Or high Matheis, with her charm severe
Of line and number, was our theme; and we
Sought to behold her unborn progeny,
And thrones reserved in Truth's celestial sphere:
While views, before attained, became more clear;
And how the One of Time, of Space the Three,
Might, in a chain Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,
Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again,
In a later age, and occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.
—William Rowan Hamilton 
Back in the first half of the nineteenth century, equations in physics were still all written out in terms of their Cartesian components. This is how Maxwell first wrote his equations of electrodynamics.1 Quaternions were one of the first attempts to develop a coordinate-free system of mathematics. Unfortunately, as the Latin root of their name2 tells us, quaternions have four components rather than three. So they are a bit awkward to use to represent ordinary three-dimensional physical quantities.3 This prompted Heaviside and Gibbs to develop the vector system that is widely used in physics and engineering today.4 Heaviside and Gibbs basically removed “by force” the fourth component of the quaternions. This then led them to dismember the original quaternionic product into two different kinds of products, known nowadays ...