On 16 October 1843, walking the Royal Canal towpath to a Royal Irish Academy meeting with his wife, the Astronomer Royal of Ireland William Rowan Hamilton solved a problem that had stumped him for years: how to multiply in three dimensions. The answer needed a fourth term, and rather than lose it he cut the formula into the stone of the bridge with a penknife – i² = j² = k² = ijk = −1, the founding equation of the quaternions.
Be clear about what’s actually here. Broom Bridge is a plain two-arch limestone bridge carrying Broombridge Road over the Royal Canal in Cabra, in the Dublin suburbs. Hamilton’s original scratching wore away long ago. What you come to see is a plaque and the spot itself. If the maths means nothing to you, this is fifteen minutes and a photograph, not an afternoon. If it does, it’s a genuine pilgrimage.
The carving and the plaque
The quaternions Hamilton invented on this towpath turned out to be the clean way to describe rotation in three dimensions, which is why they sit, unseen, inside computer graphics, games engines, robotics and aircraft navigation today. The formula on the plaque is the same one. Taoiseach Éamon de Valera, a mathematician himself before politics, unveiled the commemorative stone on 13 November 1958, the year the bridge was officially renamed in Hamilton’s honour, though almost everyone still calls it Broom Bridge. The name itself is older and has nothing to do with the maths: it remembers William Broome, a director of the Royal Canal Company who lived nearby. The canal arch dates from around 1790, the railway arch from about 1845.
Read the plaque from the towpath and keep the path clear for other users.
The Hamilton Walk
Every 16 October, mathematicians retrace Hamilton’s route from Dunsink Observatory, where he worked, to the bridge, an event known as the Hamilton Walk, or Broomsday, in a nod to Joyce’s Bloomsday. Maynooth University’s Hamilton Walk archive explains the discovery and links to the annual walk. Check the event details before travelling if you plan to join.
Getting there and what’s nearby
Take the Luas Green Line to Broombridge; the stop is beside the bridge. The adjacent Broombridge railway station is on the commuter line into Connolly. Cyclists can follow the Royal Canal Greenway, which runs past the bridge.
There is no visitor centre at the bridge, which is a road crossing, nothing more. The real reason to linger in this corner of Dublin is next door: Glasnevin Cemetery and the National Botanic Gardens are about a kilometre and a half east and make the obvious pairing, turning a short stop at the bridge into a proper half-day. Read the plaque, trace the formula once, and walk on to the gardens.