# William Rowan Hamilton, Mathematician

Created | Updated Mar 29, 2015

Have you ever had a flash of inspiration? Something you've been thinking about for weeks but never quite figuring out suddenly becomes clear to you and it's like you've been struck by lightning or touched by the hand of God. This happened to 19th-Century Irish mathematician, William Rowan Hamilton, as he was walking by a canal. He was so overcome by the feeling that he felt compelled to take out a knife and carve his discovery onto a nearby bridge. Today, mathematicians meet annually to commemorate this piece of geeky graffiti.

Hamilton was one of the greatest mathematicians of the 19th Century. He is best known today for Hamiltonian Mechanics, a reformulation of Newtonian Mechanics, and for the invention of the Hamiltonian operators used in Quantum Mechanics, but in his own time he considered his greatest achievement to be the four-dimensional complex numbers called quaternions, and it was these that he created in a flash of inspiration.

### Early Life

William was born on 4 August, 1805, to parents Sarah and Archibald Hamilton of Dominick Street in Dublin. His father had a profitable legal practice. His mother came from a family of coach makers, the Huttons. William was the fourth of nine children. His parents decided not to bring up William themselves, but to hand over the task to Archibald's brother, James Hamilton. William moved to Trim, Country Meath, about 50km from Dublin. He lived with James in the manor house known as Talbot Castle, and stayed there until he was 18. James ran a school there and William was educated in the school along with the other children.

James noticed that William was good with languages, so he taught him the classical languages of Latin, Greek and Hebrew as well as many modern languages. He also noticed that the boy had an aptitude for calculation, but he himself was not qualified to teach him any mathematics.

When William was 16, James gave him a copy *Analytical Geometry* by Bartholomew Lloyd, the Professor of Mathematics in Trinity College, Dublin. William made Mathematics the primary focus of his thought from that time onward:

Ill-omened gift! It was the commencement of my present course of mathematical reading, which has in so great a degree withdrawn my attention, I may say my affection, from the Classics.

### University

Hamilton entered Trinity College in 1823, studying mathematics. He achieved first place out of 100 entrants in the entrance exam and received a book prize for his excellence in Hebrew. The mathematics course had recently been completely redesigned and was a very modern one with all the latest developments from France and Germany. Hamilton did well – early in his second year at University, he submitted his first paper to the Royal Irish Academy, outlining a mathematical treatment of optical rays. The Academy thought the paper interesting but rejected it on the basis of lack of clarity. Hamilton worked on it throughout his time in the University, finally in 1827 resubmitting a much-expanded work as *Theory of Systems of Rays*. This was so well received that he was recommended for the position of Andrews' Professor of Astronomy at Trinity College, which also bore the title of Royal Astronomer of Ireland. This was when he was still only 21 and before he had even graduated from the college. He received his degree later the same year.

### Observatory

The position of Professor of Astronomy came with a modest salary and a residence: Dunsink Observatory, about 8km from the centre of Dublin, a spacious house with a small observatory built into the roof. Initially, Hamilton lived there with two of his sisters, Sydney and Grace. He was required to make some observations, for which he took instruction from the previous Professor of Astronomy, Brinkley, and also spent some time in Armagh Observatory. He discovered that the tasks expected of him weren't onerous, leaving him plenty of time to pursue his mathematics. In later years, his sisters and an assistant performed the observations.

Hamilton lived for the rest of his life at Dunsink.

### The Mathematics of Crystalline Diffraction

Hamilton continued his mathematical studies into optics. Two centuries earlier, Isaac Newton had put forward the theory that light was a particle, but now the scientific community was coming round to the view that light was actually a type of wave^{1}. The French physicist Augustin-Jean Fresnel had developed a way of mathematically describing the behaviour of light in a crystal. Hamilton discovered a new solution to Fresnel's equations which predicted an unobserved phenomenon: in a special type of crystal called a bi-axial crystal, when the incident light was from a particular angle, Hamilton said that the light should be refracted in a conical pattern rather than the normal straight lines.

This prediction was bizarre enough that it was worth looking into. Hamilton's friend Humphrey Lloyd, who was the Professor of Natural Philosophy at Trinity, performed the experiment and showed that light can in fact display this previously unnoticed behaviour. This was strong evidence for the wave nature of light. Hamilton received a knighthood for his work in 1835, at the age of 30, along with a royal pension of £200, while Lloyd was elected a member of the Royal Society the following year.

### Hamiltonian Dynamics

Hamilton now turned his attention to the subject of dynamics, the study of moving objects. He reworked Newton's existing theory of dynamics, developing many new approaches to mathematics to explain it. His approach concentrated on energy and momentum rather than force and speed, and is more useful in complex systems such as the analysis of planetary motion. The results were published in his *Essays on a General Method in Dynamics* but were not particularly well received at the time. Hamilton's methods did not immediately predict anything new so it was some time before their usefulness was realised.

Hamilton's Dynamics only came into its own at the start of the 20th Century with its central role in Quantum Mechanics. Anybody who has studied Quantum Mechanics will have heard of Hamiltonian operators and his name pops up in many other parts of modern physics as a result.

### Quaternions

The field of study which Hamilton was most interested in, that which he spent most time on and which he considered his greatest work, was his extension of the system of complex numbers.

In the early 19th Century the theory of complex numbers was developed, in which it is assumed that there is a number which when squared gives -1. This isn't possible for normal numbers, as both positive numbers and negative ones when squared give a positive number. The square root of minus one, normally denoted *i*, is a different sort of number from normal ones, but obeys most of the same rules. It was discovered that an algebraic treatment of the 2-dimensional plane could be done by treating each point on the plane as a multiple of 1 plus a multiple of *i*. Hamilton tried for many years to make a method of describing three-dimensional space using complex numbers, but always failed.

On 16 October, 1843, it was this problem that he was mulling over as he walked from his house in Dunsink along the path by the Royal Canal to Dublin. As he reached Brougham Bridge (nowadays known as Broom Bridge), he suddenly realised that he could solve the problem by treating it in four dimensions rather than three, and by using three separate, independent square roots of minus 1. That is, he could have three separate numbers, each of them giving -1 when squared, but not equal to each other.

I then and there felt the galvanic^{2}circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; ... I felt a problem to have been at that moment solved – an intellectual want relieved – which had haunted me for at least fifteen years before.

Hamilton was so overcome by the feeling of inspiration that, after noting his discovery in a notebook, he took out a pocket knife and scratched the following equation into the stone of the bridge:

*i*

^{2}= j^{2}= k^{2}= ijk = -1Hamilton's newly created system said that any number could be written as a group of four numbers added together, a real number, a multiple of *i*, a multiple of *j* and a multiple of *k*:

*X = w + ix + jy + kz*

He called this group of four a *quaternion* from the Latin for four, *quater*. He developed a whole algebra based on these four-dimensional numbers. One of the first interesting discoveries was that for the system to work properly, the order of multiplication was important. You didn't get the same result if you multiplied A by B as you did if when multiplying B by A. This was a big difference from standard algebra. Despite this apparent flaw, the system turned out to be fully consistent and workable.

Hamilton considered the quaternions his greatest discovery and for some years after his death they were still taught in Universities in Ireland. But the truth was, they appeared to be a bit of a dead end. The analysis of 3-dimensional space became easier when vector theory was invented, and quaternions were more-or-less forgotten. There's been a revival of interest in them recently though, because of the need to manipulate three-dimensional data for the computer graphics industry. If a three-dimensional point is represented by a quaternion, it can be rotated around an axis by multiplying by another quaternion. This is simpler than matrix multiplication, which is the main alternative method.

### Life, Love and Lack of It

Hamilton was a member of Ireland's professional class, a very restricted class of Irish society. As expected of him, he and his family would mix with others of the same class but would not expect to be friends with the lower classes. As a result, he would have had a rather limited circle of friends. On the other hand, the circles he moved in were the ones who did things, whether it was in the sciences or the arts. He was friends with poets William Wordsworth and Aubrey de Vere; with writer Maria Edgeworth; and with general all-round new woman Constance 'Speranza' Wilde, the mother of Oscar Wilde. Hamilton loved the Romantic ideals and poetry and even wrote much of it himself, although he was advised by Wordsworth that his talents lay more in mathematics than as a poet.

As a young man Hamilton was considered quite handsome, and he was fit and athletic as well. He was in good health although he suffered from double vision, and had an odd speaking voice – sometimes it was very high-pitched and sometimes a deep bass. When engaged in thinking about mathematics, he became so abstracted that he often forgot to eat, but in general he was considered a good-natured and friendly man.

At only 19, Hamilton met with a young woman, Catherine Disney^{3}, and they fell in love. But Disney's family didn't think that Hamilton had much in the line of prospects. They persuaded her to marry an elderly clergyman, William Barlow. Hamilton was deeply affected by the loss of his loved one.

Hamilton was friendly with Ellen de Vere, the sister of poet Aubrey de Vere. While visiting the de Veres at their home in Curragh Chase in the southwest of Ireland, he considered asking her to marry him, but before he had screwed up the courage to pop the question, she declared that she could 'not live happily anywhere but at Curragh'. Hamilton took this to be a rejection of his unspoken proposal.

Eventually he married a friend, Helen Bayly, but it was not a happy marriage. Although they had children, the couple did not really get on, and Bayly was frequently ill, making her unable to run the household as would have been expected of a gentleman's wife. Hamilton took to drinking heavily.

William Rowan Hamilton died on 2 September, 1865, at the age of 60, from gout brought on by excessive eating and drinking. He was buried in Mount Jerome Cemetery, in Harold's Cross, Dublin.

### Tributes, Accolades, Memorials

In addition to the knighthood which gave him the title 'Sir William', Hamilton received many accolades during his lifetime. For his work on optics, the Royal Society bestowed upon him their Royal Medal, while the Royal Irish Academy awarded him the Cunningham Medal. He was President of the Academy from 1837 until 1846. The American National Academy of Science elected him as their first honorary overseas fellow in 1865.

More recently, a crater on the Moon was named after him. The Royal Irish Academy holds an annual Hamilton Lecture, at which many famous mathematicians and physicists have spoken. Trinity College has a Hamilton Mathematics Institute. There is a Hamilton College in New York named in his honour.

In 1983, the Irish Post Office issued a commemorative 29p stamp inscribed 'Quaternions discovery by Hamilton 1843' and a copy of the inscription in his notebook of the basic multiplication equations of quaternions. The year 2005, the 200th anniversary of his birth, was named 'Hamilton Year' by the Irish government, and a commemorative €10 coin was issued.

Modern Mathematics is sprinkled with things named after Hamilton, including the Hamilton equations, the Hamiltonian, Hamilton's principle and so on.

Hamilton's graffiti is no longer visible 170 years on, but a plaque has been erected on the bridge giving details of that famous day. Every year on 16 October, a group of mathematicians and interested people gather at Dunsink Observatory and walk as far as Broom Bridge^{4}, commemorating Hamilton's moment of inspiration, when he discovered quaternions.

^{1}The modern view is that light can act like a wave or a particle depending on the situation.

^{2}An old word for 'electric'.

^{3}She does not appear to have been related to the more well-known Walt Disney, whose family came from County Carlow.

^{4}Beware if you try this on your own! Broom Bridge is considered to be Dublin's most notorious spot for thieves and bag-snatchers.