November 7, 2022

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 Integral Calculus
Silvanus P. Thompson and Martin Gardner
(St. Martin’s Press, 1998) [1910]
336 p.

I’m happy to have finally caught up with this classic, which has been read and appreciated for a century. I picked it up not for my own enjoyment but because I was casting about for a good introduction to calculus for my daughter.

The book provides a good conceptual introduction to calculus, both differential and integral. (The florid subtitle is no lie.) The technical approach is to describe calculus simply using the concept of infinitesimals, rather than limits, which is different from the way I learned calculus, but it’s intuitive and I quite liked it. The geometric meanings of derivatives and integrals are stressed, and this also helps to make the ideas clear. The phrase “fundamental theorem of calculus” does not appear (except in Martin Gardner’s annotations), but he does cover the relevant ideas. Chain rule, product rule, and quotient rule are all derived. The book sticks to single variable calculus except for one brief chapter on partial derivatives. There is a chapter on simple differential equations, and toward the end he describes how to calculate areas, volumes, and arc lengths by integration.

The focus is on practical meaning and applications; the level of rigour is right for a beginner, but would not satisfy a mathematician. (Thompson was a physicist and engineer.) In addition to a conceptual introduction, the book gets into calculating the derivatives and integrals of some elementary functions (polynomials, exponentials and logarithms, trigonometric functions), and the problems, both worked and assigned, require a decent level of algebraic competence to complete.

For my grade 8 student, I’m assigning the mainly-conceptual chapters. With another year of algebra I think she’ll be ready to tackle at least some of the more technical bits.

It’s a very nice book with a well-deserved good reputation.