In Pursuit of the Unknown: 17 Equations That Changed the World
“In Pursuit of the Unknown is a really fun read. . . . Ian Stewart is a genius . . .”
Conventional wisdom has it that adding equations to a book reduces the potential number of readers. What can be said about a book dedicated to equations? Ian Stewart knows that “equations are too important to be hidden away,” that “equations have hidden powers. They reveal the innermost secrets of nature.” And pure mathematics “reveals deep and beautiful patterns and regularities,” while applied mathematics “encodes information about the real world.”
In Pursuit of the Unknown is a really fun read. Ian Stewart takes 17 equations and instead of providing a tried-and-true linear narrative provides a more interesting, quirky, nonlinear one, going from formula to back-history to a few relevant digressions, to the formula’s impact on the world with a few more digressions along the way.
Speaking of quirky, this review must start with the introductory quote, which is Robert Recorde’s commenting on his invention of what we now call the “equals sign:”
“To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of parallels, or gemowe lines of one lengthe: ====, bicause noe .2. thynges, can be moare equalle.” —Robert Recorde, The Whetstone of Witte, 1557
—Note that gemowe comes from the old word for Gemini, meaning twins.
The book covers one equation per chapter:
1. Pythagoras’ Theorem
4. Newton’s Law of Gravity
5. The square root of –1
6. Euler’s formula for polyhedra
7. The normal distribution
8. The wave equation
9. The Fourier transform
10. The Navier-Stokes equation
11. Maxwell’s equations
12. The second law of thermodynamics
13. Relativity (E=MC^2)
14. Schrodinger’s wave equation
15. Information theory (Shannon)
16. Chaos theory
17. The Black-Scholes equation
Note that Newton genius that he was gets two entries, Calculus and the Law of Gravity, and the above list does not provide the exact chapter titles, which are sometimes punny in their own write. Each chapter is written in 15- to 30-page bite sized chunks. Each chapter’s first page displays the equation annotated with three summary paragraphs:
1. What does it tell us?
2. Why is that important?
3. What did it lead to?
And each chapter provides a serious in-depth history of that equation written for the non-scientist, lightened with digressions, anecdotes, jokes, and puns.
The chapter on Pythagoras' theorem points out evidence that the Babylonians knew about the 3-4-5 right triangle 1000 years before Pythagoras, and Pythagoras' equation was extended as a concept from right triangle to any triangle, useful for calculating distances and triangulation in surveying and maps.
In history, one idea leads to another. Geometry leads to trigonometry to algebra to coordinate geometry to Cartesian coordinates. Pythagoras leads to Euclid and Euclidean geometry, to non-Euclidean geometry and to the metrics of curved surfaces, to Gauss and the shape of space, to the curve of three-dimensional space and Riemann manifolds.
The chapter on logarithms starts out by providing a history of numbers, which began as tallying marks on the outside of clay envelopes dating back to 8000 BCE Mesopotamia.
From the history of numbers the author digresses to arithmetic, addition and subtraction and the discovery of ways to speed up calculations for multiplication and division that were made long before the invention of the adding machine. Logarithms were originally used to reduce the effort of multiplication and division as well as calculating fractions, square roots, and complex numbers. Simplification worked by first converting numbers to logarithms by use of tables and then adding and subtracting the logarithms (easier than multiplying or dividing), and then converting the logarithms back by using tables.
John Napier, a Scottish Laird first invented the use of logarithms for calculation, around 1614, and logarithms were improved by Henry Briggs, the First Professor of Geometry at Gresham College. Henry created the first table of base-10 logarithms, while the first slide rule that used logarithms was created by William Oughtred in 1630. This reviewer used a slide rule throughout high school and into college up to when the price of a calculator dropped from $300 to $30.
Why do we need logarithms now that we have inexpensive calculators and computers? Logarithms may no longer be used to speed calculations but are useful for modeling natural phenomena; for example, we measure radioactive decay by logarithmic half-life, earthquakes by the logarithmic Richter scale, and we perceive sound logarithmically and measure it by the decibel.
Calculus was invented by Isaac Newton and also independently by Gottfried Leibniz as a tool used to model the natural world. The author digresses to the controversy of who invented calculus first and firmly stands with Newton claiming that Leibniz did little with his invention while Newton used it to understand the Universe.
The power of calculus is that starting with 1000 years of astronomical observations and theories, the movement of the bodies of the Solar System could be synthesized into one simple math formula. Newton used calculus as the basis for describing his planetary orbit theory, though it is interesting to note that in Newton’s Principia, the laws of motion, motivated by the results of calculus were instead explained not by calculus but by use of classical geometric argument.
Any claim to understanding the universe may appear to infringe on the domain of those who claim to represent God. At the time of Newton’s presentation of calculus to the world, there was a religious attack on it by George Berkeley, Bishop of Cloyne. His objection was that calculus took God out of mathematics and so promoted a materialist view of the world.
The chapter on calculus also provides a history of astronomy from Aristarchus of Samos to Ptolemy, Copernicus, Kepler, and Brahe. There’s also a digression on how science works, on Poincaré and the three-body problem. Ian Stewart’s digressions often digress. He moves on to discuss science fiction and space wormholes then wanders back to Newton and gravity, addressing the closest thing to real-universe wormholes: low-energy efficient transfer orbits for manmade satellites.
The story of the square root of –1 goes back to 1545 Renaissance Italy with calculations performed by Girolamo Cardano, here called a genius and rogue. Before the acceptance of imaginary numbers by mathematicians, if you got a solution with the square root of a negative number it was believed that there was no solution. That kind of number could not exist, for where would the square root of minus one sit on a number line?
In 1673 John Wallis (who worked on a precursor to calculus) put the square root of negative numbers on the Y-axis of a plane diagram, orthogonal to the natural number line on the X-axis, in effect putting imaginary numbers into a completely different dimension. Having imaginary numbers added a new concept to the framework of existing mathematics, imaginary numbers can be used to represent many real physical concepts, including electric and magnetic fields, and air and fluid flow.
The author digresses in this chapter to cover the history of algebra starting from Diophantus circa 250 ACE, to the evolution of mathematic notation of third-order polynomials starting from the use of Greek symbols and ending with Euler’s notation in 1765, which is to all intents and purposes the modern form we use today.
The chapter on Euler’s formula for polyhedra is punningly titled “Much ado about knotting.” The importance of the formula for polyhedra is that it opens the door from geometry to topology, and the clearest introduction to topology that this reviewer has ever read.
Topology is the mathematics of shape—that a torus or donut shape, which is like the shape of a mug with a handle but fundamentally different from the shape of a sphere. Topology includes surfaces, knots and links, its study leads to Riemann’s differential geometry and the identification of topological invariants, which are geometric properties that are unchanged by continuous deformations, the study of which goes back to Gauss’ work on magnetism, on how magnetic and electric field lines link with each other. Gauss’ assistant, Mobius (yes, that Mobius) was the one who introduced the word “topology” to the world of mathematics.
Applications of topology include biology in understanding the workings of DNA, and quantum field theory (see The Shape of Inner Space by Shing-Tung Yau and Steve Nadis previously reviewed here).
The chapter on normal distribution addresses the mathematical concepts behind the bell-shaped curve. The bell-shaped curve comes from probability distributions, which are ways of measuring statistical patterns of chance events. Probability doesn’t make exact predictions but only states which figures are most likely to occur.
The mathematics of probability started with a gambling scholar, Girolamo Cardano calculating the odds and fairness in wagers, in order to know the odds better than his opponents, (which is the safest way to cheat).
The bell shape in normal distribution comes from drawing a line connecting the data points. In science the goal of line drawing, or “curve-fitting” is to minimize the errors of the distance between the line and the points when a line to retain a smooth shape needs to be drawn between points. The bell shaped curve is the shape or “distribution” suited to the mean “normal” of many repeated observations from many kinds of studies.
The bell curve started showing up in the social sciences 1835 when Adolphe Quetelet used probability theory after finding that people en masse behaved more predictably than individuals. In this chapter the author digresses to the measurement and meaning of IQ, and Ian expresses his disapproval of its abuse by social science “. . . in the social sciences models are often little better than caricatures.” And, “[i]t is a mistake to think about a mathematical model as if it were reality” and “. . . cleverness, intelligence and wisdom are not the same.”
The chapter on Navier-Stokes covers the mathematics behind fluid flow. Disturbed flow or “turbulence” can get so complicated that statistical models must be used, and Navier-Stokes is used to model complicated real world phenomena efficiently. There are many physical phenomena that act as fluids and Navier-Stokes can be used for calculating airflow, blood flow, and climate change.
The computational complexity of turbulence is similar to computational complexity in the Traveling Salesman problem (previously reviewed In Pursuit of the Traveling Salesman by William J. Cook), as such the Clay Institute offers a $ 1 million dollar prize for a guarantee that solutions of the Navier-Stokes equation exist for three dimensions.
This review provides only a sampling of what’s contained in In Pursuit of the Unknown, and, again, the author’s focus is not on the equations so much as what those equations tells us about the world.
In Pursuit of the Unknown: 17 Equations That Changed the World is more about the wonders of scientific discovery than it is about anything else, and Ian Stewart is a genius in the way he conveys his excitement and sense of wonder across. He has that valuable grasp of not only what it takes to make equations interesting, but also to make science cool.