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 awaitImpatiently thy rising over the bareExpanse of waters; fondly searching whereThy 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.

Starting at point A, and passing through each vertex exactly one time ending at

the starting vertex we can obtain the following Hamiltonian circuit:

B

A

F

G

H

C D

ET

S

K

I

J

ON

M

L

R

Q

P

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://www-gap.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