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10 October 2003

The Return of a Calculus Classic

Silvanus P. Thompson and Martin Gardner. Calculus Made Easy. Being a Very-Simplest Introduction to Those Beautiful Methods of Reckoning Which are Generally Called by the Terrifying Names of the Differential Calculus and the Integram Calculus. New York: St Martin's Press, 1998. 330 pages.

ISBN 0-312-18548-0

Reviewed by Sheldon Greaves

It has long been my suspicion that mathematics is one of those subjects that is easier to learn than it is to teach. At least this conforms with my own experience in the public schools where I learned other academic subjects with realatively little or no trouble.

But math was a different story. I was one of the early victims of the "New Math" method of teaching that left so many gibbering mathematical basket cases in its wake. The idea was sensible enough: teach kids math by introducing them to the essential concepts of math at the earliest possible age. The problem was these "essential concepts" were neither necessary for kids to begin understanding math, nor were they well taught. By the time I graduated from high school I had a deep dislike for math. Actually that's not quite true. I deeply disliked the fact that apparently I sucked at math. It was largely because of this that I elected to pursue an academic career outside the sciences, even though I have always loved science since childhood.

This dislike gradually grew so acute I actually took leave from my university studies to go back to my home town of Salem, Oregon to spend a year at the local community college taking math classes. I embarked on a one-year experiment to see if math really was something that I just could not grasp. To my suprise and delight, I managed to get as far as trig before the year was up and I had to go back to my regular studies. The difference was how the subject was taught. Community colleges have to teach welfare cases, ex-cons, high school dropouts, and others for whom elegant proofs are not a priority. They just need to be able to do the math.

But calculus eluded me, and I feared it always would. The many books I picked up to learn it didn't explain the subject well. Reading them felt like trying to listen in on a conversation where I didn't know the jargon. I eventually did sign up for a calculus class at another community college a few years ago and managed to get through the first and part of the second semester before life intervened to postpone further formal math studies. Again, to my astonishment, I understood most of it.

This was due in part to my spouse picking up an earlier edition of Silvanus Thompson's Calculus Made Easy a couple of years before I took calculus. Reading it was different from other books I had seen. It was also more fun. Despite the age of this text, it speaks to the modern reader. It has an irreverent attitude that math refugees like myself will appreciate. Thompson introduces his subject thusly:

"Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks...

Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can."

Incidentally, Thompson's self-efacement is strictly for show. He was a noted engineer, mathematician and a Fellow of the Royal Society. Out of print for some time, a new edition of Thompson's classic has appeared, updated and supplemented with new material and notes by Martin Gardner, who once penned the famous "Mathematical Games" column in Scientific American. Gardner has probably done more than anyone else in the last fifty years to raise interest in mathematics and was the perfect choice for this task. His contributions include a new introduction and three new chapters at the beginning: "What is a Function?", "What is a Derivative?", and "What is a Limit?". They provide essential material that Thompson does not cover explicitly. They definitely make understanding the subject easier. The notation in the book has also been updated to conform with modern Calculus texts. Gardner has also included--not surprisingly--an appendix on calculus and recreational math.

Most hard-core mathemeticians allegedly despise Thompson's book because he does not insist on proving everything rigorously. He simply teaches you how to do calculus and lets the whys and wherefores attend to themselves. I, myself side squarely with Thompson. Insisting that beginning students learn calculus by way of elegant and subtle proofs strikes me as comparable to insisting that students begin learning classical mythology by forcing them to read the Homeric Hymns in the original Greek. Besides, I opened more calculus textbooks than I can remember trying to get a handle on the subject, and utterly failed each time until I found Thompson.

As it turns out, Thompson's approach has several advantages (besides the fact that it's comprehensible) because the teaching of Calculus is facing new difficulties. It is now possible to go out and buy a calculator capable of solving pretty much any calculus problem a student is likely to encounter, and the cost of such a calculator will probably be less than the price of a typical calculus textbook. It's becoming more relevant for students to understand what calculus does and the kinds of problems it can solve and how to apply them. The mechanics of actually solving those problems are less critical now than they were in Thompson's day. But the fact remains that if one seeks to do any kind of serious science, calculus is still one of the most important intellectual tools in a scientist's panoply. And if you need to learn it or review it on your own, you can do no better than Thompson's book, especially with Gardner's supplements.