To Go Boldly: Hamilton, Quaternions and Space

Quaternions 1492 24 5 folio 62v

The European Space Expo is an exciting, interactive exhibition visiting Trinity College Dublin this week. Of course, space exploration would be unthinkable without the contribution of the Trinity College graduate, mathematician, poet and professor of Astronomy William Rowan Hamilton (1805-1865), perhaps best known as the inventor of Quaternions. Quaternions provide a mathematical notation for representing orientations and rotations of objects in three dimensions. They are crucial to flight dynamics and have been used in flight simulators and in the orbital mechanics of satellites.

His basic formula i² = j² = k² = ijk = -1 came in a ‘Eureka moment’ as he was walking along the Royal Canal at Cabra on 16 October, 1843. He named his new system ‘Quaternions’ because each number quadruple had four components, a totally new structure in mathematics. He described the discovery in a letter of 15 October 1858:

‘[I] felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since. I pulled out on the spot a pocket-book, which still exists, and made an entry, on which, at the very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be fifteen) years to come’

This manuscript (TCD MS 1492/24.5) is probably the ‘pocket book’ referred to in this letter, and constitutes his earliest workings of Quaternions, save for some light vandalism during the excitement of the discovery. ‘[I couldn’t] 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…’ The inscription is no longer there, but the site is now commemorated with a plaque and is the destination of the annual commemorative ‘Hamilton walk’.

Hamilton’s papers (TCD MS 1492 and 1493) are held in the Manuscripts & Archives Research Library. Prof. Rudraptna Ramnath, a consultant to NASA and Professor in the Department of Aeronautics and Astronautics at the Massachusetts Institute of Technology visited M&ARL last month to consult Hamilton’s notebooks and commented:

‘Quaternions, invented by Sir William Rowan Hamilton, have been used with great success in our work on spacecraft dynamics, in the Apollo and other space programs at Massachusetts Institute of Technology and C.S. Draper Laboratory. The use of quaternions leads to great advantages in the study of rotational motion of spacecraft.

Personally, it was a great joy for me to be able to read Hamilton’s manuscripts on quaternions and physics. [The Old Library] is a wonderful place with great treasures. I am grateful to the Trinity College Dublin mathematics and library staff for their courtesy. I also had a lot of fun in finding the bridge on which Hamilton carved out the governing laws of quaternion multiplication.’

Hosted by the Trinity College Dublin School of Physics, the spectacular Space Expo dome will be in Front Square and open to the public from 4th – 9th June.  Twitter: @SpaceExpoDublin

Estelle Gittins