Hamilton (1844 and 1853)

[39] is the foundational work on quaternions, where William Rowan Hamilton first introduced them in 1844. On page 492, he introduces the quaternion, denoted as $a + i b + j c + k d$, or $(a, b, c, d)$. He quickly establishes the multiplication rules for the basis elements $(i, j, k)$ as

\[\begin{gathered} i^2 = j^2 = k^2 = -1; \quad ij = -ji = k; \quad jk = -kj = i; \quad ki = -ik = j. \end{gathered}\]

He doesn't include the common $ijk = -1$ explicitly, but it follows from the above.

He also introduces a parameterization:

\[a = μ \cos ϱ, b = μ \sin ϱ \cos ϕ, c = μ \sin ϱ \sin ϕ \cos ψ, d = μ \sin ϱ \sin ϕ \sin ψ.\]

He then says (emphasis in original)

I would call $ϱ$ the amplitude of the quaternion; $ϕ$ its colatitude; and $ψ$ its longitude. The modulus is $μ$.

(Note that Hamilton is using the archaic meaning of the word "amplitude", which was the standard term from complex analysis in the early 1800s, but which we would now call the "argument" or "angle"; he uses "modulus" where physicists and engineers would now frequently use "amplitude" or "magnitude".) This is similar to the axis-angle representation of rotations, where $ϱ$ is the (whole, not half) angle of rotation, and $(ϕ, ψ)$ are spherical coordinates for the axis of rotation — except that this would imply that $i$ corresponds to the $z$ axis if we take the name "colatitude" seriously. And the angle $ψ$ is measured from the $j$ axis, so we would probably identify $j$ with the $x$ axis. So $i, j, k$ correspond to the axes $z, x, y$.

There is no mention of "Euler angles" in Hamilton's introduction of quaternions.

[40] is Hamilton's later monograph on quaternions, published in 1853. In the table of contents — which is really a collection of summaries of each section — we find in Lecture II §x this statement (emphasis in original):

the symbols $i, j, k$ come to denote here three rectangular vector-units (supposed usually, in these Lectures, to be in the directions of south, west, and up)

This is a left-handed system. One sometimes sees claims that the quaternions are somehow "inherently" left-handed, but this is simply not true; Hamilton could have just as easily chosen a right-handed system, and later authors generally did.

This page was generated using Literate.jl.