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March 12, 2012 11:31:22 AM

The mathematician Ian Stewart knows the famous story about equations versus book sales. Stephen Hawking's publisher told him that every equation published in *A Brief History of Time* would halve the number of books sold. One equation got in: E = mc^{2}, and maybe it really did cut the sales of the book by half. If this rule is true, Stewart is in real trouble with his newest book, *In Pursuit of the Unknown: 17 Equations That Changed the World* (Basic Books). Readers who know his work, however, know that they are in good hands. Stewart has undoubtedly written mathematical papers that would be over the heads of us other mortals, but his books for the public on the problems, range, and philosophy of mathematics are clear, funny, entertaining, and educational. His seventeen chapters include some simple equations that everyone knows; that E = mc^{2} is here, simple to write and to memorize, but pointing to complexities that most of us cannot easily comprehend, even a hundred or so years after it was developed. Some of the equations, like Schrödinger's Equation, are full of Greek letters and only physics experts will recognize them. Throughout the book, however, Stewart shows that these are equations that run our lives in our technical age. The equations may be used professionally just by the egghead experts, but in a wider sense, we all use them, every day.

As part of his introduction, Stewart tells about the equals sign, =. We don't know who invented most older symbols in mathematics, and the new ones are the work of known inventors. The inventor of the equals sign, however, was Robert Recorde, and he used it in a book published in 1557. He had found it tedious to keep repeating "is equal to," so he used two parallel lines to stand for the words. He chose the two lines because, he said, "no two things can be more equal." Stewart writes in appreciation, "Recorde chose well. His symbol has remained in use for 450 years."

The first chapter here is on the old familiar a^{2} + b^{2} = c^{2}, the Pythagorean Theorem. This is pure math, straight from Euclid, and not (as are many of the equations here) from applied mathematics or mathematical physics. But that does not mean the Pythagorean Theorem is forever locked within the mathematicians' ivory tower. One of the pleasing parts of Stewart's book is that it shows practical aspects of the equation under discussion. It's beautiful to know how those squares of sides relate, and the proofs by Euclid and by countless others are fine objects of contemplation in themselves. The theorem, however, is the foundation of the eminently practical realm of trigonometry. If you can do trigonometry, you can do surveying and make maps, and triangulation for such maps was eventually performed in the late eighteenth century. The Pythagorean theorem also was a cornerstone for the invention of coordinate geometry, those x and y axes on which can be plotted points and lines as solutions to equations. Stewart goes on to explain that as part of Euclidean geometry, the theorem also played a role in understanding non-Euclidean geometries, which Einstein was to show defined gravity. As Stewart says, "It's an astonishing progression of events. Pythagoras's equation first came into being around 3500 years ago to measure a farmer's land. Its extension to triangles without right angles, and triangles on a sphere, allowed us to map our continents and measure our planet. And a remarkable generalisation lets us measure the shape of the universe. Big ideas have small beginnings."

This is typical of his development of concepts here. When he does get to E = mc^{2}, Stewart reflects about how Pythagoras helps understand relativity, because of light paths understood as sides of triangles. Stewart's chapter on relativity demonstrates that the well-known equation showing how energy and mass were just different forms of the same thing was not, as popular thinking would have it, directly responsible for the development of the atomic bomb, especially compared to Einstein's own direct political advocacy for it. I often think that I live in a Newtonian world and that the strangeness of relativity never really touches my life, but this is not true. All the equations here are not only mathematically important, they are important for engineering and commerce. For example, Stewart explains that, in complete accord with Einstein's predictions, satellites going around the Earth have their internal clocks speeded simply by the effects of Earth's gravity. Newton never would have known of such effects, and they are tiny, a few microseconds a day. For the GPS that everyone now uses, though, if this change in time were not taken into account, it would put you on the wrong street in the next ten minutes and the wrong town the next day.

All of the equations here have improved our understanding of how nature works, and have supplied reason to wonder at how consistently mathematics underlies everything. The basic equation for calculus is here, which is responsible for most of mathematical physics. Among the simplest of equations here is Euler's formula that shows how faces, edges, and corners of a solid shape are related. Like each simple formula, it has created complications, including the powerful pure mathematics of topology, which has implications for how DNA works and why planets may move in a chaotic way. Another simple one is i^{2} = -1, indicating that minus one, which should have no square root since it is negative, does have one, the imaginary number i. Although i may be called imaginary, it is essential for understanding waves and electricity, not to mention quantum mechanics. Chaos theory is here, with an equation that helps show how the flapping of that butterfly's wing may lead to a tornado later on; that's the most famous effect, but the equation models, for instance, how a population of creatures changes over generations if they have to be curtailed by limited resources or predators. Stewart gives clear explanations, but they are relatively deep for the non-mathematician. Many people who read this book will want to take long breaks between its pithy chapters, each of which has been expanded elsewhere into many volumes.

Equations are useful for explaining the world, but like any tool they can be misused. Stewart's final chapter is on the Black-Scholes Equation, invented in 1973 and since then used to analyze the changes of price of a financial derivative. Derivatives could thus be traded before they matured. It was a useful formula as long as it was applied only when market situations fit, but it was abused. Stewart makes clear that the formula didn't cause the 2008 - 2009 financial crisis, but abuse of it, along with financial and political ineptitude and lax regulation, made for a crash that didn't have to happen.

In an epilogue, Stewart reflects that most equations aren't important: "I write them down all the time, and believe me, I know." Here are some important ones, though, equations that run our world, always in ways that the inventors of the equations could never have predicted. It may be, Stewart writes at the end, that the cellular automata famously championed by Stephen Wolfram do a better job of explaining the universe than equations do, and maybe algorithms are going to be more important than equations. His engrossing book, showing the vital importance of equations not just for explanation but as causation of historic and social change, makes clear that it will be a long time before any other modeling becomes more important.

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