Friday Reflection: Taking a Vector for a Walk
Robyn Arianrohd on the serious playfulness of mathematics
Throughlines
Our conversation with Robyn Arianrhod begins not with abstraction, but with a man and woman taking a walk. William Rowan Hamilton, on his way to preside over a meeting of the Royal Irish Academy, crossed Broome Bridge and finally saw—after years of struggle—the idea that had eluded him. In a moment equal parts drama and mild desperation, he pulled out his penknife and carved his formula into the stones of the bridge. This was the birth of quaternions, but also something broader: the beginning of a new way of understanding direction, motion, and space.
Robyn uses that moment as the hinge of Vector, but our conversation quickly makes clear that Hamilton’s epiphany was the crest of a much older wave. Long before him, Thomas Harriot, René Descartes, Isaac Newton, and others wrestled with questions that were easy to state and maddening to formalize: how do you locate a body in space? How do you describe its direction? How do you combine forces that pull in different ways? The mathematical tools we learn as children—arrows on paper, coordinate axes, the tidy “i-hat, j-hat, k-hat” of three dimensions—are the final products of centuries of confusion, argument, intuition, bitterness, and occasionally inspired leaps.
The nineteenth century produced most of the key players in this story. Augustus De Morgan, James Sylvester, Josiah Willard Gibbs, Oliver Heaviside, and James Clerk Maxwell were all grappling with how to represent the physical world mathematically. Their personalities could not have been more different, but they shared one frustration: there was no single, unified mathematical language to describe forces, fields, and directions. Hamilton’s quaternions were one attempt—bold, elegant, and slightly unwieldy. Gibbs and Heaviside built the vector calculus most scientists actually use. Maxwell wrote equations whose beauty still astonishes. And Einstein took developments in tensors to develop his own theory of relativity.
Robyn draws out the human drama behind these developments. Far from a tidy progression, the story of vectors is a long argument over symbols, diagrams, metaphors, and meanings. Mathematicians were trying to capture something inherently slippery: the “geometry of physical intuition.” The math had to feel right as well as work right. And it had to be expressive enough to describe everything from a falling apple to the propagation of an electromagnetic wave.
What makes all this more striking is how universal vectors have become. As we discussed, listeners traveling for Thanksgiving might not realize that GPS, aviation, weather forecasting, engineering, computer graphics, and gravitational wave detection all rely on vector mathematics. What began as a mathematical puzzle—how to describe a point moving in space—now undergirds the infrastructure of modern life. Yet none of these mathematicians worked on their theories because they were planning on how to design a bridge, let alone a gravity detector. They were engaged in serious playfulness.
Reflection Questions
Why did Hamilton’s Broome Bridge moment matter so much for later science and engineering?
What does the long struggle to formalize vectors reveal about how mathematical ideas evolve?
Why were quaternions both brilliant and impractical?
How did De Morgan, Gibbs, and Heaviside shape the mathematics we now take for granted?
In what sense are vectors “intuitive,” and in what sense are they deeply counterintuitive?
Why do certain mathematical notations succeed while others fail?
How might the history of vectors shape how we teach mathematics today?
Which parts of this story—scientific or personal—surprised you the most?
How do vectors illustrate the interplay between abstraction and physical reality?
Where else in modern life do we rely on mathematical concepts we rarely think about?
For Further Investigation
Robyn Arianrhod, Vector: A Surprising Story of Space, Time, and Mathematical Transformation (University of Chicago Press, 2024)
—, Thomas Harriot: A Life in Science (Oxford University Press, 2019)
—, Seduced by Logic: Émilie Du Châtelet, Mary Somerville and the Newtonian Revolution (Oxford University Press, 2012)
William Rowan Hamilton’s Papers (Royal Irish Academy)
Augustus De Morgan, Trigonometry and Double Algebra (1849)
James Clerk Maxwell, A Treatise on Electricity and Magnetism (1873)
George E. Smith, “Maxwell’s Influence on the Development of Vector Calculus” (Cambridge Companion to Maxwell)
Oliver Heaviside, Electromagnetic Theory, 3 vols. (1893–1912)
J. Willard Gibbs, Vector Analysis: A Textbook for the Use of Students of Mathematics and Physics (1901)
Related Episodes
“The Curiosities of Thomas Harriot”—my first conversation with Robyn about a remarkable Elizabethan scientist and mathematician that no one has ever heard of.
“Generations of Reason”—the story of a remarkable English extended family, and its pursuit of mathematics, science, and the divine.
“Free Creations”—Albert Einstein without the myths.
Tags: Vectors; Robyn Arianrhod; History of Mathematics; Quaternions; Hamilton; Maxwell; Physics; Historically Thinking