Tait (1867 and 1868)

[49] provided an elementary introduction to quaternions, which was full of examples and applications. He was quite disinterested in mathematics per se; his focus was on applying mathematics in general — and quaternions specifically — to physics. Recall that, at this time, vector algebra had not yet been developed, so quaternions were one of the few available tools for dealing with three-dimensional quantities. But, as such, this book was a broadly influential introduction to quaternions for many years. (The tragic quaternion-vector wars would play out over the next few decades; even though Clifford had already unified the two notions, his early death would leave a void in the field filled by decades of carnage and confusion.)

In it, he defines the quaternion basis elements $(i, j, k)$ with multiplication rules

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

Note that Tait specifically writes expressions like $xi + yj + zk$ and $\xi i + \eta j + \zeta k$; these suggest that Tait does think of $i$ as corresponding to the $x$ axis, $j$ to the $y$ axis, and $k$ to the $z$ axis — evidently unlike Hamilton himself. Moreover, Tait even says

Suppose $i$ to be drawn eastwards, $j$ northwards, and $k$ upwards.

This is consistent with our standard right-handed system, with the $k$ axis pointing out of the page.

[50] discusses the decomposition of a general rotation into three rotations about successive axes, which we now call "Euler angles". Tait uses the symbols $(ψ, θ, ϕ)$ for the three angles. He actually does so just to express that set of rotations in terms of a single quaternion.

Here the vectors $i$, $j$, $k$ in the original position of the body correspond to

$\overline{OA}$, $\overline{OB}$, $\overline{OC}$ respectively, at time $t$. The transposition is effected by — first, a rotation $ψ$ about $k$; second, a rotation $θ$ about the new position of the line originally coinciding with $j$; third, a rotation $ϕ$ about the final position of the line at first coinciding with $k'$.

So this is what would probably now be called the $z-y'-z''$ convention for Euler angles $(ψ, θ, ϕ)$: each rotation is about an axis of the body, as carried along by the preceding rotations. Our own convention is stated the other way around, as rotations about fixed axes — $γ$ about $𝐳$, then $β$ about $𝐲$, then $α$ about $𝐳$ — but the two descriptions give the same quaternion product $e^{α𝐤/2}\, e^{β𝐣/2}\, e^{γ𝐤/2}$. Tait's angles are therefore ours in the same order,

\[(α, β, γ) = (ψ, θ, ϕ),\]

and it is only when his rotations are re-expressed about the fixed axes that the list reverses, to $ϕ$, $θ$, $ψ$.

Indeed, Tait goes on to derive (somewhat laboriously) the expression for the quaternion:

\[q = \cos \frac{ϕ + ψ}{2} \cos \frac{θ}{2} + i \sin \frac{ϕ - ψ}{2} \sin \frac{θ}{2} + j \cos \frac{ϕ - ψ}{2} \sin \frac{θ}{2} + k \sin \frac{ϕ + ψ}{2} \cos \frac{θ}{2},\]

which is exactly the same as our expression from from_euler_angles(ψ, θ, ϕ).

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