
Calculus Made Easy (1914) [pdf] - luu
http://www.gutenberg.org/files/33283/33283-pdf.pdf
======
divbzero
Browsing through _Calculus Made Easy_ , I found it curious that the text
begins with “On Different Degrees of Smallness” and the _limit_ is mentioned
qualitatively but not used formally.

It reminds me of the preface to _Elementary Calculus_ [1] [2] an excellent
text that I discovered years after learning the subject using the _limit_ :

> The calculus was originally developed using the intuitive concept of an
> infinitesimal, or an infinitely small number. But for the past one hundred
> years infinitesimals have been banished from the calculus course for reasons
> of mathematical rigor. Students have had to learn the subject without the
> original intuition. This calculus book is based on the work of Abraham
> Robinson, who in 1960 found a way to make infinitesimals rigorous. While the
> traditional course begins with the difficult _limit_ concept, this course
> begins with the more easily understood infinitesimals.

[1]:
[https://www.math.wisc.edu/~keisler/calc.html](https://www.math.wisc.edu/~keisler/calc.html)

[2]:
[https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...](https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infinitesimal_Approach)

~~~
cuddlybacon
While I learned calculus, both in highschool and in university, both
introductions belabored the concept of an infinitesimal.

I remember a good portion of both classes finding the concept confusing.

EDIT: It seems like the top-level comment by user woah also finds it
confusing.

~~~
apricot
When I took my first calculus class, in the 1980s, we spent the first few
weeks of the course just computing limits. No reason was given, and no
context, just techniques.

I remember having to compute the limit of 2x+1 as x goes to 3 by writing it as
the limit of 2x + the limit of 1, and then saying the first limit is 2 times
the limit of x, and so on. Utterly boring, especially when the teacher
couldn't even answer us when we asked "if we already know we'll get 7 by
setting x to 3, why do we have to write all this?"

The whole thing almost put me off math.

There's a time and place to introduce properties of limits, but the first
month of calculus 1 isn't it.

~~~
canjobear
My first month of calculus was identical, in the 2000s.

------
jeffreyrogers
I've been relearning some math that I forgot so that I can learn general
relativity (just for fun), and this was what I worked through to brush off my
calculus. Really great book, and far superior to anything else I've seen.
There seem to be two approaches to calculus: dumbed down and mechanical
(pretty much any modern non-math major text) or abstract and proof based
(something like Spivak). This first group of books takes pages and pages to
get to the point and provides no understanding of why calculus works. The
second book is just too hard for a normal person who doesn't already know the
subject pretty well and who is trying to teach themselves. Calculus Made Easy
fits right in the sweet spot. While it isn't entirely rigorous, it does
justify everything clearly enough that you can see that the manipulations
you're learning are valid. At the same time, it always remains focused on
calculations, so you don't get lost in abstractions.

~~~
gyrfalc
This series from 3blue1brown is also an excellent (re-)introduction. Maybe not
deep enough to prepare you for relativity, though.

[https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr)

~~~
jeffreyrogers
His videos are good for giving intuition (I also liked the linear algebra
ones), but it's really easy to fool yourself into thinking you understanding
things better than you do, so you really need to work a lot of problems too.

------
nikivi
[http://calculusmadeeasy.org](http://calculusmadeeasy.org) is better rendering
of the book

------
woah
This is a good book, but the first time I read it, I was puzzled. There's all
this stuff about "infinitely small pieces" like every other calculus resource.
It wasn't until I had practice using the derivative rules and the chain rule
that calculus clicked for me.

I think calculus should be taught like "here's a set of tools to help you
rewrite your equations to make them output the slope of their graph".

Then, afterwards, you can get into what makes it all tick. Sort of like
teaching a student to write "hello world" before diving into the low level
mechanics of a compiler.

~~~
ookdatnog
> I think calculus should be taught like "here's a set of tools to help you
> rewrite your equations to make them output the slope of their graph". Then,
> afterwards, you can get into what makes it all tick.

I feel the exact opposite way. The method you're describing is how I was
taught calculus in high school, that is, with a strong focus on practicing
mechanical manipulation of formulas rather than building deep understanding.
And I just disconnected from that, I got no mental reward from procedurally
finding the derivative or the integral of a function for the 100th time. Lack
of motivation meant lack of focus, which in turn made my performance drop, and
at the time I assumed I probably just wasn't as good at mathematics as I
thought I was. That lasted until I got mathematics classes in college, where
many of my mathematics courses used a more first-principles-based approach,
and I performed well again.

> Sort of like teaching a student to write "hello world" before diving into
> the low level mechanics of a compiler.

I found Structure and Interpretation of Computer Programs
([https://mitpress.mit.edu/sites/default/files/sicp/index.html](https://mitpress.mit.edu/sites/default/files/sicp/index.html))
the most enlightening intro to programming book I ever read. Its approach is
also strongly first-principles-based.

I'm not writing this comment to say that you're wrong; the method you describe
may very well work well for you, and many others with you. Rather, I'm using
the opportunity to spread an opinion that I feel quite strongly about: the
best way to learn a topic is completely student-dependent and there is no such
thing as the one true way in which a topic should be taught. The most powerful
motivator is always intrinsic motivation, and different minds are motivated by
different challenges. Unfortunately it is hard to bring into practice in a
batch-processing based education system.

~~~
wenc
I appreciate your perspective, especially around motivation, and I think
you're right that students learn differently.

I do wonder though -- for heavily skill-based endeavors (say swimming, sushi-
cheffing, curling, etc.), the best way to learn is to do. First
principles/mental model approaches tend not work so well in these domains, at
least not isolation.

The best approach is probably a blend: learn by doing and understanding first
principles.

~~~
fiddlerwoaroof
A difference, though, is those domains are almost entirely about doing: you
don’t have to understand buoyancy or the biology of the tongue to swim or
cook. Math is almost the opposite: this day, the mechanics of math are nearly
entirely automated: there’s no real reason to manually work out answers except
insofar as it helps you understand the principles. The main reason to study
math now is so that you understand what your computer is doing, so you can
instruct it more efficiently. So, first principles in math are significantly
more important: someone who can do the mechanics of calculus or long division
but doesn’t understand what their doing is just doing slowly and unreliably
what a computer can do instantaneously.

~~~
didericis
You don’t necessarily have to start from the bottom, though; you can drill
down and get to first principles instead of starting without context at low
level abstractions. Both of the angles you two are talking about (providing
context and working through the mechanical applications vs explaining
fundamental principles that makes things work) seem addressable in the same
course if you start with understandable problems and drill down until you get
to the principles that allow you to solve those problems.

I think it would be quite difficult to do in a reasonable amount of time, but
using history as a course outline would be a cool way to explain the
motivations/context behind the principles being discussed. That might better
illustrate how math research works by explaining the gaps/interesting
questions that existed prior to certain ideas and how those ideas evolved and
lead to other gaps/questions.

The type of math courses I’ve gotten a lot out of already kind of did that,
they just usually used more of a condensed, logical narrative than a
historically based one.

~~~
fiddlerwoaroof
Definitely, although I think where to start really depends on the context.

------
dang
See also:

2018
[https://news.ycombinator.com/item?id=18250034](https://news.ycombinator.com/item?id=18250034)

2017
[https://news.ycombinator.com/item?id=14161876](https://news.ycombinator.com/item?id=14161876)

------
snapetom
I love how people discover old, hidden gems like this. Despite what we know
about how people learn and modern tools like video resources, every once in a
while, someone unearths some decades-old resource that explains things far
better than what's available now.

It reminds me of when I was learning antenna theory and propagation. I was
looking for something that started at step one. The best videos I found on
YouTube were a series created in the 1950's by the Royal Canadian Air Force.

[https://www.youtube.com/watch?v=7bDyA5t1ldU](https://www.youtube.com/watch?v=7bDyA5t1ldU)

~~~
sajforbes
A similar thing - the best explanation I've seen about how cars' differential
steering works comes from 1937.

[https://www.youtube.com/watch?v=yYAw79386WI](https://www.youtube.com/watch?v=yYAw79386WI)

------
wenc
I was a voracious reader as a kid and came across this book when I was 12, and
the prologue sounded particularly amusing to me:

\--

 _" 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.

Some calculus-tricks are quite easy. Some are enormously difficult. The fools
who write the textbooks of advanced mathematics — and they are mostly clever
fools — seldom take the trouble to show you how easy the easy calculations
are. On the contrary, they seem to desire to impress you with their tremendous
cleverness by going about it in the most difficult way.

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."_

\--

It made me believe I could really learn calculus by myself at age 12. After
reading the first few chapters, nope. Turns out calculus is built on a
foundation of algebra and geometry which I did not have until I was 16. Once I
did have the foundation though, calculus became easy and mostly mechanical.
But to a 12 year old, this book overpromised and underdelivered.

I also found the exposition a little wordy and tedious, like it was written to
explain calculus to an English major. It's the sort of book that one
appreciates in retrospect after knowing the subject, but not while learning
it. I discovered the most effective way for learning math is actually not by
reading but to mechanically work through problems to cultivate intuitions and
to develop a pattern matching schema. In doing so, one gains confidence that
one can actually solve problems. Once this confidence is achieved, going back
and delving into the underlying principles becomes so much more contextualized
and rewarding.

Learning math by reading books like this is like learning to ride a bike by
reading about it. You try to understand all the principles, but when you need
to deploy them, you find yourself unable to execute. Much better to do it the
other way around.

EDIT: but I want to soften that by saying that I appreciate not everyone
learns this way. It's just I've seen too many struggle with math even though
they've read the textbook countless times... when a simple change in stategy
would yield much better results.

~~~
kpgiskpg
Calculus Made Easy has a heap of exercises at the end of each chapter, it's
not bad in that respect.

------
lloyddobbler
This book is great. Although anytime I see it, my mind jumps to the meme about
"If the people who wrote computer programming books wrote math textbooks."

[https://i.pinimg.com/originals/71/40/36/71403664364a825e76f0...](https://i.pinimg.com/originals/71/40/36/71403664364a825e76f091ce0a6b2892.jpg)

------
del_operator
I remember entering high school and picking up this book from the library in
9th grade. It was honestly not my favorite. It being called “Calculus Made
Easy” had me feeling frustrated. I went to put it back and there was a book
next to it called Calculus and Pizza. Within the introduction I got the
intuitive feel for differentials and limits. I was able to play with the
difference formula and take a much more enthusiastic high level trip through
both differential and integral calculus. A big bonus was Calculus and Pizza
allowed me to check my algebra in the back of the book and forward those
questions to my algebra teacher. That was a few years before I was able to
officially take calculus, so I kept re-reading chapters because they were fun.

A book with an easy approach really needs to make the chapters super easy to
restart and spiral through concepts. Stories and characters didn’t hurt to
have too though.

~~~
InnerGargoyle
now i have to find this one also.

------
madballster
That is a beautiful text. Shockingly much more intuitive, better graphs and
examples than what we had in high school in the early 90s.

------
FranzFerdiNaN
If only there was a book that explained linear algebra in this way.

~~~
jlangemeier
Linear Algebra Done Right is, without a doubt, one of my favorites in this
category. It tends to be taught as an intermediate or graduate level approach,
but things clicked for me with Axler's explanations way better than about
anything else I've used.

[http://linear.axler.net/](http://linear.axler.net/)

~~~
nextos
Sadly the last edition turned their beautiful LaTeX typesetting into something
really distracting.

Aside from Axler, I wish someone picked up Halmos text and rendered it in a
more modern way. It's a really good one, a bit more advanced, but the text is
so cramped it's unpleasant to follow.

------
kpgiskpg
Funnily enough, I'm working through this book right now. The writing amuses me
to no end, from the anachronism of calling it "the calculus" to the chain rule
being referred to as a "dodge".

It goes well with 3blue1brown's calculus series on YouTube, which gives more
of a visual intuition for the concepts.

------
champagnepapi
Wow, if only I was shown this book earlier on in life ha.

~~~
DaiPlusPlus
Me too: it took until 6 months after I started differential calculus (AS-level
Pure 1 (UK)) before I grokked what the "d" in dy/dx meant (just like this
person:
[https://physics.stackexchange.com/questions/65724/difference...](https://physics.stackexchange.com/questions/65724/difference-
between-delta-d-and-delta) ) - before then I was treating the differentiation
operator as a magic, irreducible, symbol and only getting so far in the course
because I was remembering and applying the mechanics of differentiation (i.e.
by-rote) instead of really understanding what I was doing.

------
wandering-nomad
Just read initial few pages and it piqued my interest. Truly love the way it
opens and kept me engaged.

------
haider_a
For anyone interested in the history of calculus, there is an excellent paper:
Grabiner, J. (1983). Who Gave You the Epsilon? Cauchy and the Origins of
Rigorous Calculus. The American Mathematical Monthly, 90(3), 185-194.
doi:10.2307/2975545

------
julianeon
If this book appeals to you then you should see 3 brown 1 blue's videos on
YouTube. The way they show the squares with the 'little bits' added in
Calculus Made Easy is essentially repeated in 3 brown 1 blue, with graphics.

------
transitorykris
Beautiful book, and incredibly lucid. Gave me a useful intuition on
infinitesimals that never quite came out in the various standard calc texts.
Second to this imo is Serge Lang’s A Short Introduction to Calculus.

------
zozbot234
(1914), rather.

------
dunefox
Honest question: did calculus change enough in a hundred years to make this
outdated? In other words, can I use this to refresh my calc knowledge in 2020?

~~~
kwhitefoot
No calculus hasn't changed and yes if you work though all the exercises you
will probably be better at calculus as far is it is explored in the book.

------
bobobob420
This is my favorite book on mathematics of all time. I am very happy you
shared this so I can save it

~~~
dunefox
Here: [http://calculusmadeeasy.org/](http://calculusmadeeasy.org/)

------
yters
for another fundamentals approach [https://www.amazon.com/Calculus-Ground-
Jonathan-Laine-Bartle...](https://www.amazon.com/Calculus-Ground-Jonathan-
Laine-Bartlett/dp/1944918027)

------
a3n
> What one fool can do, another can. (Ancient Simian Proverb.)

------
InnerGargoyle
i think we need list of books like this on each subject which are more
explanatory, intuitive rather than mechanical procedures.

------
anthk
Is there a book like this but for metric?

~~~
kwhitefoot
Do you mean using SI instead of Imperial units? What difference would it make?
For instance Exercises IX (Chapter XI) says:

 _(4) A piece of string 30 inches long has its two ends joined together and is
stretched by 3 pegs so as to form a triangle. What is the largest triangular
area that can be enclosed by the string?_

Just replace inches with your linear measure of choice, mm, rods, poles,
chains, stadia, light years, or anything else; it makes no difference to the
problem. Thompson could have said 'units of length' instead of 'inches'; but
the whole purpose of the book is to build on easy notions that Thompson
believed everyone could manage and perhaps even be familiar with already, so
he used imperial units that were familiar to every handyman of the time
instead of metric which would be familiar to far fewer and hence distracting.

Also don't forget that there were several distinct 'metric' systems in the
past before most of the world settled on SI (MKS, CGS, etc.)

------
cullinap
great book!

~~~
scruple
It really is. I was shown this text in high school and it's the thing that got
Calculus to click for me. I've recommended it dozens of times over the years
and I've heard the same sentiment echoed nearly every time.

------
blackrock
Nice book.

This looks like a good opportunity for someone to create a website for this,
and animate it with some JavaScript code.

And insert some ads: Please click here to buy some protractors for 25% off.

