Sir William Rowan Hamilton (1805 – 1865)

William Rowan Hamilton (1805–1865) is considered to be one of the world’s greatest mathematicians and physicists. His life and work have been extensively described by others [1-4]. Hamilton was a child prodigy. He displayed an extraordinary aptitude for languages from a very young age and when he was sixteen he uncle gave him a copy of Bartholomew Lloyd’s textbook in preparation for going to university, which stimulated his interest in mathematics. In 1823 he began his studies in Trinity College Dublin and he came first place in every examination and received “optimes” (a rarely awarded distinction) during his studies.

While still an undergraduate student, in 1827 Hamilton was appointed Andrews’ Professor of Astronomy, which came with the title Royal Astronomer of Ireland. He moved to Trinity’s observatory at Dunsink but focussed on his mathematical research rather than astronomical observations. He welcomed the opportunity to base himself in this rural location on the outskirts of the city, away from the distractions of college life, where he could concentrate on his own research interests [5]. It is here that he carried out his work on quaternions, optics and dynamics, for which he received international recognition.

In 1832 Hamilton undertook a mathematical analysis of the wave surface that describes the propagation of light in a biaxial crystal [6]. Based on this study he discovered conical refraction. Hamilton’s theory predicted two manifestations of conical refraction. First, unpolarized light incident on a biaxial crystal at certain angles would be refracted to form a hollow cone of rays; this light would then emerge from the crystal in the form of a hollow cylinder. Second, rays of light travelling in certain directions within the crystal would be refracted on emergence to form a hollow cone of rays. Humphrey Lloyd, the tenth Erasmus Smith’s professor, performed experiments to confirm the findings. This was the first time that a physical effect has been mathematically predicted and subsequently experimentally verified. “Its announcement electrified the scientific community of the day. Airy called it “perhaps the most remarkable prediction that has ever been made” and it would later be commonly compared to the subsequent prediction and discovery of the planet Neptune by Adams and Leverrier” [5].  He was awarded the Royal medal of the Royal Society in 1835, Michael Faraday was also awarded the medal that year. In the same year, Hamilton was knighted at the age of 30. Conical refraction is described in the following contribution in this catalogue and is exhibited in the Fitzgerald Library.

In the following years Hamilton worked on dynamical systems and developed what is known as the Hamiltonian formalism to describe the time evolution of a system.  As remarked by David Spearman, perhaps the most important influence was that on Erwin Schrödinger, who received a thorough grounding in Hamiltonian dynamics from his professor, Friedrich Hasenöhrl [7]. The Hamiltonian formulation is a central component of Schrödinger’s wave theory description of quantum mechanics.

Hamilton is also known for his discovery of quaternions. A “ureka” moment occurred on 16th October, 1843 while walking along the Royal Canal. “An undercurrent of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work……nor could I resist the impulse –unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula.” A commemorative plaque with his famous formula for quaternion algebra; i² = j² = k² = ijk = -1 marks the location on Broome Bridge. A quaternion is a 4D complex-number, and just as a complex number was the sum of a number and an imaginary, a quaternion is the sum of a number and a line in space. Hamilton was the first to distinguish these by the terms ‘scalar’ and ‘vector’. Quaternion algebra is used for the calculation of rotations for solid bodies. At the time of their discovery their modern day applications could not have been imagined -  satellite navigation, space travel, robotics, animation and computer game programming.  Quaternions were first used by NASA to effect rotations as part of the guidance, navigation and control systems on the 1981 Space Shuttle launch and are now used for account for the altitude of almost every spacecraft [9].

It is a mark of Hamilton’s international standing that he was the first foreign member elected to the National Academy of Science (USA) in 1865. The only other Foreign Associate in Ireland has been Michael Coey, the twenty-first Erasmus Smith’s Professor [6].

Hamilton's work continues to be influential after almost two centuries. In 1943 Schrödinger is quoted in The Irish Times, “I daresay not a day passes – and seldom an hour – without somebody, somewhere on this globe, pronouncing or reading or writing or printing Hamilton’s name.” [5] This remains as true now as it was eighty years ago.


  1. Robert Perceval Graves, Life of Sir William Rowan Hamilton(3 vols, 1882–9).
  2. Thomas L. Hankins, Sir William Rowan HamiltonJohn Hopkins University Press, 1980.
  3. Sean O'Donnell, William Rowan Hamilton: portrait of a prodigy(1983)
  4. R. Wilkins, William Rowan Hamilton: Mathematical Genius, Physics World, 2005 []
  5. Eric Finch (2016), Three Centuries of Physics in Trinity College Dublin, Living Edition, p. 37.
  6. Iggy McGovern, “Sir William Rowan Hamilton”, Trinity Monday memorial discourse, 2005.
  7. G. Lunney, and D. W. Weaire, “The ins and outs of conical refraction” Europhys. News 37(3), 26–29 (2006).
  8. Spearman, Sir William Rowan Hamilton, Dictionary of Irish Biography
  9. G. O’Neill, Revising the legacy of William Rowan Hamilton, University Times, 9th February, 2022.