The Irish man who discovered quaternion algebra

This month 173 years ago, William Rowan Hamilton had his ‘eureka’ moment as he walked along the Royal Canal

Over the past 11 years, our national Maths Week has promoted awareness and appreciation of mathematics in the world around us, with hundreds of events held around the country to demonstrate the creativity and wonder to be found in this subject.

October is the anniversary of the discovery of quaternion algebra, a discovery made by an Irish man, who graffitied his idea on a bridge along the Royal Canal in the 19th century.

William Rowan Hamilton was born in Dublin in 1805. He became interested in mathematics when he lost a competition against an American child, Zerah Colburn, who was incredibly quick at computing numbers.

While Hamilton was interested in many subjects, this polymath pursued his interest in maths and chose to study classics and science in Trinity College Dublin, where he later became a professor of astronomy.


Hamilton lived at Dunsink Observatory in Dublin and, on a walk to a meeting of the Royal Irish Academy in 1843, realised a mathematical idea that had been bothering him for some time.

Hamilton was fascinated by the link between geometry and complex numbers. Complex numbers are numbers that are made up of a real number (numbers we’re used to dealing with in everyday life to count or measure, like a temperature of -4 degrees) and an imaginary number (which can be written as a real number multiplied by an “imaginary” part, symbolised by “i”).

Complex numbers could be neatly manipulated with two-dimensional geometry and Hamilton wanted to extend this mathematics to three dimensions. His attempts at using three-dimensional numbers (or triplets) was proving impossible and he had reached an impasse in his work.

Exciting idea

On this autumnal walk on October 16th, he realised that if he jumped up a dimension and worked with numbers that had four components instead of three, he could solve this problem. He called these new numbers “quaternions”.

This was one of those very rare incidents in science where a breakthrough was captured in real time. Hamilton had been working on this “eureka” moment for years and when this exciting idea took hold, he couldn’t resist the urge to etch his new equation into the stone of Broom Bridge and give life to a new system of four-dimensional numbers.

Quaternions became the first significant number system that did not obey the laws of ordinary arithmetic. Their application to three-dimensional rotations proved extremely useful in physics and was also included in Erwin Schrodinger’s construction of the mathematics of quantum mechanics. Relating to our everyday lives, quaternions are also widely used to model three-dimensional graphics and systems in sat-navs and computer games.

Hamilton’s discovery of quaternion algebra demonstrates the importance of “blue sky” research. “Blue sky” refers to research where the applications of investigations are not immediately apparent, but are driven by curiosity.

Unfortunately, much scientific research funding now demands scientists to declare the immediate, real-world applications of their findings and this leads to some topics (particularly in mathematics) being underfunded.

Nobel laureate Saul Perlmutter, who won the Nobel Prize in Physics in 2011, recently said that current funding climates for scientific research negate opportunities of discovering the kinds of surprises we sometimes get from such “blue sky” thinking.

Major breakthroughs

If we restrict ourselves to only looking for what we expect to find, we might miss out on major breakthroughs. One example of this is computational technology which, two decades ago, was developed by mathematicians and astrophysicists to detect particular stars in the night sky.

Instead of analysing passing stars, this technology is now being used to focus on cells in the blood stream to detect specific dangerous cells in cardio patients.

In the same way that these stargazers never dreamed of their technology being used to save lives, it’s doubtful that Hamilton ever dreamed his equations would be used a century later by graphics engineers or deep space exploration satellites.

Hamilton’s curiosity about numbers has helped us develop ever more complex technology and imaging in our 21st century world.

Hamilton contributed to many elements of scientific research and was the first foreign member elected to the National Academy of Sciences in the United States.

He also gave us the term “scalar” and the modern meaning of the word “vector”, which are fundamental components of the language of mathematics used throughout the world.

William Rowan Hamilton is a scientist deserving of our national celebrations and homage.

Dr Aoibhinn Ní Shúilleabháin is an assistant professor in the UCD school of mathematics and statistics