Clifford (1878)

Clifford [32] introduced Clifford algebras in 1878. The paper starts off with an introduction to quaternions — though interestingly from a "projective" perspective.^[1] This first section is a bit muddled from a modern perspective, but Clifford is using it to develop his ideas. Eventually, he gets to a definition that says

the symbols $ι₁, ι₂, ι₃$ may be taken to mean unit vectors along the axes.

But he actually settles on negative squares for these basis elements:

We are therefore obliged to write $ι₂² = -1$, and in a similar way we may find $ι₁² = ι₃² = -1$.

This differs from our modern treatments that usually assume that the basis vectors square to $+1$. To compensate for this, he goes on to identify Hamilton's quaternions with products of pairs of $ι₁, ι₂, ι₃$ that happen to be exactly the negative of modern definitions — or at least the ones we use here:

\[i = ι₂ ι₃, \quad j = ι₃ ι₁, \quad k = ι₁ ι₂.\]

Again, only in later sections does the exposition become clearer and more modern. There, he returns to quaternions briefly, and manages to describe them in exactly the way we would today, in terms of the even subalgebra of the Clifford algebra of three-dimensional space. (He even notes that they are isomorphic to the full Clifford algebra of two-dimensional space.)

Clifford — like Hamilton and Tait — failed to recognize the vital importance of conjugation (or "sandwiching") when using quaternions to represent rotations; presumably by analogy with complex numbers, he only considered left-multiplication by a single quaternion as the operation of interest. As we now understand it, that operation maps a vector into a more general odd-grade multivector, and so cannot be a rotation.

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