
Calculus Made Easy (1910) - luisb
http://calculusmadeeasy.org/
======
lordnacho
> 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!

~~~
russellbeattie
I wasn't sure I believed this, so I looked it up:
[https://www.etymonline.com/word/second](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)._

------
romwell
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:

[http://romankogan.net/math/A_paper_of_interest/A_Paper_of_In...](http://romankogan.net/math/A_paper_of_interest/A_Paper_of_Interest.pdf)

~~~
qwerty456127
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!

~~~
userbinator
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](https://news.ycombinator.com/item?id=15122031)
(and older discussion at
[https://news.ycombinator.com/item?id=8513209](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.

------
westoncb
"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.

------
nadvornix
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
:-)

~~~
silly_giraffe
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.

------
mcnichol
>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.

------
avinassh
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](https://en.wikipedia.org/wiki/Herbert_Gross)

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

[2] - [https://ocw.mit.edu/courses/mathematics/18-06-linear-
algebra...](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-
spring-2010/)

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

~~~
jackkinsella
YouTube math god 3Blue1Brown has a lovely series of videos that visualize
linear algebra:
[https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...](https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

------
dyukqu
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](https://news.ycombinator.com/item?id=14161876)

~~~
docbrown
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...](http://www.gutenberg.org/files/33283/33283-pdf.pdf?session_id=77879e5ced1f0a301dd839c55556cd8b4d880a21)

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

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

------
a-dub
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.

~~~
Jtsummers
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.

------
neuronexmachina
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."

[http://www.nlc-bnc.ca/eppp-
archive/100/201/300/cdn_medical_a...](http://www.nlc-bnc.ca/eppp-
archive/100/201/300/cdn_medical_association/cmaj/vol-162/issue-1/0079.htm)

------
aeneasmackenzie
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.

~~~
xelxebar
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.

~~~
seanjamz
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.

------
mettamage
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!

------
cimmanom
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.

------
mettamage
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

[1]

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}

~~~
gspetr
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.

~~~
mettamage
Haha I'm now looking at Khan Academy.

I went to [https://tutorme.com/](https://tutorme.com/) and went on a free
trial.

------
dopkew
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.

~~~
dopkew
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

~~~
zonethundery
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](https://en.wikipedia.org/wiki/Billion)

------
ruytlm
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](https://en.wikipedia.org/wiki/Long_and_short_scales)
[1]: [http://calculusmadeeasy.org/2.html](http://calculusmadeeasy.org/2.html)

------
JustSomeNobody
Nice. I have the Martin Gardner version of this.

------
WhitneyLand
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.

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

------
qwerty456127
This seems a bloody amazing book. Thanks.

------
craftyguy
Isn't this public domain by now?

~~~
Something1234
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.

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

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

------
davealger
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!

~~~
PhasmaFelis
What's wrong with that?

~~~
jeffrallen
"boys"

~~~
brewdad
Perhaps the girls weren't so easily frightened

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

[https://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus...](https://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf)

Calculus Made Easy seems too dumbed down to me.

~~~
tobmlt
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.

~~~
jcoffland
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.

~~~
tobmlt
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)

