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Calculus Made Easy (1910) (calculusmadeeasy.org)
378 points by luisb 4 months ago | hide | past | web | favorite | 68 comments

> Obviously 1 minute is a very small quantity of time compared with a whole week. Indeed, our forefathers considered it small as compared with an hour, and called it “one minùte,” meaning a minute fraction–namely one sixtieth–of an hour. 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.

Learned something already!

I wasn't sure I believed this, so I looked it up: https://www.etymonline.com/word/second and yep, that's generally the right history.

In Medieval Latin, pars minuta prima "first small part" was used by mathematician Ptolemy for one-sixtieth of a circle, later of an hour (next in order was secunda minuta, which became second).

From the book:

>I To deliver you from the Preliminary Terrors

>The preliminary terror [..] can be abolished once for all by simply stating what is the meaning–in common-sense terms–of the two principal symbols:

(1) d, which merely means “a little bit of.” Thus dx means a little bit of x; or du means a little bit of u.

(2) ∫, which is merely a long S, and may be called (if you like) “the sum of.”

As someone who taught Calculus, how I wish every book on the subject started like that!

If I ever have to do it again, I will use this book. Wish I had known about it earlier.

The book includes a very important chapter on compound interest, which is too often glossed over in texts used today. I wrote notes to remedy that (as an extra-credit reading project for the students), and was glad to find that the book has a similar approach:


Indeed. They never explained this at school and we were just memorizing and applying formulae having no idea of the meaning. So many years have passed an I've only realized what does the d actually mean some weeks ago and now I see this book explaining it this easy at the very beginning written in 1910!

I've seen a similar effect in various other topics too, and propose the saying "the closer a subject is to its infancy, the clearer it will be taught". Two examples of this come to mind. The first is in the automotive industry, with early videos such as https://news.ycombinator.com/item?id=15122031 (and older discussion at https://news.ycombinator.com/item?id=8513209 ) as well as the detailed yet straightforward explanations in the service manuals of the time. The second example is with early computers; I remember an engineering text from the late 50s/early 60s that managed to include a surprisingly lucid chapter on assembly language programming, essentially showing how to write programs to solve numerical problems. Later on, microprocessor and home computer user manuals would also contain such information.

Hey, there's more!

First, why "d"? Well, "d" is for "difference". As in: as x changes from x_1 to x_2, the difference (x_2 - x_1) -- when it's very small.

But wait, there's more!

The commonly used symbols for finite difference like that is the Greek letter Delta: Δ

For a list of values x_1, x_2, x_3, x_4,.. we write Δx_i = (x_i - x_{i-1}). That is, Δx_i is the i'th change. (Side note: an airline had a marketing slogan Change is Delta, which some nerd must have been immensely proud of).

Ok, bear with me for a bit more!

The symbol we use for finite sums is Σ: we write Σy_i = y_1 + y_2 + ... + y_n

Why Σ? That will have to wait a little.

Sums and differences cancel out:

ΣΔy_i = (y_1 - y_0) + (y_2 - y_1) + (y_3 - y_2) + ... + (y_n - y_{n-1}) = y_n - y_0

That is, summing up small succesive changes gives you total change. Simple?

Now apply this to the situation where the small changes in the quantity you are looking at are proportional to changes in another:

Δy_i = Δx_i * f(x_i)

Say, y is position, x is time; then f(x_i) is the speed at time x_i: as time increases a little, so does your position; the ratio of the changes is the speed. Δy_i is how much you moved from time x_{i-1} to time x_i, which is proportional to change in time Δx_i.

Note that f(x_i) = Δy_i/Δx_i here (speed = change in position div. by change in time).

Now write:

y_n - y_0 = ΣΔy_i = Σf(x_i)Δx_i

Again, just summing up small changes to get the net chnage.

NOW, what you've been waiting for!

Imagine you took infinitely many measurements. The changes become infinitely small, and the sum becomes of infinitely many things.

We need new notation for this.

But let's keep it similar. Instead of using Greek letters, let's use the same letters... in Latin.

Δx becomes dx

ΣΔy becomes S dy

And, with some sloppy handwriting of the letter S, the net change equation becomes:

y_final - y_initial = ∫ f(x)dx


f(x) = dy/dx.

You now see that Σ and the sloppy S -- ∫ -- stand for Sum.

And that, my friend, is pretty much all there is to Calculus and its symbols, fundamentally[1].


Do you realize this comment of yours is worth more than years of studying (high school + Bc.)? Perhaps schools are better at your location but the ones I've attended never explained this, just write that down, this equals that, memorize and f-ck you. Thanks G-d we have the Internet and people like you nowadays...

I think that Knuth's concrete mathematics made this connection at some point - the difference between the infinitely small abstraction of an integral and a big-sigma sum as being discreet quantities that can be measured by actual numbers that measure a thing.

I remember when I had calculus in school and the textbook was a huge book weighing like five pounds...and my mom's calculus textbook from the 1950s was this tiny little thin book. But they hadn't added anything to the subject of calculus since then!

Too true! I took Calculus in College, but I read this 2 months ago and loved it! Even better is it thought me something new - that is infinitesimal calculus.

How I wish there were books like this written for more topics!

"Most college calculus texts weigh a ton; this one does not — it just gets to the point. This is how I learned calculus: my uncle gave me a copy." - John Baez (Mathematical Physicist; people here may know him from his work with/writing on Category Theory)

—that's how I originally came across this book (reading Baez's recommended math texts for various subjects).

There's kind of a funny story/legacy behind the book too: Thompson originally published the book under "F.R.S." disguising his actual identity, but letting his peers know it was written by one of their own—a Fellow of the Royal Society—in spite of his knowing that they'd disapprove of the book.

I'd read somewhere, too (maybe in Gardner's preface?), that it remains a 'secret favorite' of many mathematicians who wouldn't welcome the social consequences of admitting this.

Hi. I am author of this website. I hope you like it. I made it public today. It was my favorite math textbook and I think it still can help a lot of people :-)

I have read through the first few chapters now and it is brilliant. The author uses excellent examples without ceremony that makes the topic far more approachable than I found it in school.

Thanks for making this resource available and giving it some exposure.

Thank you for taking the time to do this. It's one of my favorite math books, and it's always been hard to read as scanned PDFs.

You are doing good work. I’m sure it’s not the first time you’ve been told it but you probably don’t hear it enough. Keep it up.

thank you for putting it all together! this is a gem.

math formulas on my android phone, chrome, are set in tiny fonts. perhaps it is just me.

>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

This is the revolution that software development is long overdue for.

This is amazing! The fonts are easy on eyes and page renders beautifully.

If you are looking for vidoes, then check lectures by Herber Gross [0] on Youtube. These were recorded in 70s. They are in black & white, gives a feeling of watching some old beautifully shot movie. He goes into basics and gives you a taste of all derivations, by hand. Watch the first lecture by yourself [1] and you will immediately realise how good are these.

On a similar note, any similar resources like the one submitted, but for Linear Algebra? I am aware of Gilbert Strang's book [2] and vidoes [3], but I find them advanced for a beginner.

[0] - https://en.wikipedia.org/wiki/Herbert_Gross

[1] - https://www.youtube.com/watch?v=MFRWDuduuSw

[2] - https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

[3] - http://math.mit.edu/~gs/linearalgebra/

YouTube math god 3Blue1Brown has a lovely series of videos that visualize linear algebra: https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...

Sorry for the interruption; here is a thread (with 188 comments) about this book from a year back: https://news.ycombinator.com/item?id=14161876

Here is a link from the comments that includes a high-quality PDF of the text instead of page scans:http://www.gutenberg.org/files/33283/33283-pdf.pdf?session_i...

If you click through the links, the text has been reset here, only the front page is a complete scan.

Actually none of it is a scan (except the diagrams). The title page part just uses a font that looks like that.

Fun fact, the author Sylvanus P. Thompson was also one of the pioneers of what's now known as transcranial magnetic stimulation. The figure from the following article shows "Sylvanus P. Thompson eliciting retinal light flashes with a primitive magnetic stimulator, 1910."


One thing that seems to be missing (and wasn't given a lot of attention when I was in school) was the notion that the reals are continuous. Calculus made a lot more sense to me once I internalized how that one simple idea basically serves as the rug that really ties the room together.

I've often thought that an interesting treatment would start with differences and sums of integers as approximations, demonstrate their errors and then introduce reals and limits as a tool for making better theoretical models using the infinite "zoom button" continuity property of the reals.

You may want to check out Concrete Mathematics if you haven't already. "Concrete" is a punny sort of blending of "Continuous" and "Discrete" and it was while working through this text that I had a lot of calculus revelations that would've helped me out in my first couple years of college.

I like Calculus Made Easy because it uses informal infinitesimals. You can make these fully rigorous if you want and they're a much more intuitive technique than epsilon-delta.

Is it really all that intuitive, though? I mean, where does (dx)^2 = 0 come from?? Usually people say that, well, since dx is already small, then (dx)^2 is really really small, so for magical reasons it's okay to pretend that it's zero. I mean, if we're willy-nilly ignoring small things, why can't we ignore the already "infinitely small" dx?

Personally, I always found hand-waving such an infinitesimal explanation to be much more frustrating than simply building the darn things from pieces I already understand.

One thing that's problematic with this approach is the assumption that dx is a small constant. Its not, it represents a limit, specifically a value approaching 0. Look at the quantity (x + dx)^2. By expanding the terms you get x^2 + 2xdx + dx^2. Look at the last two terms, which both involve a dx. Lets look at how these compare to each other by putting the last term over the middle term so we get (dx^2 / 2xdx). Since we are in a limit, consider the value of this as dx approaches 0. You can cancel one of the dx's so you have (dx / 2x), and now you can clearly see that this limit will be 0. What this last limit shows, is that the last term is infinitely smaller than the middle term as dx shrinks. Which is why it gets "pretended" to be 0 in some math and physics classes. This is no approximation though, and can be carried along in your calculations if you choose to keep it.

The hyperreals make the concept of infinitesimals rigorous as well, while preserving intuitive ease.

eg Something like 'limit(f(x),x->a) = L' is just 'f(x) ≈ L when x ≈ a'

Wow this is so awesome. I'm sleep deprived, not really willing to learn more about any math but the prologue and intro are so captivating that I'm still reading it.

Well done!

Although my high school calculus classes didn't use this particular textbook, they introduced concepts in a very similar order and manner. I always wished college calculus had been so well explained.

I notice that I stumble over math over small but important details. I understand the big ideas, but then at chapter 4 in the book it says:

y+dy = (x+dx)^-2

is equal to

x^−2 * (1 + dx/x)^−2


To me (not that strong at math) this isn't apparent at all.

I have a couple of options here:

1. Spend a couple of hours fiddling around and trying to figure out the answer.

2. Hopefully find some app.

3. Ask a friend.

Regarding the options: I don't have a friend and I don't have an app. If you wouldn't know how to solve this, then what other strategies for understanding this are there?

[1] The LaTeX version:

y+dy &= (x+dx)^{-2} \\ &= x^{-2} \left(1 + \frac{dx}{x}\right)^{-2}

From your other post, you might have been a bit tired due to sleep deprivation, so don't be too hard on yourself for not getting it ;)

As for the expression, you just have to expand it:

  y+dy = (x+dx)^-2 
       = 1 / (x+dx)^2 
       = 1 / (x^2 + 2xdx + d^2x) 
       = 1 / [x^2 * (1 + 2dx/x + d^2x / x^2)] 
       = 1 / [x^2 * (1 + dx/x)^2]
       = [x^2 * (1 + dx/x)^2]^-1
       = x^-2 * (1 + dx/x)^-2
It looks like a bit of a jump at first, but he just skipped the expansion of the expression. When I see this kind of thing, it helps me to just mess around with both start and end to see if I can find a way to get from one to the other.

4. Try to look for a different (one you might understand better) explanation of the same concept in different sources.

These can be youtube videos, other books, math.stackexchange.com, math forums, etc.

Haha I'm now looking at Khan Academy.

I went to https://tutorme.com/ and went on a free trial.

In chapter 2:

> Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)^2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x^2.

It seems to me that the third term is actually a bit of a bit of x, rather than of x^2.

And in chapter 2:

>Now if, for such a purpose, we regard 1/1,000,000 (or one millionth) as a small quantity, then 1/1,000,000 of 1/1,000,000, that is 1/1,000,000,000,000 (or one billionth) ..

1/1,000,000,000,000 is actually one trillionth

From wikipedia[0]: A billion is a number with two distinct definitions:

1,000,000,000, i.e. one thousand million, or 109 (ten to the ninth power), as defined on the short scale. This is now the meaning in both British and American English.

Historically, in British English, 1,000,000,000,000, i.e. one million million, or 1012 (ten to the twelfth power), as defined on the long scale. This is one thousand times larger than the short scale billion, and equivalent to the short scale trillion.

[0] https://en.wikipedia.org/wiki/Billion

Historically, in British English, 1,000,000,000,000, i.e. one million million, or 1012 (ten to the twelfth power), as defined on the long scale. This is one thousand times larger than the short scale billion, and equivalent to the short scale trillion.

Check https://en.wikipedia.org/wiki/Long_and_short_scales for more

My favourite part is the use of long scale[0] when talking about a billion.[1]

Bring back the milliard!

[0]: https://en.wikipedia.org/wiki/Long_and_short_scales [1]: http://calculusmadeeasy.org/2.html

Nice. I have the Martin Gardner version of this.

Is there an updated version? The style is great, but terms like “farthing” seem like they could be replaced to make for a few less speed bumps and keep people going longer.

I know it’s a quibble and no fault of the author’s that time has passed, but smooothing out litttle bumps get more people deeper into the content.

Thank you for sharing this. Is anyone aware of a text in a similar style but for learning music? Or even programming?

This seems a bloody amazing book. Thanks.

Isn't this public domain by now?

From the website, yes it is, and exists as a pdf on project guttenberg. This is just relayed-out.

> About this edition & thanks > The text is based on the PDF version from Project Gutenberg converted to html by hand.

> Thanks to Paula Appling, Don Bindner, Chris Curnow, Andrew > D. Hwang and Project Gutenberg Online Distributed Proofreading Team for preparing the original PDF.

> The theme is borrowed from Dive Into HTML5 by Mark Pilgrim released under the CC-BY-3.0 license.

There's also an edition on the market updated by Martin Gardner, which isn't PD.

there needs to be one like this for ODE and discrete math.

Although the calculus concepts from 1910 may still be relevant, quotes like these make the book outputting -- "The preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate..." Ugh!

"may still be relevant" -> they're still relevant and you should probably known or inform yourself on this sort of thing before opining.

Besides that, there is an awful lot of literature that you must be unable of reading if you find any reference to historical norms of past times to be offputting, which is incredibly dangerous. What do you think would have happened to human progress if muslim or renaissance christian scholars felt like you about the texts from classical antiquity that they learnt so much from?

Really? That's kind of interesting. I find the language completely charming. With this quote it could be because "fifth-form" has no resonance at all with me outside of the text, or maybe I vaguely recall some novel of Fitzgerald's, or maybe Salinger. I dunno. To each her (or his) own.

What's wrong with that?


Perhaps the girls weren't so easily frightened

The Martin Gardner edition updates much of that kind of language.

The beginning calculus book for me is Gilbert Strang's Calculus.


Calculus Made Easy seems too dumbed down to me.

Do us a favor and delete your comment. This kind of talk can discourage and embarrass people who would otherwise be focused on learning.

Just link to the book, which is good, and save us your little puffing yourself up bit. I am sorry but I cannot sit by and watch someone belittle people who would want to learn.

edit: sorry, I am a bit high strung today. Defending tomorrow afternoon. Whatever though. the above is still true. We have an epidemic of "make those who would try hard feel stupid" and it needs to end.

"Please stop belittling me." - Eli Cash

Seriously, +1 on the above -- as someone who has been defeated several times by the calculus terrors preliminary and otherwise, I can say with some confidence that the last thing calc-shy students need is being made to feel dumber.

Down-to-earth books like the one in question are a boon.

Wow, that was reactionary. I'm not trying to discourage or embarrass anyone. My personal opinion is that the book is too dumbed down. It goes into the material way to slowly which makes it more difficult for me to stay focused on. That may not be other people's experience but I'm pretty sure some would agree. It's a matter of preference and I think I should be able to state mine without it being such a big deal.

Well, here you are defending the way you expressed your personal reaction to the book. Your reaction itself is of course fine. -But earlier you expressed your reaction to the book as if your particular experience of it were an absolute truth. Obviously (to both of us I have no doubt), the book is not anything in absolute terms, but you did not put it that way in your original comment. The original statement says flatly the book is "too dumbed down." This puts an implicit value judgement on anyone who might like this style of exposition. And a new learner is often _vulnerable_. So thank you for returning to clarify here.

To anyone struggling through calculus for the first time: Use what works! For all we know, Strang himself might of learned from Calculus Made Easy. He'd be in good company if so, though it seems like RPF was rather free with the calc books, if ya know what I mean. (see the other thread)

Did you read it? The book was aimed at school children who would have taken the then British matriculation. I don't think they are the target audience for Strang's book.

British matriculation is entry to university, no?

It's not a matter of dumbing down. One is classic university-level treatment, the other tries to get you up-and-running as quickly as possible.

Are you just basing that on the title, or have you actually spent some time with it?

The OP website recommend Strang's Calculus as a second text.

Please don't insult people earlier in their education than you with slurs like "dumbed down". Education is not a competition, you don't need to elevate yourself by calling others inferior.

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