Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: a Historical Reassessment

Abstract

Quaternions, invented in 1843 by the famous mathematical physicist Hamilton, largely fell out of favor long ago, being replaced by vectors, which are direct descendants of quaternions. Quaternions have become popular recently because rotation operations are simpler with quaternions than with matrices. Therefore, quaternions are widely used in computer applications involving rotations. The relation between rotations and quaternions was established shortly after the invention of the latter, but recently it has been claimed that Hamilton misunderstood the relation, and that credit should be given to Rodrigues, who had published a paper on rotations in 1840. This claim has been repeated widely, but a review of original sources shows that it is based on an incomplete reading of Hamilton's work. The same review shows, however, that Gauss had derived results similar to those of Hamilton and Rodrigues in 1819, but they remained unpublished until 1900, and thus unknown to either of them. These matters are described in a self-contained way in this article.