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Monday, February 28, 2011

While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him.

History of quaternions

Quaternion plaque on Brougham (Broom) BridgeDublin, which says:
Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication
i² = j² = k² = ijk = −1
& cut it on a stone of this bridge.
In mathematicsquaternions are a non-commutativenumber system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations. This article describes the original invention and subsequent development of quaternions.

Hamilton’s discovery

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. Hamilton sought to find a way to do the same for points inspace. Points in space can be represented by their coordinates, which are triples of numbers and have an obvious addition, but Hamilton had been stuck on defining the appropriate multiplication.
According to a letter Hamilton wrote later to his son Archibad:
Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edward and yourself used to ask me: “Well, Papa, can you multiply triples?” Whereto I was always obliged to reply, with a sad shake of the head, “No, I can only add and subtract them.”
On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. While he could not “multiply triples”, he saw a way to do so for quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:
i^2 = j^2 = k^2 = ijk = -1.\,
Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. He founded a school of “quaternionists” and popularized them in several books. The last and longest, Elements of Quaternions, had 800 pages and was published shortly after his death. See “classical Hamiltonian quaternions” for a summary of Hamilton’s work.

Precursors

Hamilton’s innovation consisted of expressing quaternionsas an algebra. The formulae for the multiplication of quaternions are implicit in the four squares formula devised by Leonard Euler in 1748; Olinde Rodrigues applied this formula to representing rotations in 1840.[1]

After Hamilton

After Hamilton’s death, his pupil Peter Tait, as well asBenjamin Peirce, continued advocating the use of quaternions. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell’s equations, were described entirely in terms of quaternions. There was a professional research association which existed from 1899 to 1913, the Quaternion Society, exclusively devoted to the study of quaternions.
From the mid 1880s, quaternions began to be displaced byvector analysis, which had been developed by Josiah Willard Gibbs and Oliver Heaviside.[2] Both were inspired by the quaternions as used in Maxwell’s A Treatise on Electricity and Magnetism, but — according to Gibbs — found that “… the idea of the quaternion was quite foreign to the subject.”[3] Vector analysis described the same phenomena as quaternions, so it borrowed ideas and terms liberally from the classical quaternion literature. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side effect of this transition is that works on classical Hamiltonian quaternions are difficult to comprehend for many modern readers because they use familiar terms from vector analysis in unfamiliar and fundamentally different ways.

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