of optics, abstract algebra,
geometry, and the sciences. Although he was a man
devoted to his studies, Hamilton had many hardships to deal
with. Through his mathematical work, he was able to invent
a game that not only provided a mental challenge but also
exemplified modern concepts used in mathematical theory.
William Rowan Hamilton (pictured right) was born in
Dublin, Ireland in 1805 was meant to cause some stir in the
mathematics world. Born at midnight, on August 3rd/4th, he
was the son of Sarah Hutton, and Archibald Hamilton.
William’s father did not have time to teach his son, for he was
away much of the time, taking care of legal business in
England. Archibald was not a very educated man, therefore it
is thought that his genius and demeanor came from his mother Sarah. (O’Connor)
At an early age, William was sent to live with his uncle, Reverend James
Hamilton in a town called Trim. He was a graduate of Trinity College, and taught
William through his own ideas
of education. Majoring in
classics, he gave William a base
for history and literature, which
led William to study many
languages. (Hughes) “At three
he could read English; at four he was thoroughly interested in geography and had begun
to read Latin, Greek and Hebrew; before he was 10 he had slaked his thirst for Oriental
languages by forming an intimate acquaintance with Sanscrit, and grounding himself in
Persian, Arabic, Chaldee, Syriac and sundry Indian dialects. Italian and French were
imbibed as a matter of course, and he was ready to give vent to his feelings in
extemporized Latin” (Turnbull). A former fellow of Trinity College had once said
astonishingly, “that he had examined in the country a child of six or seven, who read and
translated and understood Hebrew better than many candidates for fellowship” (Graves).
The child he spoke of was William Hamilton, who surprised many by his talents.
William’s Uncle James was very eccentric. It is told that he ”made a hold in the
wall of his bedroom and ran a string through it, the end of which was tied to Hamilton’s
toe. Then, early each morning he would tug on the string to wake young Hamilton and
start him on his studies” (Hughes) At age 10 he also got his hands on a Latin copy of
Euclid’s Elements and began his mathematical journey. Geometry led him to algebra and
it is also around this time that he was introduced to Zerah Colburn, an American boy
known to “perform amazing mental arithmetical feats” (O’Connor). This prodigy was
exhibited in Dublin, where Hamilton had the chance to challenge him to arithmetical
duels. Although William lost most of these battles, it sparked his interest in mathematics.
(Graves)
After diving into his newest found talent for mathematics, at age 15, it is said that
William “had mastered the ordinary amount of what was known in every department of
science” (Waller). This was also the time at which he had read Newton’s Principia, and
begun his own original investigations (Biography.com). Also, around the age of 17,
while reading LaPlace’s Celestial Mechanics, William discovered and error in LaPlace’s
work (Weisstein). This was merely the beginning of his mathematical career and many
new discoveries had yet to come on William’s part. Even early on, he was “regarded by
many as a future second Newton” (Hughes). And this is still before considering his
college education.
William attended his uncle’s alma mater Trinity College (pictured left) in Dublin,
where he continued to study mathematics. Trinity had
been known for its Continental analytic mathematics
but William “swiftly moved beyond the prescribed
curriculum and soon mastered the mathematical texts
in use at the Ecole Polytechnique. His first important
original work was in optics, rather than in pure mathematics, and, in fact, he is today
more famous for his work in dynamics that for his mathematics” (Katz). The academic
aspects of college were not his only interests at this time. With a stocky build, William
was “strong and active in gymnastics and swimming. He also had interests in reading
literature and poetry” (Hughes).
With his mind now focusing on literature and poetry, William became familiar
with the writings of William Wordsworth. He also had the opportunity to meet
Wordsworth in Scotland where they became good friends. “Wordsworth became a major
influence in Hamilton’s life… Hamilton devoted a lot of time to poetry, and constantly
sought a link between the higher levels of science and poetry” (Hughes). William often
found himself running in circles trying to relate prose to mathematics, which led
Wordsworth to convince William to put all of his talents towards science and
mathematics, rather than poetry. Below is a poem William once wrote when returning to
Ireland from England (Hughes):
My native land, appear! These eyes await
Impatiently thy rising over the bare
Expanse of waters; fondly searching where
Thy fair but hidden form lingers so late..
In thee my homeward thoughts still claim their share,
My hear, my life, to thee are dedicate.
Wordsworth was not very fond of William’s poems and found more interest in the poems
written by William’s sister Eliza. However, Hamilton still love to “compare the two,
suggesting that mathematical language was as artistic as poetry” (O’Connor).
William’s passion for poetry had stemmed from his shattered love life. Early in
his college career, William escorted his Uncle James to Summerhill where they met up
with the Disney family. Catherine Disney, the daughter, immediately caught the eye of
William, as he fell hopelessly in love with her. (O’Connor) Because he was not at the
appropriate place in time, he felt he could not propose marriage. However, less than a
year later, William was informed that Catherine would be marrying a man “fifteen years
her senior… who could offer more to Catherine than Hamilton” (O’Connor). Catherine
was William’s one true love, and this disappointment haunted William for the rest of his
life (Hughes). “In this period he turned to poetry, which was a habit that he pursued for
the rest of his life in times of anguish” (O’Connor).
To continue with his unlucky love life, he later met a woman named Ellen de
Vere, whom he bombarded with poetry and considered a marriage proposal. He
“admired her mind” yet did not propose after hearing her say that she would “not live
happily anywhere but at Curragh.” This discouraged William, who then returned home
to later find out that Ellen had also married and had moved from Curragh. (O’Connor)
This was another stroke of bad luck in William’s love life. William eventually settled
and married Helen Maria Bayly who lived not far from the observatory where William
now worked. He found her “not at all brilliant” and actually told her so (O’Connor).
Surprisingly, they had three children, the first two boys, then a girl, however, when the
daughter was born, Helen left William with the boys so that he may do his work in peace.
Helen was a “semi-invalid, suffering from ill-defined nervous complaints throughout her
life” (Hughes).
William was described to have a fascinating personality, “a buoyant cheerfulness,
and kindly human-heartedness.” He was very courteous and women found he and his
mind to be very attractive. When it came to teaching, he often went off on many
tangents, steering his students away from each topic, which leads us to wonder if his
students ever gained knowledge from his lectures. His office and house were very
disorganized, yet he knew if someone had been meddling in his work. (Hughes). For the
last third of his life, William became an alcoholic (Weisstein). These tendencies had
started once Catherine married, however carried over the rest of his life. After more
misfortune with the death of his sister and Wordsworth, William’s alcohol addiction took
a turn for the worse. “Hamilton died from a severe attack of gout shortly after receiving
the news that he had been elected the first foreign member of the National Academy of
Sciences of the USA” (O’Connor).
Although William dealt with rough matters of the heart al through his life, he
diligently worked on mathematics and science. Even on his honeymoon with Helen he
continuously worked on his third supplement to his Theory of Systems of Rays
(O’Connor). Before his marriage however, in 1827, while still an undergraduate at
Trinity, William was named Professor of Astronomy. At this time he had not had much
experience in observing, however he excepted the post even though he had not yet
obtained a fellowship to the college. (O’Connor) Through all of his efforts, which
shifted from mathematics to science, he was then knighted in 1835, which gave him the
title Sir William Rowan Hamilton. Also in that same year, he was awarded the Royal
medal of the Royal Society for his prediction of conical refraction in biaxial crystals
(Hughes). Upon the return of his wife Helen and their daughter in 1842, William had
been working on algebraic triplets, which were three-dimensional vectors. His children
were also aware of their father’s work and had been known to ask him, “Well, Papa can
you multiply triplets?” He was then to answer, “No, I can only add and subtract them”
(Turnbull) It was soon after this time that William stumbled across the true answer to his
children’s question.
“Hamilton loved to walk, in the country or along the canal when he was reading
or doing some serious thinking. It was on such an occasion that his most famous
discovery of Quaternions was made. On Monday October 16th, 1843, when he was
walking along the Royal Canal with his
wife, the idea of Quaternions came to him,
in a dramatic flash of inspiration. In his
excitement he carved the formula in the
stone of Brougham Bridge (pictured left)….
Quaternions opened up a vast new field of
mathematics. However many
mathematicians at the time found them too difficult to use. William Thomson, for
example said he could never find any instance in which they were superior to existing
methods. However, today they are the basis of computer graphics” (Hughes). At the
time, “he introduced Quaternions as a new algebraic approach to three-dimensional
geometry, and they probed to be the seed of much modern algebra” (Biography.com).
Stemming from his idea of Quaternions was his idea of Icosian Calculus. David
R. Wilkins explains that William “used [Icosian Calculus] to investigate closed edge
paths on a dodecahedron that visit each vertex exactly once”. He also adds that today
these “edge paths” are known as Hamiltonian circuits. In a paper brought forth by
William in November 1856, during the proceedings of the Royal Irish Academy, he
explained, “This calculus agrees with that of the Quaternions, in three important respects:
namely 1st, that its three chief symbols, , , , are roots of unity, as i, j, k are certain
fourth roots thereof: 2nd, that these new roots obey the associative law of multiplication;
and 3rd, that they are not subject to the commutative law, or that their places as factors
must not in general be altered in a product. And it differs from the Quaternion Calculus,
1st, by involving roots with different exponents; and 2nd by not requiring so far the
distributive property of multiplication.” Although this explanation may be hard to
follow, it led William to invent a game based on the properties of Icosian Calculus. This
game is generally referred to as the Icosian Game or the Hamiltonian Game (as in our
text).
The Icosian Game, pictured below, was a game that William sold to a London
game dealer in 1859 for 25 pounds (Weisstein). The game “consisted of a graph with 20
vertices on which pieces were to be placed in accordance with various conditions, the
overriding consideration being that a piece was always placed at the second vertex of an
edge on which the precious piece had been placed” (Katz). “The figure was the
projection on a plane of the regular pentagonal dodecahedron, and at each of the angles
were holes for receiving the ivory pins with which the game was played” (Wilkins). The
ideas that this game represented are now known as the Hamiltonian circuits. One
example of this is to construct “a path such that every vertex is visited a single time, no
edge is visited twice, and the ending point is the same as the starting point” (Weisstein).
This problem is also discussed in Howard Eves’ book as the first part of problem 13.24,
which will be address later in the paper. “The problem of finding a Hamiltonian cycle or
path in a graph is a special case of the traveling salesman problem, one where each pair
of vertices with an edge between them is considered to have distance 1, while nonedge
vertex pairs are separated by distance infinity” (Skiena). “Hamilton gave several
examples of ways in which this could be accomplished but gave no general method for
determining in cases other than his special graph whether or not such a path could be
constructed” (Katz).
In “An Introduction to the History of Mathematics”, Howard Eves describes the
Hamiltonian game and asks the reader to answer some of the suggested problems using
the representation used in the game. He explains that each of the vertices is “denoted by
letters standing for various towns.”
1. The first problem is to go “all round the world”; that is, starting from
some given town, visit every other town once and only once and return
to the initial town, where the order of the first n  5 towns may be
prescribed.
Below are three different representations of how this problem can be done, as
stated before, Hamilton provided numerous ways, therefore there are more than three
ways to do this problem. The first figure is a blank board with each vertices labeled.
A
F
G
O
B
T
N
E
P
H
Q
S
I
M
R
J
C
K
L
D
Starting at point A, and passing through each vertex exactly one time ending at
the starting vertex we can obtain the following Hamiltonian circuit:
Using the same procedure, but now starting at point D we can obtain:
Part 2 of the problem states:
Another problem suggested by Hamilton is that of starting at some first given
town, visiting certain specific towns in assigned order, then going on to every other town
once and only once and ending the journey at some second given town.
Looking at the diagram below, let A, B, and C be the towns that we would like to
visit in assigned order; A being the first, B being the second, and C being the last. You
can obtain the following:
A
B
C
References
Biography.com. “Hamilton, Sir William Rowan.” Crystal Reference 2001.
http://search.biography.com/print_record.pl?id=5379
Eves, Howard. “An Introduction to the History of Mathematics.” Saunders College
Publishing, Philadelphia. 1992
Graves, Robert Perceval. “Our Portrait Gallery – Sir William R. Hamilton” Dublin
University Magazine, Vol. 19, 1842.
Hamilton, William Rowan “Account of the Icosian Calculus”. Proceedings of the Royal
Irish Academy. 1858
Hughes, Aine. “Sir William Rowan Hamilton” 2000.
http://members.tripod.com/~Irishscientists/index.htm
Katz, Victor. “ A History of Mathematics.” Addison-Wesley, New York. 1998
O’Connor, J.J.; Robertson, E.F.. “Sir William Rowan Hamilton” 1998 http://wwwgap.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html
Skiena, Steven S. “Hamiltonian Cycle” State University of New York. 1997
http://www2.toki.or.id/book/AlgDesignManual/BOOK/BOOK4/NODE176.HTM
Turnbull, Herbert. “The Great Mathematicians” New York University Press, New York.
1961
Waller, John Francis. “Sir William Rowan Hamilton”. The Imperial Dictionary of
Universal Biography, vol.II
Weisstein, Eric. “Venn Diagram” CRC Concise Encyclopedia of Mathematics.
Chapmand and Hall/CRC, Washington D.C.. 1999
Wilkins, D.R., “The ‘Icosian Calculus’”. 1999
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian