Biography

Mathematical studies

Optics and Dynamics

Quaternions

Quotations

Hamilton Key Dates

Hamilton Links

Graphics

Math News

## Quaternions

On 16th of October 1843, a bright October day, Sir William Rowan Hamilton discovered the numbers later called Quaternions.

He had been trying to find a satisfactory way to multiply three dimensional points for several years, in such a way as to allow division. The gordian knot was battered when the idea of using four dimensions instead of three came into his mind, as he walked with his wife by the Royal Canal. Because this incident is precisely located in time and place, the event now is very well-known in the international mathematical community, and people from all over the world know about "Hamilton's Bridge". When he made the discovery, Hamilton was resting under the Brougham (or Broom) Bridge, so he took out his penknife, and scratched the fundamental formula:

**i ^{2} = j^{2} = k^{2} = ijk = -1**

into the stone right there. No trace of this can be found today, but in 1958 a commemorative plaque was erected reminding the discovery and showing the formula.

In the theory which Sir William Hamilton present to the Academy a few weeks later in November, 1843, the term quaternion was used to name a certain quadrinomial expression, of which one part was called (by analogy to ordinary algebra) the real part, while the three other parts made up together a trinomial, which (by the same analogy) was called the imaginary part of the quaternion. The square of the former part being always a positive, but the square of the latter part (or trinomial) being always a negative quantity.

This incident might be as outstanding as Archimedes' discovery while bathing, or Poincare's while stepping off a bus. The Royal Canal and the Bridge are still existing and tangible, this makes the scene so well imaginable, and let mathematicians of all times pilgrim to Broom Bridge.