Sir William Rowan Hamilton(1805–1865)

William Rowan Hamilton was born at 36 Dominick Street in Dublin on the third of August 1805, fourth child of Archibald Hamilton, a Dublin solicitor, and Sarah Hutton. He was raised from three by his uncle James Hamilton, the curate of Trim parish in County Meath, who was a phenomenal linguist; under his uncle's tuition he had learned by ten Latin, Greek and Hebrew, by twelve Italian and French, and by thirteen Persian, Arabic, Sanskrit, Hindustani, Bengali, Marathi, Syriac and Chaldean. He was identified as a mathematical prodigy at fifteen on his independent reading of Newton's Principia and Laplace's Mécanique céleste (in which he found a small algebraic error that he corrected and reported to John Brinkley, the Astronomer Royal of Ireland), and took the place at Trinity College, Dublin, in 1823 at eighteen on the open scholarship in classics and mathematics.

He took every undergraduate prize at Trinity (the Vice-Chancellor's Prizes, the Bishop Law Prize, the optime in classics and the optime in mathematics, the four-year undergraduate distinction unmatched in the history of the College), and on the basis of his undergraduate publications was elected Andrews Professor of Astronomy at Trinity College and Royal Astronomer of Ireland on the sixteenth of June 1827 in his twenty-second year, before he had taken his BA degree. He took the chair (with the associated directorship of the Dunsink Observatory five miles north-west of Dublin) and held it for the next thirty-eight years to his death. He was knighted by the Lord Lieutenant of Ireland in 1835 at thirty for his services to mathematical physics.

His central single physical work was the Hamiltonian formulation of classical mechanics, published in the two papers On a General Method in Dynamics in the Philosophical Transactions of the Royal Society in 1834 and 1835. The Hamiltonian formulation rewrote the equations of classical motion in terms of a single scalar function H (the Hamiltonian, the total energy of the system expressed in canonical position-and-momentum variables), reducing the standard Newtonian equations of motion to a pair of first-order partial differential equations. The formulation was an elegant reframing of classical mechanics; its full importance became clear only seventy years later, when Erwin Schrödinger, Werner Heisenberg and Paul Dirac built the foundational equations of quantum mechanics (1925 to 1928) on the Hamiltonian function, in the form (the Hamiltonian operator) on which every modern quantum-mechanical calculation rests.

His most-celebrated single discovery came on the afternoon of Monday the sixteenth of October 1843. He had been working for thirteen years on the algebraic problem of how to extend the complex numbers (which work in two dimensions) into a three-or-four-dimensional algebra. He had concluded that no three-dimensional algebra was possible, but a four-dimensional one might be, if the commutative law of multiplication (ab = ba) was abandoned. On the afternoon walk from the Dunsink Observatory along the Royal Canal towpath into Dublin with his wife Helen, the four-dimensional algebra suddenly resolved in his mind: a single real component plus three perpendicular imaginary components i, j and k, with the multiplication rule i squared = j squared = k squared = ijk = minus one. He stopped at the Brougham Bridge over the canal and carved the fundamental equation into the stonework with the pen-knife he was carrying. The bridge stonework was renewed in the 1950s and the original incision lost; a Royal Irish Academy memorial plaque on the bridge today marks the spot.

The quaternion algebra was the first non-commutative number system in mathematics. It was the foundational discovery of modern abstract algebra, the parent of the linear-algebra-and-vector-calculus framework that James Clerk Maxwell built on for electromagnetism in 1862 to 1865, and is the modern computational foundation of every three-dimensional rotation calculation in computer graphics, robotics, spacecraft attitude control and quantum-mechanical spin. He published the two-volume Lectures on Quaternions (1853) and the posthumous Elements of Quaternions (1866) as the foundational textbooks of the new algebra. He served as President of the Royal Irish Academy 1837 to 1846, was elected a Foreign Associate of the United States National Academy of Sciences in 1864 (the first foreign honoree), and died at the Dunsink Observatory on the second of September 1865 in his sixty-first year. The Hamilton name in modern mathematics and physics carries the weight of Hamiltonian mechanics and the Brougham Bridge quaternion equation.