Calculus Made Easy by Silvanus P. Thompson (1910) | Hacker News Reader

Calculus Made Easy by Silvanus P. Thompson (1910)

calculusmadeeasy.org

265 pointsavinassh60 comments

Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus. It can be freely downloaded at https://people.math.wisc.edu/~hkeisler/calc.html

Relevant wikipedia entry: https://en.m.wikipedia.org/wiki/Nonstandard_analysis

I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative."

But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate.

    # d is a functor. It takes a function and returns a second function.
    # The second function captures the change in f over a small distance.
    # The dx/2 business reduces artefacts of it being a finite distance.
    def d (f, dx=0.001):
        return lambda t: (f(t + dx/2) - f(t - dx/2))

    # d²x / dx²
    def second_derivative (f):
        return lambda t: d(d(f))(t) / (d(x)(t) * d(x)(t))

    def x (t):
        return t

    def cubed (t):
        return t*t*t

    print("The second derivative of cubed at 1 is near", second_derivative(cubed)(1))

Dang it! I hate re-reading after the time for editing is over and finding a stupid mistake.

The Python is correct, but the comment is not. The formula for the second derivative should be, of course:

    # d²y / dx²

Most calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.

Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function".

You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.

You're right. But then again, lots of scaffolding gets discarded when an arch gets constructed also.

Lest you disregard this as completely useless superstructure, note that basically the entirety of the theory of stochastic processes, starting with Brownian motion, is positively infested with continuous everywhere nondifferentiable functions; while the existence of a nonconstant infinitely smooth function with an identically zero Taylor series is what permits the Berezinskii-Kosterlitz-Thouless phase transition to exist. So while the weird animals of elementary real analysis are perhaps not the most important thing in the world, they are far from irrelevant to it.

It’s extremely niche (abstract, “Platonic”) to ever need to care about derivatives of non piecewise-analytic functions.

I think of wavelets and stochastic processes to be a significantly bigger niche than you probably do.

You write, "we shouldn't need the axiom of choice to define the derivative."

The good news is that we don't!

Only model-theoretic approaches, which justify the infinitesimal methods by constructing a hyperreal field, require (a weak form of) the axiom of choice [2].

However, there are axioms for nonstandard analysis which are conservative over the usual choice-free set theory ZF. The three axioms of Hrbacek and Katz presented in the article "Infinitesimal analysis without the Axiom of Choice" [1] are the best recent example: these axioms allow you to do everything that is done in Keisler's book and more (including defining the derivative), and you never need to invoke the axiom of choice to justify them.

[1] https://arxiv.org/abs/2009.04980

[2] Essentially, the set of properties satisfied by a fixed nonstandard hypernatural gives rise to a non-principal ultrafilter over the naturals. The axiom of choice is necessary to prove the existence of non-principal ultrafilters in (choice-free) set theory, but the existence of non-principal ultrafilters is not sufficient to prove the axiom of choice.

Yeah, yeah. My comment was my reaction 30 years ago.

I find it a mildly interesting intellectual exercise that you can do NSA with weaker axioms than choice. But for all cases I care about, I can already prove it with NSA without ANY additional axioms!

How is this possible? From Shoenfield's absoluteness theorem, you can prove that all statements that an be made in the Peano Axioms that can be proven in ZFC, are also true in ZF. (Note, they must be statable in PA, but not necessarily provable there.) But PA can encode any statement we can make about computation. So take any calculation we can talk about that can be approximated on a computer. We can rewrite it in PA. We can prove it using NSA. We then know that it is true in ZF. And we know that it is true without any additional axioms beyond ZF!

That which we can actually calculate in any useful way can all be calculated on a computer. And therefore NSA can prove anything about Calculus that I care about without needing any axiom beyond ZF.

But in the end this is using a mathematical sledgehammer to drive in a thumb tack. Many approaches to Calculus do not require assertions about the existence of sets that we cannot construct, even in principle. Even though I understand how NSA works, I'd prefer to use any of those.

I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.

> I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.

I'd say that limits in the sense that you mean (as opposed to category-theoretic limits) are precisely the domain of calculus or, if one wishes so to call it (because one is proving things!), analysis. For example, many US universities, mine included, regard the computation of infinite series as part of Calculus II.

And non-standard analysis is useful elsewhere in math as well. Here's an example:

https://discreteanalysisjournal.com/article/87772-a-simple-c...

More importantly for this audience, the idea of limits lies at the heart of numerical analysis. Explicating the quantifiers of the definition of a limit is the first step in obtaining control over any estimator of your data. This, among other things, is why I am perpetually baffled at the people in this audience who say that limits is something "they will never need". Limits, their algebra (which subsumes all of high school algebra and inequalities) and their delicate analysis is what makes a ton of numerical algorithms work; Higham's classic has it all, and is perfectly clear about it all.

I think that Big O / little o are both more approachable and provide a richer understanding than limits. See, for example, https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... to see Donald Knuth agreeing.

As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks.

It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?

My calculus teacher tried to talk me out of taking the AP Calculus AB exam because I wasn’t doing that well in his class. “Calculus the Easy Way” which teaches in the setting of a mythical fantasy world where applied problems are solved with calculus unlocked the secret of calculus for me. I ended up scoring a “4” which was a big triumph for me, as it placed me out of taking calculus with all the people who did not want to be taking calculus.

https://www.thriftbooks.com/w/calculus-the-easy-way-easy-way...

Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people. Same approach of situating Trigonometry in a mythical fantasy land. I remember Trigonometry as a chore of trying to memorize double-angle formulas and such, but this book really helped connect trig in an intuitive way.

I didn't know he had a calculus book, I'll have to check it out now.

> Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people.

> but this book really helped connect trig in an intuitive way.

To me, understanding math in an intuitive way is actually how "maths people" think about math. The numbers and formulas are just a means to an end to get there.

When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth's days, they called “second minùtes” (i.e.: small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called.

A "third minute" (of 1/60 second) could be useful in talking about human reaction times. We could say that people commonly have a reaction time of 7 to 15 thirds, and that alcohol consumption may slow that reaction time by a further 8 thirds or more.

Or, speedrunners and professional musicians can practice motions (relative to their own motions or to an ongoing rhythm) that are accurate to about 1 third.

I assume the terminology would be too easy to confuse with ⅓, so maybe they would have to be called "terces" or "tertians" or "tertiaries" or something. (Specifically, we want to say (1/60)³ hour rather than (1/3) minute!) And maybe it's not a great thing to introduce a non-decimal unit when we're also trying to get rid of them in many other contexts.

I should follow up and say that I learned that this unit has already been referred to under the name "tierce", although it's not at all common.

It could also be useful for measuring latency of Internet connections, although these have commonly gotten faster over time and many latencies can be under a tierce. On the other hand, the round-trip latency to a satellite in geostationary orbit is about 14 tierces.

Consider Analysis 1 by Terry Tao. It walks you from the very beginning and also uses a conversational style. It's one of the most pedagogically smooth or gentle books I've ever read.

I graduated from engineering school more than 30 years ago, and learned how to apply calculus without understanding why it worked. This book helped me to finally understand.

I love projects like this, but I really wish instead of converting text resources to web sites that these projects would produce epub outputs. It's great for distribution, offline reading, and scaling to different display sizes, aspect ratios, and resolutions.

I wonder if there will be a time when textbooks will be created in digital-first format, instead of being mere replicas of what print books are. It doesn’t have to be static text and images on A4 pages.

The page links to a pdf version[0], which can easily be converted to epub using Calibre[0], which is free and open-source.

[0] https://www.gutenberg.org/ebooks/33283

[1] https://calibre-ebook.com/

I'm familiar with those tools, but they don't preserve the formatting of the source document (when PDF) and certain types of embedded resources don't transfer properly.

Some PDF documents don't contain the source text as digitized text, either. It's just a bundle of scanned images.

Epubs are horrible for technical documents

Only if they exist as poor conversions. Properly formatted epub documents are worth their weight in effort.

Epub documents _are_ HTML files in a zip archive - I'd argue that if a web site is a proper presentation of the source material, an epub is even better.

Additionally, PDF documents _are_ worse for any kind of document technical or otherwise. PDF documents are primarily Postscript instructions without any relation or hierarchy to the included elements. Epub documents/HTML provide semantic relationships and hints to the content.

I think that what previous poster is pointing is that technical documents as epub are often difficult where they're no textual contents. Images may be the easiest case. Things are a bit more clunky when code source are involved and it's weird with complex formulas.

A competent stylesheet will sidestep all those issues. Admittedly, I do wish MathML were in place to be used online and in epub documents as that would make the problem much more easily solvable.

Epub3 (not previous versions) does support MathML. The main concern is that most e-readers are not as capable as main browsers we use. Very few e-readers have good support of Epub3 features, and maths rendering is limited when available.

For codes, `pre` tag should be enough, stylesheet just adding some enlightenment. What I'm pointing is that many source code are more than 80 columns wide and that often make you scroll like with PDF. Physical readers are best for content that use aspect-ratio of pocket paper books and as far I can remember coding paper books are more (at least twice) wider and may even suffer that problem (they're often hacks I dislike to split lines.)

To short, Epub per se is not the limiting factor. Vendors must update their hard and soft (even then, things will improve only when everybody has upgraded.) Authors also must take more care to technical contents for that media (some of them convert formulas and codes to image, oooch…)

EPUB works fine for documents that mostly consist of easily reflowable text. For documents with lots of equations, diagrams, tables/charts, footnotes, etc. I would much rather have a PDF.

My biggest mistake as a SWE (now in my 30s) was not learning higher level mathematics and allowing what knowledge I did possess to wither on the vine.

What better hobby to pick for those cold winter evenings, than to do some integrals!

Edit: I can recommend this book for a self-guided study

https://archive.org/details/zeldovich-higher-mathematics-for...

The author was a Soviet nuclear physicist (who participated in the creation of the H-bomb), so his main point isn't rigor. It can be a nice change of perspective from standard American texts.

Although Soviet-era books were notoriously terse & difficult to digest, they had some very aesthetic typesetting. I have owned a few (Problems in Physics by Irodov, & another by Krotov) and they all share similar design aesthetics.

Late 30s here. I keep feeling like I should learn math better, but damn, I just never need it. It’s much easier to learn stuff I need. As it is, I’ve lost everything back to about 8th grade math because I’ve never used any of it, so it’s just as gone as all the French I used to know but never found an excuse to use.

[edit] and I’m dreading my kids getting past elementary school math because they’re gonna be like “why the hell am I spending months of my life on quadratic equations?” and I’m not gonna have an answer, because IDK why we did that either. At least I have answers for calculus, even if they’re not much good (“so you can do physics stuff”, “right, but will I ever need to do physics stuff?”, “uhhh… unless you really want to, no.”)

Quadratics are useful-- finding dimensions of things in the plane; relating area and constrained side lengths, etc. They come up a lot if you want to solve problems.

And good luck taking on calculus without being super solid in the mathematical tools you use against quadratics -- factoring, completing squares, manipulation of binomials, pairing up like terms, etc.

The quadratic equation is completing the square while inexplicably avoiding all the intuition of completing the square. For example, to solve x^2 + 6x + 5 = 0, you would re-write it as (x^2 + 6x + 9) - 4 = 0, which is (x + 3)^2 - 4 = 0 and hence equivalent to (x + 3)^2 = 4, so that x + 3 = ±2 and hence x = 3 ± 2 is 1 or 5. Euclid thought of things this way, though his language is, of course, very different to modern language; see, for example, Proposition 6 of Book II (http://aleph0.clarku.edu/~djoyce/elements/bookII/propII6.htm...).

That's the same answer as the quadratic formula, but makes a lot more sense to me! Of course I've cooked the numbers so that you don't wind up with surds in the answer, but those are just complications in bookkeeping, not in concept.

Learning mathematics teaches you general concepts of critical thinking.

For example, the way to solve a quadratic is to reduce it to a form one knows how to solve (via competing the square).

The specifics of the mathematics are not the prize, the methods of thinking are.

If this is true, there is surely a way to present material honing that specific skill as a game or series of games that would generate more intrinsic interest from more kids than math does.

I don’t think the methods of thinking per se are why we teach math, though. Might be part of it, but if that’s all, I think we could do a lot better for a lot less effort for all concerned. I think it’s because the math itself is useful. If in fact the point were to teach methods of thinking, I doubt we’d teach it as we do math—why would we, when it generates such resistance and loathing from so many students?

If you don't read anything else in this, read the "Epilogue and Apologue".

> What one fool can do, another can.

This quote helped me get over my fear of math. It's probably the fewest words that have made the largest impact in my life.

I think it would be fun to go through all of these books in a group, but to also do worksheets with something like XCas, Maxima, Fricas, etc.

Does anyone know any other good resources for learning calculus at home? Preferably ones that show how and why. I did some in college but I've forgotten just about everything, and now I'm finding myself needing it again.

> Does anyone know any other good resources for learning calculus at home?

Professor Leonard:

Calculus I - https://www.youtube.com/playlist?list=PLF797E961509B4EB5

Calculus II - https://www.youtube.com/playlist?list=PLDesaqWTN6EQ2J4vgsN1H...

Calculus III - https://www.youtube.com/playlist?list=PLDesaqWTN6ESk16YRmzuJ...

I consider him one of the best lecturers in math education, at least for these subjects. And in particular because he is very detailed in his explanations. He points out that most students who struggle with Calculus struggle because they (never mastered | forgot | whatever) their basic Algebra. So he does a very thorough job of explaining all of the subtle algebraic manipulations that go on as he works through derivatives, integrals, etc.

TBH, I think a person who wanted to learn the equivalent of high-school algebra could just about doing it by watching his Calc I series... and treat any Calculus they learn as "found money." But assuming you remember at least a little algebra and really want to learn Calculus, I think he's one of the best at teaching it.

Note that most of his lectures are live lectures to an actual class, so IMO the best way to approach it is to pretend you're right there in class. Listen, take notes, and then when he puts an example on the board pause the video and work through the example. Just restart the video when you finish the problem or if you get stuck.

If you want to work additional problems, go on Amazon or Alibris or whatever and buy a cheap used copy of one of the enormous Calculus books, and/or a Shaum's Outlines book on Calculus, or one of those "1001 solved problems in $SUBJECT" books... or some combination of all of the above.

Also as a side-note, speaking for myself, I find that I can follow his material find at 1.25x speed, so I pretty much always watch on 1.25x. I could probably manage 1.5x if I really tried, but the time savings from just doing 1.25x is enough to make me happy. YMMV, of course.

The 3blue1brown "Essence of Calculus" series is a great (re-)introduction to the material, focusing primarily on intuition and understanding. https://youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVR...

Would anyone have any recommendations for books/textbooks of this style and the comment's Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler on Algebra/PreCalc & Trig?

I've always wanted to learn math but my teachers could never explain it to me in a way that clicked and any textbook I've read couldn't either. These two above really seem to be in my wheelhouse.

Perhaps Axler's "Precalculus: A Prelude to Calculus"?

I picked this book up when I was learning Algebra to see what the hype was about as a kid. It was dull. I went back to the library stacks after some weeks and found Calculus and Pizza to be even easier to swallow.

Calculus Made Easy is a good book. It made me appreciate even easier books when you need them and have enthusiasm for learning a topic.

A newer text is:

_Make: Calculus: Build models to learn, visualize, and explore_ by Joan Horvath and Rich Cameron

https://www.goodreads.com/book/show/61739368-make

It's part of a series with matching books on Geometry and Trigonometry.

I love this book!!! I also just finished writing a visual/geometric intro to Calculus and the derivative which you can find here: https://photonlines.substack.com/p/visualizing-the-derivativ...

Another book on calculus for beginners I really liked [1].

[1] - https://www.amazon.com/Short-Calculus-Original-Undergraduate...

Has anyone been through the calculus courses on khan academy? What did you think?

I went through them a long time ago. It’s not the most in-depth resource, but you will learn enough calculus for when you need to actually apply it or even just pass tests. Also learning math on KA is really fun, there’s something they just get right. I’d definitely recommend giving it a try.

If you don’t feel satisfied after going through the courses, you can always pick up a book afterwards to dig deeper.

This is the book I learned calculus from (a paperback copy) several years ago. Fairly enjoyable - but I've only found uses for calculus a handful of times as a developer sadly.

Thank You. Already loving reading the text. Life long struggle with Calculus ʃ the beast

My classical high school used this book

Want a simple to boost Calculus scores globally? Make the notion standardized. I passed Calculus one and this notion is unrecognizable to me.

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