Each chapter begins in the same way. First,the formula is stated, with labels attached to each term to say what it means. Then come short paragraphs to explain what the equation says, why this matters, and what it led to. The main body of the chapter then explores these ideas in more detail.
The equations range widely in scope and complexity. Some, like Pythagoras's theorem, will be familiar to practically everyone from their schooldays, at least by name. Others are more complex and challenging, coming from relativity theory and quantum physics. But even the seemingly 'old-fashioned' topics offer plenty of surprises, as in the case of Newton's law of gravity.
.[T]he planets and moons of the Solar System are tied together by a network of tubes, whose mathematical definition requires many more dimensions than four. The tubes provide energy-efficient routes from one world to another. They can be seen only through mathematical eyes, because they are not made of matter; their walls are energy levels. If we could visualise the ever-changing landscape of gravitational fields that controls how the planets move, we would be able to see the tubes, swirling along with the planets as they orbit the Sun.
There is certainly a great deal of interesting information in this book. But I didn't find Stewart's explanations of the mathematics to be always as easy to understand as I expected. A good illustration of this is his discussion of logarithms in Chapter Two. This starts with a perhaps over-long historical introduction and then embarks on an algebraic analysis of how logarithms work. Contrast this with what Steven Strogatz does with the same subject in a book I reviewed earlier, Infinite Powers.
Strogatz avoids algebra and instead develops logarithms smoothly from a list of powers of 10. The logic is easy to follow. He then goes on to explain the natural logarithm, e, by means of an imaginary bank compound interest scheme; this works brilliantly. Stewart, on the other hand, simply mentions the natural logarithm in passing but doesn't explain it at all.
Readers of this book will presumably vary widely in their levels of mathematical expertise, but those whose recollections of school algebra are hazy will find some of it hard going. But at least it is a vivid illustration of the truth of Galileo's famous aphorism: mathematics is the language in which God chose to write the laws of the universe.