History of quaternions

Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840,[1] but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.

In 1843, Hamilton knew that the complex numbers could be viewed as points in a plane and that they could be added and multiplied together using certain geometric operations.

Points in space can be represented by their coordinates, which are triples of numbers and have an obvious addition, but Hamilton had difficulty defining the appropriate multiplication.

According to a letter Hamilton wrote later to his son Archibald: Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edwin and yourself used to ask me: "Well, Papa, can you multiply triples?"

As mathematical terminology has grown since that time, and usage of some terms has changed, the traditional expressions are referred to as classical Hamiltonian quaternions.

In 1840, Olinde Rodrigues used spherical trigonometry and developed a formula closely related to quaternion multiplication in order to describe the new axis and angle of two combined rotations.

[3][4]: 9 The special claims of quaternions as the algebra of four-dimensional space were challenged by James Cockle with his exhibits in 1848 and 1849 of tessarines and coquaternions as alternatives.

The variety of fonts available led Hoüel to another notational innovation: A designates a point, a and a are algebraic quantities, and in the equation for a quaternion A is a vector and α is an angle.

[7] William K. Clifford expanded the types of biquaternions, and explored elliptic space, a geometry in which the points can be viewed as versors.

Thus in England, when Arthur Buchheim prepared a paper on biquaternions, it was published in the American Journal of Mathematics since some novelty in the subject lingered there.

For instance, Thomas Kirkman and Arthur Cayley considered the number of equations between basis vectors which would be necessary to determine a unique system.

Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold [the three imaginary units], why should you stop there?

He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion.

Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including physicists Murray Gell-Mann in 2002, Steven Weinberg in 2005, Frank Wilczek in 2007, and mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.