Narrator: In Victorian London, Prime Minister William Gladstone visited the Royal Institution to witness a demonstration by the brilliant scientist Michael Faraday. After watching Faraday reveal the strange, invisible connections between electricity and magnetism, Gladstone, a man of practical politics and economics, was unimpressed. He asked, "But after all, what use is it?" Faraday, with what must have been a mix of frustration and foresight, allegedly replied, "Why, sir, one day you will tax it." That seemingly abstract science, described by a set of elegant equations, would eventually become the foundation for radio, television, and the entire modern communications grid—all of which are, of course, heavily taxed. This single exchange captures a profound truth: equations are not just scribbles on a page; they are the invisible engines that have shaped human civilization. In his book, In Pursuit of the Unknown: 17 Equations That Changed the World, mathematician Ian Stewart unveils the stories behind the formulas that have defined our history, from ancient geometry to the volatile world of modern finance.

Narrator: One of the first and most famous equations in history, Pythagoras's theorem, represents a monumental shift in human thought. For millennia, knowledge was tied to the physical world. The Babylonians, for instance, knew of the relationship between the sides of a right-angled triangle long before Pythagoras. A clay tablet from around 1000 BC, known as YBC 7289, shows a square with its diagonal marked with a stunningly accurate approximation of the square root of 2. They understood this specific geometric case because it was practical for surveying and building.

However, the formalization of this relationship into the algebraic equation a² + b² = c² was revolutionary. It detached the concept from any single, physical triangle and transformed it into a universal, abstract truth. This equation became a bridge between the tangible world of geometry and the abstract world of algebra, allowing for the calculation of distance in a coordinate system. It laid the groundwork for trigonometry, which became essential for navigation, astronomy, and mapping the world. By turning a physical observation into a symbolic rule, Pythagoras's theorem established a core principle: that the universe could be understood not just through observation, but through the language of mathematics.

Narrator: For centuries, the sheer scale of calculation was a barrier to scientific progress. Astronomers spent months on tedious, error-prone multiplication to map the heavens. Then, in the early 17th century, John Napier invented logarithms, a technique so powerful that the French mathematician Pierre-Simon de Laplace claimed it "doubled the life of the astronomer." By converting multiplication into addition, logarithms made complex calculations manageable and accelerated the pace of discovery.

Around the same time, another, even more profound tool for managing complexity was being forged. During the Great Plague of 1665-1666, a young Isaac Newton retreated to his family farm. In that period of intense, isolated thought, he developed calculus. Before Newton, mathematics was largely static, excellent at describing fixed shapes and quantities. Calculus provided a way to describe a world in motion. It gave humanity the language to talk about instantaneous rates of change—the speed of a falling apple at a precise moment, the changing slope of a curve, or the flow of heat through a metal bar. By providing a method to model and predict dynamic systems, calculus became the indispensable tool of physics and engineering, allowing us to understand everything from planetary orbits to the vibrations of a skyscraper.

Narrator: Newton did not stop with calculus. He used it to formulate his Law of Universal Gravitation, an equation that declared the same force pulling an apple to the ground also holds the Moon in orbit around the Earth. For the first time, a single mathematical law united the heavens and the Earth, revealing a universe governed by predictable, universal principles. This idea of a "clockwork cosmos" that could be understood through mathematics dominated scientific thought for over two centuries.

This unification was mirrored in the 19th century by James Clerk Maxwell. Building on Michael Faraday's groundbreaking experiments, Maxwell formulated a set of four equations that wove electricity and magnetism into a single, inseparable force: electromagnetism. His equations did more than just describe existing knowledge; they made a startling prediction. They showed that electromagnetic disturbances should travel as waves at a specific speed—the speed of light. This revealed that light itself was an electromagnetic wave and predicted the existence of an entire spectrum of invisible waves, including the radio waves that would later fulfill Faraday's prophecy to Gladstone.

Narrator: By the turn of the 20th century, the clockwork universe began to show cracks. Two revolutionary theories emerged that forced humanity to abandon its classical intuitions. The first was Albert Einstein's theory of relativity, encapsulated in the deceptively simple equation E=mc². This formula revealed a stunning equivalence between mass and energy, unlocking the power of the atom and forever changing warfare and energy production. But relativity did more; it fused space and time into a single four-dimensional fabric and described gravity as the curvature of this fabric, an idea so strange it took an observation of starlight bending during a solar eclipse to convince the world.

At the same time, physicists exploring the subatomic realm discovered a world that was even stranger. Schrödinger's equation, the central formula of quantum mechanics, described a world built on probability and uncertainty. It suggested that particles could exist in multiple states at once—a "superposition"—only collapsing into a definite state when observed. To illustrate the absurdity of this, Erwin Schrödinger devised his famous thought experiment of a cat in a box that is simultaneously alive and dead until the moment of observation. This quantum weirdness, once a philosophical puzzle, is now the foundation of all modern electronics, from lasers to the semiconductors in our phones and computers.

Narrator: The 20th century saw equations move beyond the physical world to describe more abstract concepts. Claude Shannon, a mathematician at Bell Labs, wanted to quantify the very idea of "information." His information theory provided a mathematical definition of information and established the fundamental limits of communication. This work led directly to the development of error-correcting codes, the unsung heroes of the digital age. These codes are why a scratched CD can still play perfectly and why NASA can receive clear signals from the Voyager space probe, now billions of miles away, using a transmitter with less power than a refrigerator light bulb.

Yet, just as information theory provided tools for digital certainty, another field revealed the universe's inherent unpredictability. Chaos theory, born from simple-looking formulas like the logistic equation, showed that deterministic systems—those following exact rules with no randomness—could still produce behavior that was complex, unpredictable, and seemingly random. This is the "butterfly effect," where a tiny change in initial conditions can lead to vastly different outcomes. Chaos theory destroyed the last vestiges of the simple, clockwork universe, showing that nature is a delicate, wobbly balance, full of complex dynamics that are impossible to predict far into the future.

Narrator: The power of equations to shape the world carries immense risk, a lesson learned with devastating consequences in the financial markets. In 1973, economists Fischer Black and Myron Scholes developed the Black-Scholes equation, a formula for pricing financial options. It was hailed as a breakthrough, a way to manage risk and create new, sophisticated financial products.

The equation worked by making several simplifying assumptions about the market, such as that price fluctuations follow a normal distribution, or bell curve. For a time, it seemed to have the Midas touch, unleashing a multi-trillion-dollar derivatives market and making fortunes for those who used it. But the model's assumptions were flawed. It failed to account for the rare, catastrophic events that real-world markets experience. When the housing market collapsed in 2008, the complex instruments built upon the Black-Scholes model imploded, triggering a global financial crisis. The equation that was designed to manage risk had, through its misuse and the false sense of security it created, amplified it to a catastrophic degree.

Narrator: Ian Stewart's journey through seventeen equations reveals that mathematics is far more than a classroom exercise. It is the primary language we use to understand, predict, and ultimately shape our reality. The book's single most important takeaway is that equations are not passive descriptions; they are active tools that, once unleashed, can build new worlds of possibility, from global communication networks to the digital economy.

However, the story of the Black-Scholes equation serves as a powerful, modern-day cautionary tale. The power of these formulas is a double-edged sword. They can illuminate the universe and build technological marvels, but they can also create blind spots and systemic risks when their limitations are ignored. As we stand on the cusp of developing new equations to govern artificial intelligence, climate models, and other complex systems, the ultimate question is not what our equations can do, but whether we have the wisdom to wield their immense power responsibly.