# Is Calculus Made Easy by Silvanus P. Thompson a good book for first-time calculus learners?

Specifically the one updated by Martin Gardner. I'm not studying as part of a high school or college course (I, in the near future, will though) just as a personal project.

• I don't know enough about the book to make this an answer, and although I hear it praised once in a while I've looked at it and was unimpressed. I learned calculus from Hughes Hallet, which seems to be written in the same spirit but much improved, so I recommend that. Aug 2 at 19:24
• @Thierry: I took a look at Hughes Hallett (6th ed.), and it seems to me like a very typical modern, commercial text. What is there about it that seems to you to be similar to Thompson? Weighing in at 1244 pages, and covering two years of calculus, it seems to me like the antithesis of Thompson.
– user507
Aug 2 at 20:39
• Take a look at William Chen's lecture notes, they are well organized and reasonably rigorous while understandable. Aug 2 at 21:12
• Before seriously getting into calculus, I recommend a strong foundation in complicated algebra, precalculus problems and likely trigonometry. Since I have worked one on one with hundreds of Calculus students, the most common difficulty is students lacking a strong math foundation. Aug 3 at 19:27
• What is your intent? University Calculus courses range from a bare introduction to Calculus, common for business students to Engineering Calculus, which is often considered a "weed out course". Your math background and long term goals will determine the rigor which you need to ultimately develop in Calculus. I would consider a larger context essential before answering your question. Even important enough to ignore other answers. Aug 3 at 19:31

Yes. The book is well written and fun. It cuts out a ton of the ridiculous cruft that has clogged up the current crop of commercial freshman calculus texts. IIRC Gardner moderates some of the dated and sexist language.

Be aware of how this book's approach relates to the history and to currently fashionable ways of presenting calculus. It uses infinitesimals rather than limits. Both infinitesimals and limits are perfectly fine ways of introducing calculus. Historically, there were some concerns that infinitesimals were inherently inconsistent, but it turns out that that's not the case.

When teaching yourself something intellectually challenging, I would always suggest that you find multiple sources of information rather than just focusing on one book. There is a text by Keisler that is free online and introduces freshman calculus using infinitesimals, but with a bit more of the modern machinery. Compared to Thompson, it's boring and pedantic and slow and lacking in personality, but it's worth having on hand while working through Thompson. You may also be interested in my own textbook Fundamentals of Calculus, which introduces both infinitesimals and limits while focusing mainly on techniques and applications rather than foundational issues.

After giving the book another look I have to expand on my comment above and make it stronger: no, this isn't a good book for a first-time learner, and in fact I think it's a terrible choice. I looked at the second edition (not the Gardner update), but unless Gardner practically rewrote the book I don't think this matters much.

At the very beginning, derivatives are explained by simply stating that $$dx$$ is small and so $$(dx)^2$$ and higher powers are, like, really really small and they don't matter so we ignore them. I'm not being unfair either, this is the actual essence of his argument. It leads to wonderful chains of reasoning such as $$y+dy=x^2+2x\cdot dx+(dx)^2$$ but let's drop the small term (no, not that one...the really small one dummy) and actually $$y+dy=x^2+2x\cdot dx$$. As a real world example, we're told to think of $$dx$$ as a flea on an ox (which would bother the ox) and $$(dx)^2$$ as a flea on the flea (which wouldn't bother the ox). But wait, we can make that flea $$dx$$ as small as we wish, and a flea as small as we wish certainly wouldn't bother an ox. What's a beginning student supposed to make of this?

Thompson and anyone who has already learned calculus can make sense of it in hindsight, but how can it be viewed as anything but nonsense by a first-time reader? Some of the other books listed in comments are far better imo. I already recommended Hughes Hallet, and I can also highly recommend What is Calculus About by Sawyer and Calculus: An Intuitive and Physical Approach by Kline (in Dave L. Renfro's link).

• Synthetic differential calculus is based on the idea of (dx)^2=0 and is fully rigorous. And I think Thompson does a fairly good job at justifying this idea. So I find it unfair to claim that it can only be viewed as nonsense. Aug 3 at 12:25
• Everybody from Newton and Leibniz onwards thought about calculus this way, until some mathematicians decided to make everything more complicated just to handle special situations of no great practical importance. Even the mathematicians eventually figured out how to make "infinitesimal numbers" rigorous, but educators haven't caught up with that yet. Thompson won't teach you how to answer exam questions about epsilon-delta proofs (and most people who use calculus will never jump through those hoops again for the rest of their working life), but it will teach you calculus. Aug 3 at 13:08
• @alephzero I have no doubt they thought about calculus at least in some vaguely similar way once they were comfortable justifying it (and I don't mean $\epsilon-\delta$ type justifications either). The relevant question is did they learn calculus the way Thompson presents it? Whole generations of students who knew algebra saw an algebraic expression and were ok just discarding terms because they knew they were small enough? And they knew that you can drop them starting at the second power and not the third or fourth or tenth? Right away they just knew that? That's hard to believe. Aug 3 at 21:00
• “Right away they just knew that?” No, of course not — no more than students learning calculus with limits “just know” which limits go to zero and which don’t. They knew it first because their teachers or textbooks told them, do it this way; and later because, having worked with it that way for a while, they found it was a clear, coherent, and conceptually meaningful framework. You’re certainly right that discarding the square is a bit of a conceptual leap; but it isn’t any bigger or stranger than plenty of other such leaps we teach students to make (e.g. $i^2 = -1$). Aug 3 at 21:09
• @PeterLeFanuLumsdaine When I was first learning calculus, I could know a limit was zero because I could use previous definitions to verify it myself. I didn't "know" because a book told me that it was and to accept it because it will all work out in the end. That's the difference between Thompson and my recommendations. (Your comment was kind of general so I wonder if you've recently looked at Thompson or, even better, learned from it? The latter possibility would sure be a good rebuttal to my answer!) Aug 4 at 3:48

The fact that the book uses infinitesimals (presented informally) rather than limits is definitely a big deal, and I think you will have to read the book for yourself to see if it works for you. When I was exposed to the idea of $$dx$$ as some kind of infinitely small quantity, my philosophical confusion completely blocked me. But the idea of a limit, even with its complicated $$\varepsilon$$-$$\delta$$ definition, made sense. But lots of other people quickly see how to manipulate and interpret $$dx$$ and $$dy$$ viewed as infinitesimals, and you might be like them.

Just be aware that, when you take calculus, you will likely be taught using limits.

In a sense it is OK. In the sense that almost any book (except for one that is very difficult, and assigned to a weaker student and/or one without strong motivation to prevail) is OK.

It's really more about your sticktoitiveness than anything else, what you get out of it. I'm working on a language study right now. I have several different texts available on my shelf. And there are plusses and minuses. But really, the key thing that will determine if I prevail is if I stick to it or not. (I know several times I have not!)

I would probably advise a more standard book. But particularly one that has answers to the exercises (not worked solutions per se, just answers). You need to do your own drill and to check yourself. The intermediate review grammar I am self studying from has all the exercise answers. It's designed for usage. Not for acceptance by teacher committees that pan books with drill answers. And it sure as heck improves my understanding to see that I missed something and then check why (often figuring out my own mistake, but in a few cases, prompting a question of someone, which I keep a "parking lot" for.)

I like Granville although it is a tiny bit dry. But it uses simple words. And has short sections with minimal text (not in a proofy "intuitive to most casual observer" way but more in a KISS way). You can get one of the older editions free online in a pdf. Although really cost should not be a discriminating factor. Think of the value of your time invested. There is a huge cheap market for old books on Amazon, so just buying an old edition is a no brainer.

I also like Thomas Finney prior to about 1982 (when Thomas retired). Studied with their AP version in early 80s. [You can see the blue kaleidoscope cover of the book in the movie Stand and Deliver.] It is slightly more theoretical than Granville though, without being some baby real analysis craziness. I probably benefited some from partially working Granville first.

I also recommend to look at the Schaum's Outline. In general this sort of series is very useful for adult learners looking to review an old topic (minimal text for one thing, having answers, etc.) More of a businessman's "80-20" attitude versus the fussy complete-ism that you get from high end types on SE. Some posters here pan the series, but I think they are much more user friendly for a self studier. (And, like dieting, giving up is the key danger, so picking something user friendly is a good way to go. In that sense, the book you listed may be nice just because it is a little more amusing and so you may stick with it. You could even try an older politically incorrect version to provoke your laughs--either at or with, your choice. ;)

I have not used it, but one text I hear referred to is "Calculus for the Practical Man". Maybe also similar in being pretty gentle and marketed at self studiers and especially those who are a bit more dilettantish.

• Regarding "Calculus for the Practical Man", I have a Dover version of Calculus Refresher for Technical Men by A. Albert Klaf (1944, reprinted by Dover in 1956). The photo where I linked looks like my copy, but my copy has $3.00 in the upper right corner, and thus is an older printing (I've had it since around 1973; I think I got it at one of those well-stocked bookstore/candy-kitchens that used to present in larger U.S. cities, in this case Charlotte), but not as old as some that (continued) Aug 3 at 16:31 • have$2.00 in the upper right corner (amazon.com copy 1 and amazon.com copy 2). However, I never got much out of this book, either when I was first trying to learn what calculus was about (for that, I thank a library copy of Ferrar's "Calculus for Beginners" that I managed to get about 1/4 the way through in early 1974 while I was still working on finishing Dolciani's Algebra 2 and Trig. text), or later when teaching calculus and looking through various books for ideas and examples that might be worth using. Aug 3 at 16:31
• This doesn't say anything that specifically relates to Thompson.
– user507
Aug 4 at 19:34

Also, get Steven Strogatz's "Infinite Powers" Book. It doesn't have any work problems. He very carefully explains many of the fundamental ideas of calculus. I was fuzzy on some of the ideas, and his book really cleared some items up. Calculus has a "tortoise and a hare", aspect to it. If you take it slowly you will learn more than if you try to race through it.