On 16 October 1843 the Irishman William Rowan Hamilton was walking with his wife along the Royal Canal in Dublin when he suddenly solved a problem that had tormented him for fifteen years. He took a penknife from his pocket and cut the formula into a stone of Brougham Bridge:

$$i^2=j^2=k^2=ijk=-1.$$

The carving wore away long ago, but there is now a plaque on the spot, and every year mathematicians walk to the bridge on 16 October. Rarely has a notation had such a well-documented birth.

A number that points

Hamilton’s problem came from an earlier discovery. In 1797 the Norwegian-Danish surveyor Caspar Wessel had shown that complex numbers can be read as directions in the plane. Multiplying by $-1$ is an about-turn ($180^\circ$); multiplying by the imaginary unit, which Wessel called $\varepsilon$ and we now call $i$, is a quarter turn ($90^\circ$). And the product of two directions has as its angle the sum of the angles:

$$i\cdot i=-1\quad\text{because}\quad 90^\circ+90^\circ=180^\circ.$$

It is the same miracle as Napier’s logarithms, now with angles: multiplying becomes adding. Hamilton wanted to do the same in three-dimensional space. For years he tried with three-part numbers, $a+bi+cj$, and never managed to make the calculations close. The solution that day on the bridge was to accept four parts, $q=a+bi+cj+dk$, and also to accept something that seemed absurd: that the order of multiplication matters.

$$ij=k,\qquad ji=-k.$$

It is the same surprise as Galois’s table, now with numbers. Hamilton called his creatures quaternions, from the Latin quaternio, “a set of four”.

Scalar, vector, versor

Hamilton was good with names. In an 1846 paper he split the quaternion into two parts and christened each:

The algebraically real part may receive the name of the scalar part, or simply the scalar of the quaternion. […] The algebraically imaginary part, being geometrically constructed by a straight line, which has, in general, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.

— William Rowan Hamilton, Philosophical Magazine (1846)

Scalar comes from scala, a ladder or graduated rule: a number marked on a scale. Vector comes from vehere, to carry: the vector is “the one that carries” a point to another. Later Hamilton called a quaternion that only rotates, without stretching, a versor (from vertere, to turn). All three words survived, even after quaternions went out of fashion.

The notation war

Hamilton and his Scottish disciple Peter Guthrie Tait wanted to rewrite all of physics in quaternions. James Clerk Maxwell, who in the 1860s was unifying electricity and magnetism, looked on with interest but was wary: to describe an electric field, the three parts $bi+cj+dk$ were enough, and the scalar part only got in the way. In the 1865 paper in which he presented the theory, Maxwell wrote the equations of electromagnetism component by component: twenty equations in twenty unknowns. In the Treatise on Electricity and Magnetism of 1873 he used part of the quaternion notation, but without adopting it in full.

Two people, without any arrangement, solved the problem at the same time. In the United States the physicist Josiah Willard Gibbs privately printed, between 1881 and 1884, a pamphlet for his students, Elements of Vector Analysis. In England the telegraph engineer Oliver Heaviside, who had taught himself everything he knew, did the same in the pages of the magazine The Electrician. Both threw away the scalar part of the quaternions, kept only the vector and created two different products, each with its own sign:

$$\begin{aligned}\vec a\cdot\vec b&\quad\text{dot product: gives a number;}\\ \vec a\times\vec b&\quad\text{cross product: gives a vector.}\end{aligned}$$

With this notation Maxwell’s twenty equations fitted into four:

$$\begin{aligned} \nabla\cdot\vec E&=\frac{\rho}{\varepsilon_0}, & \nabla\times\vec E&=-\frac{\partial\vec B}{\partial t},\\ \nabla\cdot\vec B&=0, & \nabla\times\vec B&=\mu_0\vec J+\mu_0\varepsilon_0\frac{\partial\vec E}{\partial t}. \end{aligned}$$

It is perhaps the most spectacular example of active notation in physics. The content is Maxwell’s, but now the symmetry between the electric and magnetic fields is in plain sight: two equations with $\nabla\cdot$ (on the left), two with $\nabla\times$ (on the right). It was by looking at this symmetry that physicists at the start of the twentieth century, Einstein among them, began asking the questions that led to relativity.

Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector Analysis; a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann.

— Peter Guthrie Tait, preface to the 3rd edition of An Elementary Treatise on Quaternions (1890)

Tait lost the war. Physicists and engineers adopted Gibbs’s and Heaviside’s vectors because they were easier to use on the problems they had. Quaternions were nearly forgotten for a century, until they came back through an unexpected door: today they are used to compute rotations in satellites, robots and computer games, precisely because they keep Wessel’s and Hamilton’s rule that turning and then turning is multiplying.

How do you write a vector?

The dispute left traces in the ways of writing a vector. The old texts on this site gathered some of them:

Author Year Writing
Jean-Robert Argand 1806 $\overline{AB}$
August Möbius 1827 $AB$
Hermann Grassmann 1844 $v_1e_1+v_2e_2+v_3e_3$
William Rowan Hamilton 1840s $\alpha=xi+yj+zk$
Josiah Willard Gibbs 1881 Greek letters and, later, bold letters
early twentieth century, especially in France the arrow: $\vec a$, $\overrightarrow{AB}$

Each script highlights something. Argand’s bar says “segment from $A$ to $B$”. Grassmann’s sum says “combination of three basic directions”, and it is the one that scales best to four, five or a hundred dimensions. The bold $\mathbf v$ of physics books is clean, but cannot be handwritten. The arrow, which can be handwritten, is the only one that draws the object itself: a vector is an arrow. It is an icon, in the sense Peirce will give us in chapter 2.

Exercise: order matters in quaternions

Using $ij=k$ and $ji=-k$, compute $(1+i)(1+j)$ and $(1+j)(1+i)$.

$$(1+i)(1+j)=1+j+i+ij=1+i+j+k,$$

$$(1+j)(1+i)=1+i+j+ji=1+i+j-k.$$

The results differ in the sign of $k$. And $ijk$? Grouping, $(ij)k=k\cdot k=-1$, exactly the formula on the bridge.

Exercise: Gibbs's two products

For $\vec a=(1,2,0)$ and $\vec b=(3,-1,0)$:

  • the dot product is a number: $\vec a\cdot\vec b=1\cdot3+2\cdot(-1)+0\cdot0=1$;
  • the cross product is a vector: $\vec a\times\vec b=(2\cdot0-0\cdot(-1),\ 0\cdot3-1\cdot0,\ 1\cdot(-1)-2\cdot3)=(0,0,-7)$.

The first measures how far the vectors point “the same way”; the second points out of the plane of the two, like the force in the right-hand rule. Two different signs for two different operations: that is what Gibbs gained by separating what the quaternion mixed together.

Sources: Michael J. Crowe, A History of Vector Analysis (1967) · Florian Cajori, A History of Mathematical Notations · William Rowan Hamilton, MacTutor · Oliver Heaviside, MacTutor.

Chapter index · ← Galois and Cayley: an algebra for things that are not numbers · Differential equations: the law in an anagram, ∂ and ∇ →