Ian Stewart’s In Pursuit of the Unknown: 17 Equations That Changed the World presents 17 mathematical ideas that helped people describe, calculate, or predict things in fields ranging from geometry and astronomy to communications and finance. They are not 17 directly comparable equations, nor an objective ranking: the selection includes a theorem, mathematical tools, physical laws, a probability distribution, a measure, and broader theories and models. Their influence comes from what people could do with them, alongside measurement, engineering, and other human work—not from equations acting alone.
What the 17 entries have in common—and what they do not
The chapter sequence in Stewart’s book is a useful guide to its scope, not a definitive list agreed on by all historians or mathematicians. Some entries are reusable tools, such as logarithms and calculus; others describe particular domains, such as fluid motion or quantum mechanics. Some labels, including relativity, information theory, and chaos theory, cover families of ideas rather than one uniquely canonical equation.
An equation can condense a relationship into a form that supports calculation or prediction. But what it predicts depends on the model’s assumptions and domain: the familiar Pythagorean relationship, for example, applies to right triangles in flat Euclidean geometry, not every possible geometry. The examples below explain the role of each entry without treating the selection as a single measure of historical importance.
The 17 equations and ideas
1. Pythagoras’s theorem
For a right triangle in Euclidean geometry, the square of the hypotenuse equals the sum of the squares of the other two sides: a2 + b2 = c2. The relationship lets you calculate a missing side from the other two and underpins familiar ways of measuring straight-line distance. Its assumptions matter: spherical geometry, for instance, does not use the same rule in the same way.
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2. Logarithms
A logarithm answers the question, “To what power must this base be raised to produce this number?” Because logarithms turn multiplication into addition and exponentiation into multiplication, they make some calculations easier to organize. They also provide a language for quantities that span very different scales. Logarithms are a mathematical tool, rather than a single physical law.
3. Calculus
Calculus brings together methods for describing change and accumulation. Differential calculus represents how quickly a quantity changes; integral calculus represents accumulated quantity, such as the total built up from many small contributions. Together, these methods let people formulate and solve problems involving continuously changing quantities. “Calculus” names a family of methods, not one equation.
4. Newton’s law of gravity
In its familiar classical form, the law says that two masses attract with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. It gives a mathematical model for gravitational attraction and allows calculations about how that force varies with mass and separation. It is a classical model; it should not be confused with every later account of gravity or treated as universally sufficient in every setting.
5. The square root of minus one: complex numbers
The symbol i is defined by i2 = −1. Introducing it extends the number system beyond the real numbers, making it possible to express solutions and relationships that cannot be represented using real numbers alone. Complex numbers are not merely a special-purpose curiosity: they provide a versatile mathematical language used across multiple technical fields.
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6. Euler’s formula for polyhedra
For a convex polyhedron, Euler’s formula relates its vertices, edges, and faces: V − E + F = 2. It shows that these counts are connected by a stable structural relationship, rather than varying independently. The result is a bridge between geometric objects and a more general study of shape and connectivity; the stated formula has conditions, including the convex-polyhedron setting.
7. The normal distribution
The normal distribution is a bell-shaped probability distribution described by its center and spread. It gives a mathematical way to model how values are distributed around an average when the model’s conditions are appropriate. It is a distribution, not a universal rule that real-world measurements must follow; choosing it requires attention to the data and assumptions involved.
8. The wave equation
The wave equation describes how a wave-like quantity changes across space and time. It provides a framework for calculating how disturbances propagate in systems where its assumptions apply. “The wave equation” can refer to particular formulations for particular settings, so it is more useful to think of it as a model of wave propagation than as one formula that governs every kind of wave in every medium.
9. The Fourier transform
The Fourier transform expresses a signal or function in terms of its component frequencies. This changes the way a problem is represented: patterns that are difficult to see in the original signal may become easier to analyze by frequency. It is a mathematical method for analyzing signals, not by itself a claim about what produced them.
10. The Navier–Stokes equation
Navier–Stokes equations model the motion of fluids by relating fluid velocity and changes in that motion to forces and fluid properties. They give a mathematical framework for studying flows, but a model’s output depends on its formulation, conditions, and how the equations are solved. The name commonly refers to a family of equations for fluid dynamics rather than a single context-free expression.
11. Maxwell’s equations
Maxwell’s equations describe how electric and magnetic fields relate to electric charge and current and to one another. Taken together, they provide a compact framework for classical electromagnetism. Their significance lies in the relationships they express and the calculations they enable; any particular prediction still depends on the situation being modeled and the relevant conditions.
12. The second law of thermodynamics
The second law constrains how entropy behaves in thermodynamic processes. In broad terms, it gives a direction to processes that would otherwise be permitted by energy accounting alone. It is a physical principle, not simply another name for energy conservation, and its precise mathematical expression depends on how the system and process are described.
13. Relativity
Relativity is a broad framework for describing space, time, motion, and—in general relativity—gravity. It is not one equation. Different formulations answer different questions, so compressing all of relativity into a single famous expression would conceal important distinctions. The book’s entry points to a wide-reaching theory, rather than one standalone formula with one narrow use.
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14. Schrödinger’s equation
Schrödinger’s equation describes how a quantum state changes. It is central to one mathematical framework for quantum mechanics, connecting a system’s state with its evolution. It does not make quantum systems behave like classical objects; understanding what a solution means also requires the broader concepts and interpretation of quantum theory.
15. Information theory
Information theory supplies mathematical tools for describing information and uncertainty, including ways to quantify how much uncertainty is associated with possible outcomes. It helps frame questions about representing and communicating information. The chapter label refers to a field and its concepts, not one equation that alone explains every communication system.
16. Chaos theory
Chaos theory studies certain nonlinear systems whose behavior can be highly sensitive to their starting conditions. In such systems, a small difference in the initial state can grow into a substantial difference in later behavior, complicating long-range prediction even when the system follows definite rules. Chaos is not a synonym for randomness, and the label covers a body of ideas rather than one universal equation.
17. The Black–Scholes equation
The Black–Scholes equation is a mathematical model associated with pricing financial options. It represents a particular financial problem in equation form, making its assumptions part of the result rather than an optional detail. It is a model in finance, not a general law guaranteeing a correct market price in every circumstance.
How to compare their influence
These entries are most meaningfully compared by asking what problem each helps represent, what it allows someone to calculate or predict, and where its assumptions limit its use. A general mathematical tool such as calculus can serve many kinds of problems; a domain-specific model such as the Black–Scholes equation addresses a narrower one. That difference is not a score of historical importance. The book’s 17 examples are a curated route through mathematics’ connections to science, technology, and human activity, not proof that each idea independently caused a particular invention or change.
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