Infinite Powers

Strogatz wants to tell you what calculus is for before telling you how to do it, and for a reader like me, who passed the exams long ago without ever seeing the point, that order is exactly right.

One idea

The book is organised around what he calls the Infinity Principle: to understand something curved or changing, cut it into infinitely many simple pieces, deal with each piece, then put them back together. Everything else in the book is that idea applied again and again.

He starts with Archimedes, which surprised me. Archimedes squeezes a circle between polygons with more and more sides to pin down pi, and finds the area of a parabolic segment by filling it with an endless sequence of triangles. That is integration, two thousand years early, done with geometry instead of symbols.

The long middle

The middle chapters walk through Galileo and Kepler, who found patterns in motion and orbits without the tools to explain them, then Fermat and Descartes, who gave us coordinates and turned curves into equations. By the time Newton and Leibniz arrive, you feel the pieces were lying on the table and somebody finally assembled them.

I liked that he does not turn the Newton–Leibniz priority dispute into the main story. It gets its due, but the emphasis stays on what each of them saw: Newton thinking in terms of motion and rates, Leibniz in terms of notation that does the thinking for you. Having used Leibniz’s notation for years without noticing how clever it is, that was a small revelation.

Where it earns its keep

The last third is about applications, and this is where the book stops being history. Differential equations explain why things oscillate, decay and spread. Fourier showed that complicated waves can be broken into simple ones, which is quietly behind audio compression and medical imaging. There is a chapter on how a simple model of viral dynamics helped show that HIV was being produced and cleared far faster than anyone thought, which helped make the case for combination therapy. I had no idea calculus had a role in that.

Reservations

Sometimes the enthusiasm runs ahead of the explanation. A few chapters list applications without giving you enough to see how the calculus does the work. And readers who already know the material will find the pace slow.

But I am not that reader. What I got from it is a sense that calculus is not a collection of rules for derivatives and integrals; it is one way of seeing, repeated in many places. I wish someone had told me that at seventeen.