Hamilton and the square root of minus one

Abstract

Quaternions, objects consisting of a scalar and a vector, sound like a mysterious concept from the past. In the nineteenth century, the theory of quaternions was praised as one of the most brilliant achievements in mathematical physics. The originator of this theory, Hamilton, surely one of the greatest scientists in that area, spent about 18 years in discussing all kinds of algebraic and geometric properties of quaternions. His research was communicated to the Philosophical Magazine in three series of papers comprising a total of 29 contributions. In this commentary, these three series of papers are revisited concentrating primarily on the algebraic properties of quaternions.

Keywords:

• geometry
• modelling
• statistical physics
• structure
• theoretical
• transformations

Acknowledgments

I am grateful to Dr S.L. Altmann for a critical reading of the manuscript.

Notes

1 An exceedingly interesting history accompanies the theory of quaternions as was pointed out in the introduction of Ref. [30], in which also a short biography of Hamilton is compiled.

2 Quotations are indicated by italics, original equations by his name.

3 Mostly likely derived from the Latin word ‘quaternio’ meaning a set of four.

4 Monday, 16 October 1843, accompanied by Lady Hamilton, he was walking past Broome Bridge in Dublin, when suddenly he realized that three imaginary units were needed to solve the problem he had in mind. He abruptly stopped his walk and carved Equation (Hamilton 6Hamilton 6 ) in the stone of the bridge [30].

5 Such a theory has to include many more famous names such as Euler, Rodrigues, etc. and of course generations of group theoreticians.