*Infinite Powers: How Calculus Reveals the Secrets of the Universe* by Steven H. Strogatz is a delightful, useful and masterful book that describes and demystifies the central idea(s) of Calculus in a way that most audiences, regardless of their prior background or knowledge of mathematics, will understand and enjoy. Strogatz, the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University, is the author of several books about mathematics, some intended for specialized audiences (*Nonlinear Dynamics and Chaos*) and some for general audiences (*The Calculus of Friendship*, *The Joy of X*, *Sync*) as well as the host of a popular podcast (of course!) titled *The Joy of X* in association with Quanta magazine. He is an acclaimed and accomplished communicator and *Infinite Powers* is an excellent demonstration of his talents. In 2020 the Museum of Mathematics announced it was sponsoring an annual prize in his honor, the Steven H. Strogatz Prize for Math Communication.

*Infinite Powers* is wonderful in the many ways it can be read by different readers. It is both an easy-to-read exegesis on “the importance of Calculus” for casual but curious readers (like parents and their high school children) as well as an informative and interesting historical reference for college students and instructors of mathematics. I believe that even people who (incorrectly) view themselves as “not math” people will be able to generally follow the arguments Strogatz makes and they will definitely enjoy the surprising stories he tells about the many different ways mathematical discoveries are made, and how they have changed our world.

As an applied mathematician, one of the joys of this book is the way Strogatz makes a compelling argument about how connected mathematics is to many aspects of everyday life and society itself, from cell phones to microwave ovens to the Declaration of Independence. As someone who thinks the history of mathematics should be a required course for all undergraduate mathematics majors, I am happy to see Strogatz present so many interesting details about the history of the Calculus, which ends up being a sneaky way of telling the history of mathematics, as well.

The key idea in Infinite Powers is presented early in the Introduction to the book as The Infinity Principle:

To shed light on any continuous shape, object, motion, process, or phenomenon—no matter how wild and complicated it may appear—reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.

For most people familiar with Calculus, this representation of its main idea will not be surprising. But for other readers, Strogatz does a beautiful job of summarizing and explaining this central concept of Calculus. In fact, it’s very possible that for students who have only seen Calculus from the perspective of an Advanced Placement curriculum or in a somewhat mechanical or long-forgotten college Calculus course, Strogatz’s description may be enlightening. He uses the evocative term “harnessing infinity” as a shortened version of the Infinity Principle throughout the rest of the text. One of the most interesting ways of experiencing *Infinite Powers* is as a history of Calculus. However, if you’re expecting a(nother) treatise on the notorious dispute between Sir Isaac Newton and Gottfried Wilhelm Leibniz about who should receive credit for its invention, Strogatz’s book will surprise you. First, he starts his history of Calculus by discussing the history of infinity, so he begins with Zeno and his paradoxes. The discussion of infinity continues with a deep dive into the work of Archimedes (described as ``the man who harnessed infinity”), all the time making it clear that the reason infinity is being harnessed is to solve specific, practical problems. Fans of Newton or Leibniz don’t need to be worried about their hero being dissed, since Strogatz includes plenty of material from each genius that validates each one’s claim to greatness. He then takes the unusual position that neither Leibniz or Newton “invented” Calculus; Strogatz’ view is that calculus (in the form of “harnessing infinity”) has been used for centuries, by Archimedes and others to solve problems and provide answers to important questions.

In the latter sections of the book, Strogatz goes on to describe several ways the infinity principle can be used to analyze natural phenomena and has led to important scientific developments. He links differential equations to the discovery of life-saving HIV treatments, Fourier analysis to the invention of the microwave oven and provides many more examples of the “unreasonable effectiveness” of mathematics. These are equal parts informative and inspiring for anyone who considers mathematics an important part of their life.

Overall, Strogatz’s *Infinite Powers* is a compelling, engaging read, one which readers of all different levels of interest in, aptitude for, or aversion to mathematics will enjoy.

Ron Buckmire is a Professor of Mathematics at Occidental College.