William Rowan Hamilton

The life of William Rowan Hamilton

Timeline 1805Born on the night of 3 to 4 August in Dublin, the son of a solicitor. 1808Goes to live in Trim with his uncle James, a clergyman and schoolmaster, who teaches him languages. 1818Takes on the American calculating prodigy Zerah Colburn; the defeat leads him to mathematics. 1823Enters Trinity College Dublin in first place. 1827While still a student, is appointed Royal Astronomer of Ireland. Visits Wordsworth in the Lake District. 1832Predicts conical refraction, observed soon afterwards by Humphrey Lloyd. 1833Marries Helen Bayly. Presents complex numbers as couples. 1834Publishes his general method in dynamics. 1835Is knighted and becomes Sir William. 1837Becomes President of the Royal Irish Academy. 1843On 16 October discovers quaternions and carves the formula on a bridge. 1853Publishes the Lectures on Quaternions. 1857Invents the Icosian game. 1865Is the first foreigner elected to the National Academy of Sciences of the United States. Dies on 2 September at Dunsink. ◀ Previous Follow the journey Next ▶ 1805–1808 DublinHe is born on the night of 3 to 4 August 1805, in Dominick Street, the son of a solicitor. At three he is sent to live with an uncle. 1808–1823 TrimHe grows up with his uncle James, a clergyman and schoolmaster, who teaches him languages. At thirteen he claims to know a dozen, from Latin to Persian. 1823–1827 Trinity CollegeHe studies in Dublin and wins the top prizes in classics and in science. While still a student he presents his theory of systems of rays. 1827 Rydal MountHe visits William Wordsworth in the Lake District. The two become lifelong friends, and the poet advises him to leave poetry for science. 1827–1865 Dunsink ObservatoryAt 21 he becomes Royal Astronomer of Ireland. There he writes his optics, his mechanics and his quaternions. He dies at the observatory on 2 September 1865. 1843 Broom BridgeOn 16 October, walking with his wife along the Royal Canal, he finds the rule for quaternions and carves it on a stone of the bridge. Enable JavaScript to see the map. The stages are listed below. From Dublin to Dunsink, with a detour to Wordsworth’s house in the Lake District. Map: © OpenStreetMap contributors

October 27, 2026 Read

Systems of rays and conical refraction

At the beginning of the nineteenth century optics was a collection of cases: the plane mirror, the curved mirror, the lens, the prism, each with its own formulas. Hamilton, still a student, asked himself whether there was not a single idea behind them all. The result was the Theory of Systems of Rays, read at the Royal Irish Academy in 1827 and extended over the following years. One function to govern all rays The starting point is a principle of Fermat: light goes from one point to another along the path that takes the least time, or, more precisely, a stationary time. Hamilton took that time as a function of the endpoints. For two points $A$ and $B$, let $V(A, B)$ be the “optical length” of the light’s path between them: the distance travelled in each medium, multiplied by the refractive index of that medium. He called $V$ the characteristic function.

October 27, 2026 Read

Complex numbers as couples

By the early nineteenth century complex numbers had been in use for three hundred years, since Cardano and Bombelli, and Euler had worked wonders with them. Wessel, Argand and Gauss had drawn them as points of the plane. But the unease remained: what, after all, is $\sqrt{-1}$? A number whose square is negative seemed a useful fiction, not an object of mathematics. Hamilton settled the question without drawing anything. In the essay Theory of Conjugate Functions, or Algebraic Couples, presented to the Royal Irish Academy in 1833 and 1835 and published in 1837, he proposed that a complex number is an ordered couple of real numbers, and nothing more.

October 27, 2026 Read

Hamilton's mechanics

In 1788 Lagrange had turned Newton’s mechanics into a set of equations valid in any coordinates. Hamilton admired that book, which he called “a kind of scientific poem”. In 1834 and 1835, in the two essays On a General Method in Dynamics, he took the idea further, bringing into mechanics what he had learned from rays of light. Position and momentum Lagrange described a system by its positions $q$ and velocities $\dot q$. Hamilton replaced the velocity with the momentum $p$ (for a simple particle, mass times velocity, $p = m\dot q$) and put at the centre of everything a single function, the Hamiltonian $H(q, p)$, which in most cases is the total energy, kinetic plus potential:

October 27, 2026 Read

Quaternions: the formula on the bridge

After showing that complex numbers are couples of real numbers and that their multiplication rotates the plane, Hamilton wanted to do the same in space. He was looking for triplets $a + bi + cj$ with a multiplication that respected the law of the moduli: the size of the product should be the product of the sizes. He tried for years, above all in 1843, so much so that his children asked him every morning whether he could multiply triplets yet.

October 27, 2026 Read

Rotating in space with quaternions

Complex numbers of modulus 1 rotate the plane. Hamilton hoped his numbers would do the same in space, and they do, but in a subtler way than he first imagined. The recipe Write a point of space $(x, y, z)$ as the pure quaternion $v = xi + yj + zk$. To rotate by an angle $\theta$ about an axis given by a unit vector $u = u_1 i + u_2 j + u_3 k$, take

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Vectors and scalars: Hamilton's words

In a quaternion $q = a + bi + cj + dk$ there are two parts of different kinds. The number $a$ stays put on the line of real numbers; the part $bi + cj + dk$ points in a direction in space. In 1846 Hamilton named them: the first is the scalar, because it can be read off a scale, from less to more; the second is the vector, from the Latin vector, “carrier”, because it carries one point to another. Both words caught on, and today they are used by people who have never heard of quaternions.

October 27, 2026 Read

The Cayley–Hamilton theorem

Every quaternion $q = a + v$, with scalar part $a$ and vector part $v$, satisfies a quadratic equation with real coefficients. Since $v^2 = -\lvert v \rvert^2$ (see vectors), $$q^2 = a^2 + 2av + v^2 = a^2 - \lvert v \rvert^2 + 2av,$$and so $$q^2 - 2a\,q + \lvert q \rvert^2 = 0, \qquad \lvert q \rvert^2 = a^2 + \lvert v \rvert^2.$$A quaternion is a root of a real polynomial of degree 2, just as a complex number $z$ is a root of $z^2 - 2\operatorname{Re}(z)\,z + \lvert z \rvert^2 = 0$. In the Lectures on Quaternions of 1853 Hamilton went further and showed that certain linear functions of quaternions also satisfy a polynomial equation of their own, which he used to compute their inverses.

October 27, 2026 Read

The Icosian game and Hamiltonian cycles

In 1856, still busy with non-commutative algebras, Hamilton invented the icosian calculus: three symbols $\iota$, $\kappa$ and $\lambda$ obeying the rules $$\iota^2 = 1, \qquad \kappa^3 = 1, \qquad \lambda^5 = 1, \qquad \lambda = \iota\kappa.$$Today we would say they generate the group of rotations of the icosahedron, the solid with twenty triangular faces. Hamilton presented the calculus at a meeting of the British Association in Dublin in 1857, and noticed that it described paths along the edges of the dodecahedron, the solid with twelve pentagonal faces and twenty vertices.

October 27, 2026 Read

The legacy of William Rowan Hamilton

The Hamiltonian principle has become the cornerstone of modern physics, the thing with which a physicist expects every physical phenomenon to be in conformity. — Erwin Schrödinger (1943) Hamilton believed quaternions would be his great work, the language of the physics of the future. He was wrong in the detail and right in the essentials. Physics adopted Gibbs’s and Heaviside’s vectors, but Hamilton’s mechanics, which he thought less important, became the basis of almost everything that came after.

October 27, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons; the Kirkwood etching was reduced in size and the conical refraction diagram had its colour palette reduced, with no other editing. Portrait of Hamilton, Library of Congress: source page · public domain. Oil portrait of Hamilton, unknown author: source page · public domain. Etching by J. Kirkwood, Wellcome Collection: source page · CC BY 4.0. Irish stamp of 1943: source page · public domain. The plaque on Broom Bridge, photo by Cone83: source page · CC BY-SA 4.0. Broom Bridge, photo by Johnflan: source page · CC BY-SA 4.0. Dunsink Observatory, photo by David Malone: source page · CC BY-SA 2.5. Trinity College Dublin, photo by JoachimKohler-HB: source page · CC BY-SA 4.0. Wordsworth on Helvellyn, painting by Benjamin Robert Haydon: source page · public domain. Internal conical refraction, by Thomas Preston and William Edward Thrift: source page · public domain. Replica of the Icosian game at the Matemateca IME-USP, Estes Objethos Atelier, photo by Alisson Ricardo: source page · CC BY-SA 4.0. Figures made for the blog The cover and the interactive figures for the pendulum’s phase space, rotations with quaternions and the Icosian game were made for this trail and work in all four languages of the site.

October 27, 2026 Read