A Quaternion Journey

$Mathematicians$

Hamilton was aware that the Quaternions satisfies the listed axioms. The following list locates several of Hamilton's proofs (all from Chapter 1). As mentioned, he used vector and versor notation in Elements of Quaternions.

Definition of multiplication: Section 10, Paragraphs 181, 182.

Proof of (1a): Section 12.

Proof of (1b): Section 10, Paragraphs 181, 182.

Falsity of (2b): Section 10, 183.

Proof of (3b): Section 10, 183.

Proof of (5b): Section 11, 187.

Proof of (6): Section 13.
Hamilton had hoped that his new discovery would be more than only an elegant algebra. In fact, he and his peers found many physical applications for the Quaternions.

Hamilton was convinced that in the Quaternions he had found a natural algebra of three-dimensional space. The quaternion seemed to him to be more fundamental than any coordinate representation of space, because operations on quaternions were independent of any given coordinate system. The scalar part of the quaternion caused difficulty in any geometrical representation and Hamilton tried without notable success to interpret it as an extra-spatial unit. The geometrical significance of the quaternion became clearer when Hamilton and A. Cayley independently showed that the quaternion operator rotated a vector about a given axis [Hankins 91].

Hamilton himself divided the quaternion into a real part and complex part which he called a vector. The multiplication of two such vectors according to the rules for quaternions gave a product consisting again of a scalar part and a complex part. If

and

The scalar part, which he wrote as , is recognizable as the negative of the modern dot product. The vector part, which he wrote as , is recognizable as the modern cross product. Nevertheless, Gibbs' and Heaviside's vector analysis was a more useful tool for solving problems in applied mathematics. The once-active theory of Quaternions has since been relegated to the standard skew field example in algebra texts.