# Calculus for the curious

Some weeks ago I got involved in a discussion on Twitter about, which books to recommend on the history of calculus. Somebody chimed in that Steven Strogatz’s new book would tell you all that you needed to know about the history of calculus. I replied that I couldn’t comment on this, as I hadn’t read it. To my surprise Professor Strogatz popped up and asked me if I would like to have a copy of his book. Never one to turn down a freebee, I naturally said yes. Very soon after a copy of Infinite Powers: The Story of Calculus The Language of the Universe arrived in the post and landed on my to read pile. Having now read it I can comment on it and intend to do so.

For those, who don’t know Steven Strogatz, he is professor of applied mathematics at Cornell University and the successful author of best selling popular books on mathematics.

First off, Infinite Powers is not a history of calculus. It is a detailed introduction to what calculus is and how it works, with particular emphasis on its applications down the centuries, Strogatz is an applied mathematician, presented in a history-light frame story. Having said this, I’m definitely not knocking, what is an excellent book but I wouldn’t recommend it to anybody, who was really looking for a history of calculus, maybe, however, either as a prequel or as a follow up to reading a history of calculus.

The book is centred on what Strogatz calls The Infinity Principle, which lies at the heart of the whole of calculus:

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, analyse those, and then add the results back together to make sense of the original whole.

Following the introduction of his infinity principle Strogatz gives a general discussion of its strengths and weakness before moving on in the first chapter proper to discuss infinity in all of its guises, familiar material and examples for anybody, who has read about the subject but a well done introduction for those who haven’t. Chapter 2 takes us  into the early days of calculus, although it didn’t yet have this name, and introduces us to The Man Who Harnessed Infinity, the legendary ancient Greek mathematician Archimedes and the method of exhaustion used to determine the value of π and the areas and volumes of various geometrical forms. Astute readers will have noticed that I wrote early days and not beginning and here is a good example of why I say that this is not a history of calculus. Although Archimedes put the method of exhaustion to good use he didn’t invent it, Eudoxus did. Strogatz does sort of mention this in passing but whereas Archimedes gets star billing, Eudoxus gets dismissed in half a sentence in brackets. The reader is left completely in the dark as to who, why, what Eudoxus is/was. OK here, but not OK in a real history of calculus. This criticism might seem petty but there are lots of similar examples throughout the book that I’m not going to list in this review and this is why the book is not a history of calculus and I don’t think Strogatz intended to write one; the book he has written is a different one and it is a very good one.

After Archimedes the book takes a big leap to the Early Modern Period and Galileo and Kepler with the justification that, “When Archimedes died, the mathematical study of nature nearly died along with him. […] In Renaissance Italy, a young mathematician named Galileo Galilei picked up where Archimedes had left off.” My inner historian of mathematics had an apoplectic fit on reading these statements. They ignore a vast amount of mathematics, in particular the work in the Middle Ages and the sixteenth century on which Galileo built the theories that Strogatz then presents here but I console myself with the thought that this is not a history of calculus let alone a history of mathematics. However, I’m being too negative, let us return to the book. The chapter deals with Galileo’s terrestrial laws of motion and Kepler’s astronomical laws of planetary motion. Following this brief introduction to the beginnings of modern science Strogatz moves into top gear with the beginnings of differential calculus. He guides the reader through the developments of seventeenth century mathematics, Fermat and Descartes and the birth of analytical geometry bringing together the recently introduced algebra and the, by then, traditional geometry. Moving on he deals with tangents, functions and derivatives. Strogatz is an excellent teacher he introduces a new concept carefully, explains it, and then shows how it can be applied to an everyday situation.

Having laid the foundations Strogatz move on naturally to the supposed founders of modern calculus, Leibnitz and Newton and their bringing together of the strands out of the past that make up calculus as we know it and how they fit together in the fundamental theorem of calculus. This is interwoven with the life stories of the two central figures. Again having introduced concepts and explained them Strogatz illustrates them with applications outside of pure mathematics.

Having established modern calculus the story moves on into the eighteenth century.  Here I have to point out that Strogatz perpetuates a couple of myths concerning Newton and the writing of his Principia. He writes that Newton took the concept of inertia from Galileo; he didn’t, he took it from Descartes, who in turn had it from Isaac Beeckman. A small point but as a historian I think an important one. Much more important he seems to be saying that Newton created the physics of Principia using calculus then translated it back into the language of Euclidian geometry, so as not to put off his readers. This is a widely believed myth but it is just that, a myth. To be fair it was a myth put into the world by Newton himself. All of the leading Newton experts have over the years very carefully scrutinised all of Newton’s writings and have found no evidence that Newton conceived and wrote Principia in any other form than the published one. Why he rejected the calculus, which he himself developed, as a working tool for his magnum opus is another complicated story that I won’t go into here but reject it he did[1].

After Principia, Strogatz finishes his book with a random selection of what might be termed calculus’ greatest hits, showing how it proved its power in solving a diverse series of problems. Interestingly he also addresses the future. There are those who think that calculus’ heyday is passed and that other, more modern mathematical tools will in future be used in the applied sciences to solve problems, Strogatz disagrees and sees a positive and active future for calculus as a central mathematical tool.

Despite all my negative comments, and I don’t think my readers would expect anything else from me, given my reputation, I genuinely think that this is on the whole an excellent book. Strogatz writes well and fluidly and despite the, sometimes, exacting content his book is a pleasure to read. He is also very obviously an excellent teacher, who is very good at clearly explaining oft, difficult concepts. I found it slightly disappointing that his story of calculus stops just when it begins to get philosophical and logically interesting i.e. when mathematicians began working on a safe foundation for the procedures that they had been using largely intuitively. See for example Euler, who made great strides in the development of calculus without any really defined concepts of convergence, divergence or limits, but who doesn’t appear here at all. However, Strogatz book is already 350-pages-long and if, using the same approach, he had continued the story down to and into the twentieth century it would probably have weighed in at a thousand plus pages!

Despite my historical criticisms, I would recommend Strogatz’s book, without reservations, to anybody and everybody, who wishes to achieve a clearer, deeper and better understanding of what calculus is, where it comes from, how it functions and above all, and this is Strogatz’s greatest strength, how it is applied to the solution of a wide range of very diverse problems in an equally wide and diverse range of topics.

[1] For a detailed analysis of Newton’s rejection of analytical methods in mathematics then I heartily recommend, Niccolò Guicciardini, Reading the Principia, CUP, 1999, but with the warning that it’s not an easy read!

Filed under Book Reviews, History of Mathematics

### 17 responses to “Calculus for the curious”

1. Todd Timberlake

You mentioned that he talks about Kepler’s laws of planetary motion, but does he say anything about Kepler’s Nova Stereometria, in which Kepler adopts Archimedes’ methods and extends them to a variety of new shapes? That’s another nice example of calculus-like thinking prior to the development of what we now consider calculus. Kepler was certainly making use of Strogatz’s infinity principle.

• He does but only in passing, in two lines. I once wrote a whole blog post about it here

2. Hi Thony, Thanks for the very kind and careful review. You are certainly right — I did not intend the book to be a history of calculus! Far from it. I deliberately left out a LOT of great work by many people, because it seemed like the right thing to do for my intended reader. But it saddened me greatly to give such short shrift, or none at all, to Eudoxus, Euler, Cauchy, Riemann, and all the rest. Sigh…

But I want to push back on one thing you wrote: I don’t believe I ever claimed (or at least I certainly did not intend to claim!) that “Newton created the physics of Principia using calculus then translated it back into the language of Euclidian geometry, so as not to put off his readers. ” I agree with you that that’s a myth, having read Guicciardini’s book that you cite. I was persuaded by his point that Newton came to doubt his own earlier calculus as “specious algebra” (see p. 196 of Infinite Powers). On p.236, I explain that Newton couched his Principia in the reassuring language of geometry because it was the gold standard of rigor and certainty (but perhaps I should have stressed he used his geometric version of calculus to convince himself, not just his readers). Anyway, just a small quibble. Thanks again for taking the time to read the book and write such a kind and thoughtful review!
Warm regards, Steve

• Newton did use three distinct approaches to calculus in published works:

De analysi per aequationes numero terminorum infinitas used infinitesimals;

Methodus fluxionum et serierum infinitarum used fluxions; and

De quadratura curvarum used prime & ultimate ratios & sums.

I have to thank Rich Cochrane of City Lit, London for this information. I took his History and Philosophy of Mathematics course, which included a term on calculus, in 2016/17. He goes into a good deal of detail on these different interpretations.

His web site is at http://artgeometry.com/

3. Mike from Ottawa

Just one of the reasons the HistSciHulk posts are my favourites is that they seldom have me ordering a book as a result. You’ll be hearing from my bankers, Mr Christie!

4. Reblogged this on Project ENGAGE.