The word "equation" has a scary ring. Equations are filled with numbers, letters and strange undecipherable symbols. It is a word we associate with the high school math class that we loathed and the smarty-pants kid who always raised his hand and knew the answers. For these reasons, we avoid equations. And we are fortunate that they play no part in our lives.
Except that they do, asserts Robert Crease in his book "The Great Equations." Crease is a philosophy professor at Stony Brook University who has written extensively about science and mathematics. According to him, equations have shaped modern society and have had a huge effect on all of us. To substantiate that claim, he tells the fascinating back stories of 10 equations and shows how modern life is built on them.
He includes, for example, one of the best-known equations in the world (after Einstein's E=mc2, which is also in the book). It is Isaac Newton's law of universal gravitation. This equation tells us that all bodies attract all other bodies and that the degree of attraction is related to the masses of the bodies divided by the square of the distance between them.
The equation shows that gravitational attraction is the force that causes objects to fall to Earth. Until Newton came along, the theory that best explained falling objects was proposed by the Greek philosopher Aristotle about 2,000 years earlier. Things want to be in their natural places, he said. A rock's place was in the ground, so it naturally fell toward the ground. This is teleological thinking. It assumes that the processes of nature have a goal, a purpose.
No way, said Newton. Rocks don't want anything. They are just obeying a set of mathematical laws. As Galileo said long before Newton came along, nature is a book written in "the language of mathematics." And the sentences we construct with that language are equations -- such as the one that tells us why and how fast a rock will fall.
Crease contends that the influence of Newton's laws of motion and universal gravitation "extended well beyond science -- to education, philosophy, theology, and other areas of human culture." Voltaire, the great humanist thinker of that era, realized the importance of the new discoveries to society. He summarized their revolutionary nature in a single short sentence. "[Newton] is our Columbus," he wrote in 1732, "he led us to a new world."
Many historians agree with Voltaire. In their monumental series "The Story of Civilization," Will and Ariel Durant identified three books that shaped the mind of modern Europe. All three were books about science. Prominent among them was Newton's masterpiece, "The Principia," in which he revealed his groundbreaking laws. (The other two were "On the Revolutions of Heavenly Spheres" by Nicolaus Copernicus and Charles Darwin's "The Origin of Species.")
Without Newton's laws, engineers could not have put space stations into orbit, the Eagle would never have landed and robot rovers would not be roaming Mars. As impressive as those results are, they are small potatoes compared with what grew out of the equations dreamed up by James Clerk Maxwell.
Maxwell is not as well-known as Newton or Einstein. To physicists, however, he is on par with those giants. In 1865, he published a paper, "A Dynamical Theory of the Electromagnetic Field," that contained 20 mind-boggling equations. The paper described how a combination of changing electric and magnetic fields produces a special type of wave, called an electromagnetic wave. When Maxwell calculated how fast those waves moved, he found they traveled at the speed of light. This led him to the stunning insight that light itself must be a form of electromagnetic radiation.
This conclusion, which follows directly from Maxwell's equations, has transformed society. Those dry, almost unintelligible scribblings, Crease writes, are the "structural foundation of the modern era, embodied in ... radio, radar, television, microwave, and wireless communication." The Nobel laureate physicist Richard Feynman said that "the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison." These days, Maxwell's original 20 equations have been distilled to four. They are commemorated on geeky T-shirts favored by physicists and math-class whiz kids.
Crease investigates some equations for reasons other than their direct utility. One of these is Euler's equation. Leonhard Euler was a prolific 18th-century Swiss mathematician. He was a genius who read equations effortlessly, "just as men breathe, as eagles sustain themselves in the air." Euler's equation shows the relationship among three different types of numbers -- rational, irrational and imaginary. It also includes two fundamental constants: pi -- the ratio of the circumference of a circle to its diameter -- and e -- the base of a special type of logarithm. Feynman encountered this equation when he was 14 years old. It was, he wrote in his diary, "the most remarkable formula in math."
To most of us, Euler's equation is not easy to read or understand. But to mathematicians, it is beautiful, and to a few it is self-evident. The great German mathematician Friedrich Gauss once said that anyone to whom Euler's equation "is not obvious is not a mathematician."
That supercilious comment may remind you of that kid with all the answers in your high school math class. But don't let either Euler or the kid discourage you. There's still hope.
Mathematics can be learned later in life. Einstein did his best to avoid difficult math when he began work on his theory of relativity. But he changed his mind as he grew older. He was middle-aged when he began to dig into and finally master the high-powered math he needed to complete his theory. As he was wrapping up his work, he confided to a friend, "I have gained enormous respect for mathematics."
This book will instill in most nonmathematicians a similar respect for the subject. It also will give you some strong talking points if you ever run into that math-class smarty-pants again.
Phillip Manning is a Chapel Hill writer; his book reviews and essays on science are available at www.scibooks.org.