**Infinite Powers**

By Steven Strogatz

Houghton Mifflin Harcourt, Boston, 2019

Okay, I admit it, even for me this is a geeky book.

Infinite Powers: How Calculus Reveals the Secrets of the Universe is about the history of calculus and its impact on science, technology, and society.

The author, Steven Strogatz, Professor of Applied Mathematics at Cornell University, has done a great job telling both the historical and the mathematical sides of the story (yes, the book has lots of equations) in an way that is interesting and approachable. He traces the development of calculus, starting with its first inklings in ancient Greece, through its full development by Newton and Leibniz in the late seventeenth century, right up to modern applications in physics, medicine, biology and other fields.

First of all, what exactly is calculus? The short answer is that calculus is a branch of mathematics devoted to the study of change, particularly smooth or continuous change. *Differential* calculus is used to study rates of change, and *integral* calculus is focused on the accumulated results of change.

Calculus developed from studies of curves, motion, and change. Strogatz says that calculus is concerned with three fundamental problems:

- Given a curve, find its slope at every point
- Given a curve’s slope everywhere, find the curve
- Given a curve, find the area under it

Today, we’re less interested in curves as geometric shapes, as classical geometers were. We’re more interested in the physical process that gave rise to the curve.

“In the early seventeenth century, before calculus arrived, [such] curves were viewed as geometrical objects. They were considered fascinating in their own right. Mathematicians wanted to quantify their geometrical properties. Given a curve, they wanted to figure out the slope of its tangent line at each point, the arc length of the curve, the area beneath the curve, and so on. In the twenty-first century, we are more interested in the function that produced the curve, which models some natural phenomenon or technological process that manifested itself in the curve. The curve is data but something deeper underlies it. Today we think of the curve as footprints in the sand, and a clue to the process that made it. That process – modeled by a function – is what we are interested in, not the traces it left behind.” [p. 145]

But you could also think of calculus as an approach for thinking about and solving certain kinds of mathematical and scientific problems. Strogatz calls this approach 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.” [p. xvi]

It’s simple, right? For example, to calculate the area of a complicated shape, say the area underneath a parabolic curve, slice up the area into an infinite number of infinitely thin rectangular strips, figure out how to calculate the area of one of the strips, then add up all the strips. Easy-peasy!

Of course, the tricky part is that the area of one of those infinitely thin strips should be zero. And adding up a bunch of strips whose area is zero should give you a total area of zero, which doesn’t seem like the right answer.

This dance with infinity is what makes calculus a little mind-bending at first. It turns out this dance has been going on for a long, long time.

Strogatz traces the development of calculus, and the accompanying challenge of dealing with infinity, starting with Zeno (495 BC – 430 BC) and his paradoxes, and Archimedes (287 BC – 212 BC) and his surprisingly modern methods for determining the circumference of a circle. Strogatz takes us through the contributions of Kepler and Galileo, Descartes and Fermat, and of course Newton and Leibniz. This development culminates in the Fundamental Theorem of Calculus which links the area under a curve at any point, to the function that produced that curve to the slope of the curve at that same point. It’s one of the most powerful mathematical results ever produced.

But it’s not just dry history. What makes *Infinite Powers* so interesting is that in each chapter Strogatz also explores modern problems or technologies that depend on calculus for their solution, or even their existence. For example, the methods of calculus, cutting up a problem into in infinite number of tiny sub-problems, is used today in CT scanning where x-rays are used to scan tissue in thin slices from many different angles. The results are then reassembled to produce diagnostic images of tumors deep inside the brain. Or take computer animation where the contours of a human face are built up from millions of tiny polygons.

But the influence of calculus hasn’t stopped there. Newton used calculus to derive his famous three laws of motion. But as Strogatz argues, perhaps Newton’s greatest legacy is the idea of a logical universe, one describable by natural laws, expressed in the language of calculus, which apply universally, both on earth and in the heavens. We take this idea of a universe governed by natural laws for granted today, but in Newton’s time it was shocking and revolutionary.

This idea in turn influenced Enlightenment thinking about determinism, liberty and human rights. Strogatz points out that you can even see the influence of Newtonian thinking in the US Declaration of Independence, which contains in its opening paragraph an appeal to Natural Law.

It really is hard to overstate the impact of calculus on mathematics, science and technology, and Strogatz does a good job conveying his enthusiasm for both the subject and its applications.

I know this isn’t the sort of book that will appeal to everyone, but if you feel like dipping your toe into a popular science book, give *Infinite Powers* a try. Even if you skim over the mathematical details and just read for the history, you might be pleasantly surprised.